Chemical equation - Wikipedia

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class="vector-toc-list"> </ul> </li> <li id="toc-Notation_variants" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notation_variants"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Notation variants</span> </div> </a> <ul id="toc-Notation_variants-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Balancing_chemical_equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Balancing_chemical_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Balancing chemical equations</span> </div> </a> <button aria-controls="toc-Balancing_chemical_equations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Balancing chemical equations subsection</span> </button> <ul id="toc-Balancing_chemical_equations-sublist" class="vector-toc-list"> <li id="toc-Inspection_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inspection_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Inspection method</span> </div> </a> <ul id="toc-Inspection_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-System_of_linear_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#System_of_linear_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>System of linear equations</span> </div> </a> <ul id="toc-System_of_linear_equations-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Matrix_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrix_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Matrix method</span> </div> </a> <ul id="toc-Matrix_method-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ionic_equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ionic_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ionic equations</span> </div> </a> <ul id="toc-Ionic_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Typesetting" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Typesetting"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Typesetting</span> </div> </a> <ul id="toc-Typesetting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Chemical equation</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 64 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-64" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">64 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A9_%D9%83%D9%8A%D9%85%D9%8A%D8%A7%D8%A6%D9%8A%D8%A9" title="معادلة كيميائية – Arabic" lang="ar" hreflang="ar" data-title="معادلة كيميائية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Ecuaci%C3%B3n_qu%C3%ADmica" title="Ecuación química – Asturian" lang="ast" hreflang="ast" data-title="Ecuación química" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BC%D0%B8%D0%BA_%D1%82%D0%B8%D0%B3%D0%B5%D2%99%D0%BB%D3%99%D0%BC%D3%99" title="Химик тигеҙләмә – Bashkir" lang="ba" hreflang="ba" data-title="Химик тигеҙләмә" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A5%D1%96%D0%BC%D1%96%D1%87%D0%BD%D0%B0%D0%B5_%D1%9E%D1%80%D0%B0%D1%9E%D0%BD%D0%B5%D0%BD%D0%BD%D0%B5" title="Хімічнае ўраўненне – Belarusian" lang="be" hreflang="be" data-title="Хімічнае ўраўненне" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BC%D0%B8%D1%87%D0%BD%D0%BE_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5" title="Химично уравнение – Bulgarian" lang="bg" hreflang="bg" data-title="Химично уравнение" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Hemijska_jedna%C4%8Dina" title="Hemijska jednačina – Bosnian" lang="bs" hreflang="bs" data-title="Hemijska jednačina" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equaci%C3%B3_qu%C3%ADmica" title="Equació química – Catalan" lang="ca" hreflang="ca" data-title="Equació química" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BC%D0%B8%D0%BB%D0%BB%D0%B5_%D1%82%D0%B0%D0%BD%D0%BB%C4%83%D1%85" title="Химилле танлăх – Chuvash" lang="cv" hreflang="cv" data-title="Химилле танлăх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Chemick%C3%A1_rovnice" title="Chemická rovnice – Czech" lang="cs" hreflang="cs" data-title="Chemická rovnice" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Reaktionsskema" title="Reaktionsskema – Danish" lang="da" hreflang="da" data-title="Reaktionsskema" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Reaktionsgleichung" title="Reaktionsgleichung – German" lang="de" hreflang="de" data-title="Reaktionsgleichung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Reaktsiooniv%C3%B5rrand" title="Reaktsioonivõrrand – Estonian" lang="et" hreflang="et" data-title="Reaktsioonivõrrand" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A7%CE%B7%CE%BC%CE%B9%CE%BA%CE%AE_%CE%B5%CE%BE%CE%AF%CF%83%CF%89%CF%83%CE%B7" title="Χημική εξίσωση – Greek" lang="el" hreflang="el" data-title="Χημική εξίσωση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ecuaci%C3%B3n_qu%C3%ADmica" title="Ecuación química – Spanish" lang="es" hreflang="es" data-title="Ecuación química" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Ekuazio_kimiko" title="Ekuazio kimiko – Basque" lang="eu" hreflang="eu" data-title="Ekuazio kimiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D9%87_%D8%B4%DB%8C%D9%85%DB%8C%D8%A7%DB%8C%DB%8C" title="معادله شیمیایی – Persian" lang="fa" hreflang="fa" data-title="معادله شیمیایی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89quation_chimique" title="Équation chimique – French" lang="fr" hreflang="fr" data-title="Équation chimique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Cothrom%C3%B3id_cheimiceach" title="Cothromóid cheimiceach – Irish" lang="ga" hreflang="ga" data-title="Cothromóid cheimiceach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ecuaci%C3%B3n_qu%C3%ADmica" title="Ecuación química – Galician" lang="gl" hreflang="gl" data-title="Ecuación química" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%99%94%ED%95%99%EB%B0%98%EC%9D%91%EC%8B%9D" title="화학반응식 – Korean" lang="ko" hreflang="ko" data-title="화학반응식" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%94%D5%AB%D5%B4%D5%AB%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4%D5%B6%D5%A5%D6%80" title="Քիմիական հավասարումներ – Armenian" lang="hy" hreflang="hy" data-title="Քիմիական հավասարումներ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B0%E0%A4%BE%E0%A4%B8%E0%A4%BE%E0%A4%AF%E0%A4%A8%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="रासायनिक समीकरण – Hindi" lang="hi" hreflang="hi" data-title="रासायनिक समीकरण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kemijska_jednad%C5%BEba" title="Kemijska jednadžba – Croatian" lang="hr" hreflang="hr" data-title="Kemijska jednadžba" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Persamaan_reaksi" title="Persamaan reaksi – Indonesian" lang="id" hreflang="id" data-title="Persamaan reaksi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazione_chimica" title="Equazione chimica – Italian" lang="it" hreflang="it" data-title="Equazione chimica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%94_%D7%9B%D7%99%D7%9E%D7%99%D7%AA" title="משוואה כימית – Hebrew" lang="he" hreflang="he" data-title="משוואה כימית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BC%D0%B8%D1%8F%D0%BB%D1%8B%D2%9B_%D1%82%D0%B5%D2%A3%D0%B4%D0%B5%D1%83%D0%BB%D0%B5%D1%80" title="Химиялық теңдеулер – Kazakh" lang="kk" hreflang="kk" data-title="Химиялық теңдеулер" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Ekwasyon_chimik" title="Ekwasyon chimik – Haitian Creole" lang="ht" hreflang="ht" data-title="Ekwasyon chimik" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BC%D0%B8%D1%8F%D0%BB%D1%8B%D0%BA_%D1%82%D0%B5%D2%A3%D0%B4%D0%B5%D0%BC%D0%B5" title="Химиялык теңдеме – Kyrgyz" lang="ky" hreflang="ky" data-title="Химиялык теңдеме" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-ln mw-list-item"><a href="https://ln.wikipedia.org/wiki/Mokokano" title="Mokokano – Lingala" lang="ln" hreflang="ln" data-title="Mokokano" data-language-autonym="Lingála" data-language-local-name="Lingala" class="interlanguage-link-target"><span>Lingála</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/K%C3%A9miai_egyenlet" title="Kémiai egyenlet – Hungarian" lang="hu" hreflang="hu" data-title="Kémiai egyenlet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B0%E0%B4%BE%E0%B4%B8%E0%B4%B8%E0%B4%AE%E0%B4%B5%E0%B4%BE%E0%B4%95%E0%B5%8D%E0%B4%AF%E0%B4%82" title="രാസസമവാക്യം – Malayalam" lang="ml" hreflang="ml" data-title="രാസസമവാക്യം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Persamaan_kimia" title="Persamaan kimia – Malay" lang="ms" hreflang="ms" data-title="Persamaan kimia" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%93%E1%80%AC%E1%80%90%E1%80%AF%E1%80%97%E1%80%B1%E1%80%92_%E1%80%8A%E1%80%AE%E1%80%99%E1%80%BB%E1%80%BE%E1%80%81%E1%80%BC%E1%80%84%E1%80%BA%E1%80%B8" title="ဓာတုဗေဒ ညီမျှခြင်း – Burmese" lang="my" hreflang="my" data-title="ဓာတုဗေဒ ညီမျှခြင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Reactievergelijking" title="Reactievergelijking – Dutch" lang="nl" hreflang="nl" data-title="Reactievergelijking" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8C%96%E5%AD%A6%E5%8F%8D%E5%BF%9C%E5%BC%8F" title="化学反応式 – Japanese" lang="ja" hreflang="ja" data-title="化学反応式" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Ligning_(kjemi)" title="Ligning (kjemi) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Ligning (kjemi)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Kimyoviy_tenglamalar" title="Kimyoviy tenglamalar – Uzbek" lang="uz" hreflang="uz" data-title="Kimyoviy tenglamalar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B0%E0%A8%B8%E0%A8%BE%E0%A8%87%E0%A8%A3%E0%A8%95_%E0%A8%B8%E0%A8%AE%E0%A9%80%E0%A8%95%E0%A8%B0%E0%A8%A8" title="ਰਸਾਇਣਕ ਸਮੀਕਰਨ – Punjabi" lang="pa" hreflang="pa" data-title="ਰਸਾਇਣਕ ਸਮੀਕਰਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/R%C3%B3wnanie_reakcji" title="Równanie reakcji – Polish" lang="pl" hreflang="pl" data-title="Równanie reakcji" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Equa%C3%A7%C3%A3o_qu%C3%ADmica" title="Equação química – Portuguese" lang="pt" hreflang="pt" data-title="Equação química" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Ecua%C8%9Bie_chimic%C4%83" title="Ecuație chimică – Romanian" lang="ro" hreflang="ro" data-title="Ecuație chimică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Ruranakuy_paqtachani" title="Ruranakuy paqtachani – Quechua" lang="qu" hreflang="qu" data-title="Ruranakuy paqtachani" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BC%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5" title="Химическое уравнение – Russian" lang="ru" hreflang="ru" data-title="Химическое уравнение" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Chemical_equation" title="Chemical equation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Chemical equation" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Chemick%C3%A1_rovnica" title="Chemická rovnica – Slovak" lang="sk" hreflang="sk" data-title="Chemická rovnica" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kemijska_ena%C4%8Dba" title="Kemijska enačba – Slovenian" lang="sl" hreflang="sl" data-title="Kemijska enačba" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%BE%D8%A7%D9%88%DA%A9%DB%8E%D8%B4%DB%95%DB%8C_%DA%A9%DB%8C%D9%85%DB%8C%D8%A7%DB%8C%DB%8C" title="ھاوکێشەی کیمیایی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ھاوکێشەی کیمیایی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A5%D0%B5%D0%BC%D0%B8%D1%98%D1%81%D0%BA%D0%B5_%D1%98%D0%B5%D0%B4%D0%BD%D0%B0%D1%87%D0%B8%D0%BD%D0%B5" title="Хемијске једначине – Serbian" lang="sr" hreflang="sr" data-title="Хемијске једначине" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Hemijska_jedna%C4%8Dina" title="Hemijska jednačina – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Hemijska jednačina" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Reaktioyht%C3%A4l%C3%B6" title="Reaktioyhtälö – Finnish" lang="fi" hreflang="fi" data-title="Reaktioyhtälö" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Reaktionsformel" title="Reaktionsformel – Swedish" lang="sv" hreflang="sv" data-title="Reaktionsformel" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Ekwasyong_kimikal" title="Ekwasyong kimikal – Tagalog" lang="tl" hreflang="tl" data-title="Ekwasyong kimikal" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AF%87%E0%AE%A4%E0%AE%BF%E0%AE%AF%E0%AE%BF%E0%AE%AF%E0%AE%B1%E0%AF%8D_%E0%AE%9A%E0%AE%AE%E0%AE%A9%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="வேதியியற் சமன்பாடு – Tamil" lang="ta" hreflang="ta" data-title="வேதியியற் சமன்பாடு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tagda_takrurant" title="Tagda takrurant – Kabyle" lang="kab" hreflang="kab" data-title="Tagda takrurant" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BC%D0%B8%D0%BA_%D1%82%D0%B8%D0%B3%D0%B5%D0%B7%D0%BB%D3%99%D0%BC%D3%99" title="Химик тигезләмә – Tatar" lang="tt" hreflang="tt" data-title="Химик тигезләмә" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B0%E0%B0%B8%E0%B0%BE%E0%B0%AF%E0%B0%A8_%E0%B0%B8%E0%B0%AE%E0%B1%80%E0%B0%95%E0%B0%B0%E0%B0%A3%E0%B0%BE%E0%B0%B2%E0%B1%81" title="రసాయన సమీకరణాలు – Telugu" lang="te" hreflang="te" data-title="రసాయన సమీకరణాలు" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kimyasal_denklem" title="Kimyasal denklem – Turkish" lang="tr" hreflang="tr" data-title="Kimyasal denklem" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A5%D1%96%D0%BC%D1%96%D1%87%D0%BD%D0%B5_%D1%80%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D0%BD%D1%8F" title="Хімічне рівняння – Ukrainian" lang="uk" hreflang="uk" data-title="Хімічне рівняння" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DA%A9%DB%8C%D9%85%DB%8C%D8%A7%D8%A6%DB%8C_%D9%85%D8%B3%D8%A7%D9%88%D8%A7%D8%AA" title="کیمیائی مساوات – Urdu" lang="ur" hreflang="ur" data-title="کیمیائی مساوات" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_tr%C3%ACnh_h%C3%B3a_h%E1%BB%8Dc" title="Phương trình hóa học – Vietnamese" lang="vi" hreflang="vi" data-title="Phương trình hóa học" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8C%96%E5%AD%A6%E6%96%B9%E7%A8%8B%E5%BC%8F" title="化学方程式 – Wu" lang="wuu" hreflang="wuu" data-title="化学方程式" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8C%96%E5%AD%B8%E6%96%B9%E7%A8%8B%E5%BC%8F" title="化學方程式 – Cantonese" lang="yue" hreflang="yue" data-title="化學方程式" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8C%96%E5%AD%A6%E6%96%B9%E7%A8%8B%E5%BC%8F" title="化学方程式 – Chinese" lang="zh" hreflang="zh" data-title="化学方程式" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Symbolic representation of a chemical reaction</div><style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">October 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>A <b>chemical equation</b> is the symbolic representation of a <a href="/wiki/Chemical_reaction" title="Chemical reaction">chemical reaction</a> in the form of symbols and <a href="/wiki/Chemical_formula" title="Chemical formula">chemical formulas</a>. The <a href="/wiki/Reactant" class="mw-redirect" title="Reactant">reactant</a> entities are given on the left-hand side and the <a href="/wiki/Product_(chemistry)" title="Product (chemistry)">product</a> entities are on the right-hand side with a plus sign between the entities in both the reactants and the products, and an arrow that points towards the products to show the direction of the reaction.<sup id="cite_ref-goldbook_1-0" class="reference"><a href="#cite_note-goldbook-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The chemical formulas may be symbolic, <a href="/wiki/Structural_formula" title="Structural formula">structural</a> (pictorial diagrams), or intermixed. The coefficients next to the symbols and formulas of entities are the absolute values of the <a class="mw-selflink-fragment" href="#Structure">stoichiometric numbers</a>. The first chemical equation was diagrammed by <a href="/wiki/Jean_Beguin" title="Jean Beguin">Jean Beguin</a> in 1615.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Structure">Structure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=1" title="Edit section: Structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A chemical equation (see an example below) consists of a list of reactants (the starting substances) on the left-hand side, an <a href="/wiki/Arrow_(symbol)" title="Arrow (symbol)">arrow symbol</a>, and a list of products (substances formed in the chemical reaction) on the right-hand side. Each substance is specified by its <a href="/wiki/Chemical_formula" title="Chemical formula">chemical formula</a>, optionally preceded by a number called <b>stoichiometric coefficient</b>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> The coefficient specifies how many entities (e.g. <a href="/wiki/Molecule" title="Molecule">molecules</a>) of that substance are involved in the reaction on a molecular basis. If not written explicitly, the coefficient is equal to 1. Multiple substances on any side of the equation are separated from each other by a <a href="/wiki/Plus_sign" class="mw-redirect" title="Plus sign">plus sign</a>. </p><p>As an example, the equation for the reaction of <a href="/wiki/Hydrochloric_acid" title="Hydrochloric acid">hydrochloric acid</a> with <a href="/wiki/Sodium" title="Sodium">sodium</a> can be denoted: </p> <dl><dd><style data-mw-deduplicate="TemplateStyles:r1123817410">.mw-parser-output .template-chem2-su{display:inline-block;font-size:80%;line-height:1;vertical-align:-0.35em}.mw-parser-output .template-chem2-su>span{display:block;text-align:left}.mw-parser-output sub.template-chem2-sub{font-size:80%;vertical-align:-0.35em}.mw-parser-output sup.template-chem2-sup{font-size:80%;vertical-align:0.65em}</style><span class="chemf nowrap">2HCl + 2Na → 2NaCl + H<sub class="template-chem2-sub">2</sub></span></dd></dl> <p>Given the formulas are fairly simple, this equation could be read as "two H-C-L plus two N-A yields<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> two N-A-C-L and H two." Alternately, and in general for equations involving complex chemicals, the chemical formulas are read using <a href="/wiki/IUPAC_nomenclature" class="mw-redirect" title="IUPAC nomenclature">IUPAC nomenclature</a>, which could <a href="/wiki/Verbalisation" title="Verbalisation">verbalise</a> this equation as "two hydrochloric acid molecules and two sodium atoms react to form two <a href="/wiki/Formula_unit" title="Formula unit">formula units</a> of <a href="/wiki/Sodium_chloride" title="Sodium chloride">sodium chloride</a> and a <a href="/wiki/Hydrogen" title="Hydrogen">hydrogen</a> gas molecule." </p> <div class="mw-heading mw-heading3"><h3 id="Reaction_types">Reaction types</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=2" title="Edit section: Reaction types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Different variants of the arrow symbol are used to denote the type of a reaction:<sup id="cite_ref-goldbook_1-1" class="reference"><a href="#cite_note-goldbook-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><table> <tbody><tr> <td style="text-align: center; padding-right: 0.5em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }"></span></td> <td>net forward reaction </td></tr> <tr> <td style="text-align: center; padding-right: 0.5em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightleftarrows }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇄</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightleftarrows }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdbaea31afa0894aa82079b04d6a5e7566da7195" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle \rightleftarrows }"></span></td> <td>reaction in both directions<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td style="text-align: center; padding-right: 0.5em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightleftharpoons }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo class="MJX-variant" stretchy="false">⇌</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightleftharpoons }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c37b981df851b9e54e489e017b1481e37d418f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightleftharpoons }"></span></td> <td><a href="/wiki/Chemical_equilibrium" title="Chemical equilibrium">equilibrium</a><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td style="text-align: center; padding-right: 0.5em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span></td> <td><a href="/wiki/Stoichiometric" class="mw-redirect" title="Stoichiometric">stoichiometric</a> relation </td></tr> <tr> <td style="text-align: center; padding-right: 0.5em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"></span></td> <td><a href="/wiki/Resonance_(chemistry)" title="Resonance (chemistry)">resonance</a> (not a reaction) </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="State_of_matter">State of matter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=3" title="Edit section: State of matter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To indicate <a href="/wiki/Physical_state" class="mw-redirect" title="Physical state">physical state</a> of a chemical, a symbol in parentheses may be appended to its formula: (s) for a solid, (l) for a liquid, (g) for a gas, and (aq) for an <a href="/wiki/Aqueous_solution" title="Aqueous solution">aqueous solution</a>. This is especially done when one wishes to emphasize the states or changes thereof. For example, the reaction of aqueous hydrochloric acid with solid (metallic) sodium to form aqueous sodium chloride and hydrogen gas would be written like this: </p> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1123817410"><span class="chemf nowrap">2HCl(aq) + 2Na(s) → 2NaCl(aq) + H<sub class="template-chem2-sub">2</sub>(g)</span></dd></dl> <p>That reaction would have different <a href="/wiki/Thermodynamic" class="mw-redirect" title="Thermodynamic">thermodynamic</a> and <a href="/wiki/Chemical_kinetics" title="Chemical kinetics">kinetic</a> properties if gaseous <a href="/wiki/Hydrogen_chloride" title="Hydrogen chloride">hydrogen chloride</a> were to replace the hydrochloric acid as a reactant: </p> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1123817410"><span class="chemf nowrap">2HCl(g) + 2Na(s) → 2NaCl(s) + H<sub class="template-chem2-sub">2</sub>(g)</span></dd></dl> <p>Alternately, an arrow without parentheses is used in some cases to indicate formation of a gas ↑ or precipitate ↓. This is especially useful if only one such species is formed. Here is an example indicating that hydrogen gas is formed: </p> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1123817410"><span class="chemf nowrap">2HCl + 2Na → 2 NaCl + H<sub class="template-chem2-sub">2</sub> ↑</span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Catalysis_and_other_conditions">Catalysis and other conditions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=4" title="Edit section: Catalysis and other conditions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Baker-Venkataraman-Rearrangement_V.1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Baker-Venkataraman-Rearrangement_V.1.svg/290px-Baker-Venkataraman-Rearrangement_V.1.svg.png" decoding="async" width="290" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Baker-Venkataraman-Rearrangement_V.1.svg/435px-Baker-Venkataraman-Rearrangement_V.1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Baker-Venkataraman-Rearrangement_V.1.svg/580px-Baker-Venkataraman-Rearrangement_V.1.svg.png 2x" data-file-width="806" data-file-height="333" /></a><figcaption>The <a href="/wiki/Baker%E2%80%93Venkataraman_rearrangement" title="Baker–Venkataraman rearrangement">Baker–Venkataraman rearrangement</a> needs a <a href="/wiki/Base_(chemistry)" title="Base (chemistry)">base</a> as a <a href="/wiki/Catalyst" class="mw-redirect" title="Catalyst">catalyst</a>.</figcaption></figure> <p>If the reaction requires energy, it is indicated above the arrow. A capital Greek letter <a href="/wiki/Delta_(letter)" title="Delta (letter)">delta (Δ)</a> or a <a href="/wiki/%E2%96%B3" class="mw-redirect" title="△">triangle (△)</a><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>e<span class="cite-bracket">]</span></a></sup> is put on the reaction arrow to show that energy in the form of heat is added to the reaction. The expression <span class="texhtml mvar" style="font-style:italic;">hν</span><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>f<span class="cite-bracket">]</span></a></sup> is used as a symbol for the addition of energy in the form of light. Other symbols are used for other specific types of energy or radiation. </p><p>Similarly, if a reaction requires a certain medium with certain specific characteristics, then the name of the acid or base that is used as a medium may be placed on top of the arrow. If no specific acid or base is required, another way of denoting the use of an acidic or basic medium is to write H<sup>+</sup> or OH<sup>−</sup> (or even "acid" or "base") on top of the arrow. Specific conditions of the temperature and pressure, as well as the presence of catalysts, may be indicated in the same way. </p> <div class="mw-heading mw-heading3"><h3 id="Notation_variants">Notation variants</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=5" title="Edit section: Notation variants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Acid-CatAmideHydrolMarch.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Acid-CatAmideHydrolMarch.png/370px-Acid-CatAmideHydrolMarch.png" decoding="async" width="370" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Acid-CatAmideHydrolMarch.png/555px-Acid-CatAmideHydrolMarch.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Acid-CatAmideHydrolMarch.png/740px-Acid-CatAmideHydrolMarch.png 2x" data-file-width="1430" data-file-height="920" /></a><figcaption>This illustration of a mechanism for acid-catalyzed <a href="/wiki/Hydrolysis" title="Hydrolysis">hydrolysis</a> of an <a href="/wiki/Amide" title="Amide">amide</a> puts some reactants and products above the arrows and to their branches to allow chaining of the chemical "equations".</figcaption></figure> <p>The standard notation for chemical equations only permits all reactants on one side, all products on the other, and all stoichiometric coefficients positive. For example, the usual form of the equation for <a href="/wiki/Dehydration_reaction" title="Dehydration reaction">dehydration</a> of <a href="/wiki/Methanol" title="Methanol">methanol</a> to <a href="/wiki/Dimethylether" class="mw-redirect" title="Dimethylether">dimethylether</a> is: </p> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1123817410"><span class="chemf nowrap">2 CH<sub class="template-chem2-sub">3</sub>OH → CH<sub class="template-chem2-sub">3</sub>OCH<sub class="template-chem2-sub">3</sub> + H<sub class="template-chem2-sub">2</sub>O</span></dd></dl> <p>Sometimes an extension is used, where some substances with their stoichiometric coefficients are moved above or below the arrow, preceded by a plus sign or nothing for a reactant, and by a minus sign for a product. Then the same equation can look like this: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {2CH3OH->[{\overset {}{\ce {-H2O}}}]CH3OCH3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>CH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>OH</mtext> <mrow class="MJX-TeXAtom-REL"> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>O</mtext> </mrow> <mrow /> </mover> </mrow> </mpadded> </mover> </mrow> <msubsup> <mtext>CH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <msubsup> <mtext>OCH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {2CH3OH->[{\overset {}{\ce {-H2O}}}]CH3OCH3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9322a7fd3bb1fb28dab61d3c7fa287ab389d3c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-top: -0.386ex; width:27.57ex; height:5.343ex;" alt="{\displaystyle {\ce {2CH3OH->[{\overset {}{\ce {-H2O}}}]CH3OCH3}}}"></span></dd></dl> <p>Such notation serves to hide less important substances from the sides of the equation, to make the type of reaction at hand more obvious, and to facilitate chaining of chemical equations. This is very useful in illustrating multi-step <a href="/wiki/Reaction_mechanism" title="Reaction mechanism">reaction mechanisms</a>. Note that the substances above or below the arrows are not <a href="/wiki/Catalyst" class="mw-redirect" title="Catalyst">catalysts</a> in this case, because they are consumed or produced in the reaction like ordinary reactants or products. </p><p>Another extension used in reaction mechanisms moves some substances to branches of the arrow. Both extensions are used in the example illustration of a mechanism. </p><p>Use of negative stoichiometric coefficients at either side of the equation (like in the example below) is not widely adopted and is often discouraged.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {2 CH3OH \;-\; H2O -> CH3OCH3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>CH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>OH</mtext> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mspace width="thickmathspace" /> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>O</mtext> <mo stretchy="false">⟶</mo> <msubsup> <mtext>CH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <msubsup> <mtext>OCH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {2 CH3OH \;-\; H2O -> CH3OCH3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bd3a1101f75d22efd346977da25d7af801f3836" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.135ex; height:2.843ex;" alt="{\displaystyle {\ce {2 CH3OH \;-\; H2O -> CH3OCH3}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Balancing_chemical_equations">Balancing chemical equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=6" title="Edit section: Balancing chemical equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Stoichiometry" title="Stoichiometry">Stoichiometry</a></div> <p>Because no <a href="/wiki/Nuclear_reaction" title="Nuclear reaction">nuclear reactions</a> take place in a chemical reaction, the <a href="/wiki/Chemical_element" title="Chemical element">chemical elements</a> pass through the reaction unchanged. Thus, each side of the chemical equation must represent the same number of atoms of any particular element (or <a href="/wiki/Nuclide" title="Nuclide">nuclide</a>, if different <a href="/wiki/Isotope" title="Isotope">isotopes</a> are taken into account). The same holds for the total <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a>, as stated by the <a href="/wiki/Charge_conservation" title="Charge conservation">charge conservation</a> law. An equation adhering to these requirements is said to be <b>balanced</b>. </p><p>A chemical equation is balanced by assigning suitable values to the stoichiometric coefficients. Simple equations can be balanced by inspection, that is, by trial and error. Another technique involves solving a <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of linear equations</a>. </p><p>Balanced equations are usually written with smallest <a href="/wiki/Natural_number" title="Natural number">natural-number</a> coefficients. Yet sometimes it may be advantageous to accept a fractional coefficient, if it simplifies the other coefficients. The introductory example can thus be rewritten as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {HCl + Na -> NaCl + 1/2 H2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>HCl</mtext> <mo>+</mo> <mtext>Na</mtext> <mo stretchy="false">⟶</mo> <mtext>NaCl</mtext> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {HCl + Na -> NaCl + 1/2 H2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80e085e0596a151cce3a1c5ff82a23fa9cf11535" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.436ex; height:3.509ex;" alt="{\displaystyle {\ce {HCl + Na -> NaCl + 1/2 H2}}}"></span></dd></dl> <p>In some circumstances the fractional coefficients are even inevitable. For example, the reaction corresponding to the <a href="/wiki/Standard_enthalpy_of_formation" title="Standard enthalpy of formation">standard enthalpy of formation</a> must be written such that one molecule of a single product is formed. This will often require that some reactant coefficients be fractional, as is the case with the formation of <a href="/wiki/Lithium_fluoride" title="Lithium fluoride">lithium fluoride</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {Li(s) + 1/2F2(g) -> LiF(s)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Li</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mtext>s</mtext> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <msubsup> <mtext>F</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mtext>g</mtext> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">⟶</mo> <mtext>LiF</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mtext>s</mtext> <mo stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {Li(s) + 1/2F2(g) -> LiF(s)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59965a014de31ada99dbcba42504eba58769a8e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.306ex; height:3.509ex;" alt="{\displaystyle {\ce {Li(s) + 1/2F2(g) -> LiF(s)}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Inspection_method">Inspection method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=7" title="Edit section: Inspection method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Combustion_reaction_of_methane.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Combustion_reaction_of_methane.jpg/290px-Combustion_reaction_of_methane.jpg" decoding="async" width="290" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Combustion_reaction_of_methane.jpg/435px-Combustion_reaction_of_methane.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Combustion_reaction_of_methane.jpg/580px-Combustion_reaction_of_methane.jpg 2x" data-file-width="1009" data-file-height="451" /></a><figcaption>As seen from the equation <span class="nowrap"><span class="chemf nowrap">CH<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">4</sub></span></span></span> + 2 <span class="chemf nowrap">O<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sub></span></span></span> →</span> <span class="nowrap"><span class="chemf nowrap">CO<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sub></span></span></span> + 2 <span class="chemf nowrap">H<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sub></span></span>O</span>,</span> a coefficient of 2 must be placed before the <a href="/wiki/Oxygen" title="Oxygen">oxygen</a> gas on the reactants side and before the <a href="/wiki/Properties_of_water" title="Properties of water">water</a> on the products side in order for, as per the law of conservation of mass, the quantity of each element does not change during the reaction</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:H3PO4_balancing_chemical_equation_phosphorus_pentoxide_and_water_becomes_phosphoric_acid.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/H3PO4_balancing_chemical_equation_phosphorus_pentoxide_and_water_becomes_phosphoric_acid.gif/290px-H3PO4_balancing_chemical_equation_phosphorus_pentoxide_and_water_becomes_phosphoric_acid.gif" decoding="async" width="290" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/1c/H3PO4_balancing_chemical_equation_phosphorus_pentoxide_and_water_becomes_phosphoric_acid.gif 1.5x" data-file-width="387" data-file-height="208" /></a><figcaption><a href="/wiki/Phosphorus_pentoxide" title="Phosphorus pentoxide">P<sub>4</sub>O<sub>10</sub></a> + 6 <a href="/wiki/Water" title="Water">H<sub>2</sub>O</a> → 4 <a href="/wiki/Phosphoric_acid" title="Phosphoric acid">H<sub>3</sub>PO<sub>4</sub></a><br />This chemical equation is being balanced by first multiplying H<sub>3</sub>PO<sub>4</sub> by four to match the number of P atoms, and then multiplying H<sub>2</sub>O by six to match the numbers of H and O atoms.</figcaption></figure> <p>The method of inspection can be outlined as setting the most complex substance's stoichiometric coefficient to 1 and assigning values to other coefficients step by step such that both sides of the equation end up with the same number of atoms for each element. If any <a href="/wiki/Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fractional</a> coefficients arise during this process, the presence of fractions may be eliminated (at any time) by multiplying all coefficients by their <a href="/wiki/Lowest_common_denominator" title="Lowest common denominator">lowest common denominator</a>. </p> <dl><dt>Example</dt></dl> <p>Balancing of the chemical equation for the complete <a href="/wiki/Combustion" title="Combustion">combustion</a> of <a href="/wiki/Methane" title="Methane">methane</a> </p> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {{\mathord {?}}\,{CH4}+{\mathord {?}}\,{O2}->{\mathord {?}}\,{CO2}+{\mathord {?}}\,{H2O}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>?</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>CH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>?</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>O</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo stretchy="false">⟶</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>?</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>CO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>?</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>O</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {{\mathord {?}}\,{CH4}+{\mathord {?}}\,{O2}->{\mathord {?}}\,{CO2}+{\mathord {?}}\,{H2O}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/437bb1049e67918fbad8e83489abd367eb7794c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.198ex; height:2.843ex;" alt="{\displaystyle {\ce {{\mathord {?}}\,{CH4}+{\mathord {?}}\,{O2}->{\mathord {?}}\,{CO2}+{\mathord {?}}\,{H2O}}}}"></span></dd></dl></dd></dl></dd></dl> <p>is achieved as follows: </p> <ol><li>A coefficient of 1 is placed in front of the most complex formula (<a href="/wiki/Methane" title="Methane">CH<sub>4</sub></a>): <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {1{CH4}+{\mathord {?}}\,{O2}->{\mathord {?}}\,{CO2}+{\mathord {?}}\,{H2O}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>CH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>?</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>O</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo stretchy="false">⟶</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>?</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>CO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>?</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>O</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {1{CH4}+{\mathord {?}}\,{O2}->{\mathord {?}}\,{CO2}+{\mathord {?}}\,{H2O}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16afae70635fb5854647f6f964a6f4e0538fdb96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.264ex; height:2.843ex;" alt="{\displaystyle {\ce {1{CH4}+{\mathord {?}}\,{O2}->{\mathord {?}}\,{CO2}+{\mathord {?}}\,{H2O}}}}"></span></dd></dl></li> <li>The left-hand side has 1 <a href="/wiki/Carbon" title="Carbon">carbon</a> atom, so 1 molecule of <a href="/wiki/Carbon_dioxide" title="Carbon dioxide">CO<sub>2</sub></a> will balance it. The left-hand side also has 4 <a href="/wiki/Hydrogen" title="Hydrogen">hydrogen</a> atoms, which will be balanced by 2 molecules of <a href="/wiki/Water" title="Water">H<sub>2</sub>O</a>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {1{CH4}+{\mathord {?}}\,{O2}->1{CO2}+2H2O}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>CH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>?</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>O</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo stretchy="false">⟶</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>CO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>O</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {1{CH4}+{\mathord {?}}\,{O2}->1{CO2}+2H2O}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e853c398b48546cbf247c4b33439e7bd193939c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.394ex; height:2.843ex;" alt="{\displaystyle {\ce {1{CH4}+{\mathord {?}}\,{O2}->1{CO2}+2H2O}}}"></span></dd></dl></li> <li>Balancing the 4 <a href="/wiki/Oxygen" title="Oxygen">oxygen</a> atoms of the right-hand side by 2 molecules of <a href="/wiki/Dioxygen" class="mw-redirect" title="Dioxygen">O<sub>2</sub></a> yields the equation <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {1 CH4 + 2 O2 -> 1 CO2 + 2 H2O}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>CH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>O</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo stretchy="false">⟶</mo> <mn>1</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>CO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>O</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {1 CH4 + 2 O2 -> 1 CO2 + 2 H2O}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dcef39b057cc4f23c77325bcf43dda36aecc244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.459ex; height:2.843ex;" alt="{\displaystyle {\ce {1 CH4 + 2 O2 -> 1 CO2 + 2 H2O}}}"></span></dd></dl></li> <li>The coefficients equal to 1 are omitted, as they do not need to be specified explicitly: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {CH4 + 2 O2 -> CO2 + 2 H2O}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>CH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>O</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo stretchy="false">⟶</mo> <msubsup> <mtext>CO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>O</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {CH4 + 2 O2 -> CO2 + 2 H2O}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff89c129e6222f641b40a710d97016377395720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.36ex; height:2.843ex;" alt="{\displaystyle {\ce {CH4 + 2 O2 -> CO2 + 2 H2O}}}"></span></dd></dl></li> <li>It is wise to check that the final equation is balanced, i.e. that for each element there is the same number of atoms on the left- and right-hand side: 1 carbon, 4 hydrogen, and 4 oxygen.</li></ol> <div class="mw-heading mw-heading3"><h3 id="System_of_linear_equations">System of linear equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=8" title="Edit section: System of linear equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For each chemical element (or nuclide or unchanged <a href="/wiki/Moiety_(chemistry)" title="Moiety (chemistry)">moiety</a> or charge) <span class="texhtml mvar" style="font-style:italic;">i</span>, its conservation requirement can be expressed by the <a href="/wiki/Equation_(mathematics)" class="mw-redirect" title="Equation (mathematics)">mathematical equation</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j\,\in \,{\text{reactants}}}\!\!\!\!\!a_{ij}s_{j}\ =\!\!\!\!\!\sum _{j\,\in \,{\text{products}}}\!\!\!\!\!a_{ij}s_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mspace width="thinmathspace" /> <mo>∈</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>reactants</mtext> </mrow> </mrow> </munder> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mspace width="thinmathspace" /> <mo>∈</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>products</mtext> </mrow> </mrow> </munder> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j\,\in \,{\text{reactants}}}\!\!\!\!\!a_{ij}s_{j}\ =\!\!\!\!\!\sum _{j\,\in \,{\text{products}}}\!\!\!\!\!a_{ij}s_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54de8b6d4497051472a1f18b71c820f9d88228a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.019ex; width:25.992ex; height:5.843ex;" alt="{\displaystyle \sum _{j\,\in \,{\text{reactants}}}\!\!\!\!\!a_{ij}s_{j}\ =\!\!\!\!\!\sum _{j\,\in \,{\text{products}}}\!\!\!\!\!a_{ij}s_{j}}"></span></dd></dl> <p>where </p> <dl><dd><span class="texhtml mvar" style="font-style:italic;">a<sub>ij</sub></span> is the number of atoms of element <span class="texhtml mvar" style="font-style:italic;">i</span> in a molecule of substance <span class="texhtml mvar" style="font-style:italic;">j</span> (per formula in the chemical equation), and</dd> <dd><span class="texhtml mvar" style="font-style:italic;">s<sub>j</sub></span> is the stoichiometric coefficient for the substance <span class="texhtml mvar" style="font-style:italic;">j</span>.</dd></dl> <p>This results in a <a href="/wiki/Homogeneous_system_of_linear_equations" class="mw-redirect" title="Homogeneous system of linear equations">homogeneous system of linear equations</a>, which are readily solved using mathematical methods. Such system always has the all-zeros <a href="/wiki/Trivial_solution" class="mw-redirect" title="Trivial solution">trivial solution</a>, which we are not interested in, but if there are any additional solutions, there will be infinite number of them. Any non-trivial solution will balance the chemical equation. A "preferred" solution is one with <a href="/wiki/Integer" title="Integer">whole-number</a>, mostly positive<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>g<span class="cite-bracket">]</span></a></sup> stoichiometric coefficients <span class="texhtml mvar" style="font-style:italic;">s<sub>j</sub></span> with <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> equal to one. </p> <div class="mw-heading mw-heading4"><h4 id="Example">Example</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=9" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let us assign variables to stoichiometric coefficients of the chemical equation from the previous section and write the corresponding linear equations: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {{\mathit {s}}_{1}{CH4}+{\mathit {s}}_{2}{O2}->{\mathit {s}}_{3}{CO2}+{\mathit {s}}_{4}{H2O}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext class="MJX-tex-mathit" mathvariant="italic">s</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>CH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext class="MJX-tex-mathit" mathvariant="italic">s</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>O</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo stretchy="false">⟶</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext class="MJX-tex-mathit" mathvariant="italic">s</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>CO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext class="MJX-tex-mathit" mathvariant="italic">s</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>O</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {{\mathit {s}}_{1}{CH4}+{\mathit {s}}_{2}{O2}->{\mathit {s}}_{3}{CO2}+{\mathit {s}}_{4}{H2O}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07f85bf087629c1dc5eb9a008bf0d368c8c27bb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.282ex; height:2.843ex;" alt="{\displaystyle {\ce {{\mathit {s}}_{1}{CH4}+{\mathit {s}}_{2}{O2}->{\mathit {s}}_{3}{CO2}+{\mathit {s}}_{4}{H2O}}}}"></span></dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \;\;\;{\begin{aligned}{\text{C:}}&&s_{1}&=s_{3}\\{\text{H:}}&&4s_{1}&=2s_{4}\\{\text{O:}}&&2s_{2}&=2s_{3}+s_{4}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>C:</mtext> </mrow> </mtd> <mtd /> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>H:</mtext> </mrow> </mtd> <mtd /> <mtd> <mn>4</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>O:</mtext> </mrow> </mtd> <mtd /> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \;\;\;{\begin{aligned}{\text{C:}}&&s_{1}&=s_{3}\\{\text{H:}}&&4s_{1}&=2s_{4}\\{\text{O:}}&&2s_{2}&=2s_{3}+s_{4}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c640991c1043dc3c4a13f7d505ca8edea382d5dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:26.808ex; height:8.843ex;" alt="{\displaystyle \quad \;\;\;{\begin{aligned}{\text{C:}}&&s_{1}&=s_{3}\\{\text{H:}}&&4s_{1}&=2s_{4}\\{\text{O:}}&&2s_{2}&=2s_{3}+s_{4}\end{aligned}}}"></span> </p><p>All solutions to this system of linear equations are of the following form, where <span class="texhtml mvar" style="font-style:italic;">r</span> is any <a href="/wiki/Real_number" title="Real number">real number</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s_{1}&=r\\s_{2}&=2r\\s_{3}&=r\\s_{4}&=2r\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>r</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>r</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{1}&=r\\s_{2}&=2r\\s_{3}&=r\\s_{4}&=2r\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f57d023d55b7039d2b76fe1bd7c9798daefa864c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:8.206ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}s_{1}&=r\\s_{2}&=2r\\s_{3}&=r\\s_{4}&=2r\end{aligned}}}"></span></dd></dl> <p>The choice of <span class="nowrap"><span class="texhtml mvar" style="font-style:italic;">r</span> = 1</span> yields the preferred solution, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s_{1}&=1\\s_{2}&=2\\s_{3}&=1\\s_{4}&=2\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{1}&=1\\s_{2}&=2\\s_{3}&=1\\s_{4}&=2\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d722453a7a635491c21ac24622249ff29236c4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:7.157ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}s_{1}&=1\\s_{2}&=2\\s_{3}&=1\\s_{4}&=2\end{aligned}}}"></span></dd></dl> <p>which corresponds to the balanced chemical equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {CH4 + 2 O2 -> CO2 + 2 H2O}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>CH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>O</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo stretchy="false">⟶</mo> <msubsup> <mtext>CO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>O</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {CH4 + 2 O2 -> CO2 + 2 H2O}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff89c129e6222f641b40a710d97016377395720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.36ex; height:2.843ex;" alt="{\displaystyle {\ce {CH4 + 2 O2 -> CO2 + 2 H2O}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Matrix_method">Matrix method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=10" title="Edit section: Matrix method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The system of linear equations introduced in the previous section can also be written using an efficient <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> formalism. First, to unify the reactant and product stoichiometric coefficients <span class="texhtml mvar" style="font-style:italic;">s<sub>j</sub></span>, let us introduce the quantity </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{j}={\begin{cases}-s_{j}&{\text{for a reactant}}\\+s_{j}&{\text{for a product}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for a reactant</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for a product</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{j}={\begin{cases}-s_{j}&{\text{for a reactant}}\\+s_{j}&{\text{for a product}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34f37597ef3260d97e950cf1d8bda2b734a840c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.295ex; height:6.176ex;" alt="{\displaystyle \nu _{j}={\begin{cases}-s_{j}&{\text{for a reactant}}\\+s_{j}&{\text{for a product}}\end{cases}}}"></span></dd></dl> <p>called <a href="/wiki/Stoichiometric_number" class="mw-redirect" title="Stoichiometric number">stoichiometric number</a>,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>h<span class="cite-bracket">]</span></a></sup> which simplifies the linear equations to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j=1}^{J}a_{ij}\nu _{j}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j=1}^{J}a_{ij}\nu _{j}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb0c3c8854b7cf17e9b32282edcd25d1f941648f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:12.768ex; height:7.676ex;" alt="{\displaystyle \sum _{j=1}^{J}a_{ij}\nu _{j}=0}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">J</span> is the total number of reactant and product substances (formulas) in the chemical equation. </p><p>Placement of the values <span class="texhtml mvar" style="font-style:italic;">a<sub>ij</sub></span> at row <span class="texhtml mvar" style="font-style:italic;">i</span> and column <span class="texhtml mvar" style="font-style:italic;">j</span> of the composition matrix </p> <dl><dd><span class="texhtml"><b>A</b> =</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,J}\\a_{2,1}&a_{2,2}&\cdots &a_{2,J}\\\vdots &\vdots &\ddots &\vdots \end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>J</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>J</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,J}\\a_{2,1}&a_{2,2}&\cdots &a_{2,J}\\\vdots &\vdots &\ddots &\vdots \end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6789e87f694a1c16978fe2ae6a5116651efc17c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:24.707ex; height:11.509ex;" alt="{\displaystyle {\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,J}\\a_{2,1}&a_{2,2}&\cdots &a_{2,J}\\\vdots &\vdots &\ddots &\vdots \end{bmatrix}}}"></span></dd></dl> <p>and arrangement of the stoichiometric numbers into the stoichiometric vector </p> <dl><dd><span class="texhtml"><b>ν</b> =</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\nu _{1}\\\nu _{2}\\\vdots \\\nu _{J}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\nu _{1}\\\nu _{2}\\\vdots \\\nu _{J}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/922a6418eae4382161cda874fa4e7378464845f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:6.273ex; height:13.843ex;" alt="{\displaystyle {\begin{bmatrix}\nu _{1}\\\nu _{2}\\\vdots \\\nu _{J}\end{bmatrix}}}"></span></dd></dl> <p>allows the system of equations to be expressed as a single <a href="/wiki/Matrix_equation" class="mw-redirect" title="Matrix equation">matrix equation</a>: </p> <dl><dd><span class="texhtml"><b>Aν</b> = <b>0</b></span></dd></dl> <p>Like previously, any nonzero stoichiometric vector <span class="texhtml"><b>ν</b></span>, which solves the matrix equation, will balance the chemical equation. </p><p>The set of solutions to the matrix equation is a <a href="/wiki/Linear_space" class="mw-redirect" title="Linear space">linear space</a> called the <a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">kernel</a> of the matrix <span class="texhtml"><b>A</b></span>. For this space to contain nonzero vectors <span class="texhtml"><b>ν</b></span>, i.e. to have a positive <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimension</a> <var style="padding-right: 1px;">J</var><sub>N</sub>, the columns of the composition matrix <span class="texhtml"><b>A</b></span> must not be <a href="/wiki/Linear_independence" title="Linear independence">linearly independent</a>. The problem of balancing a chemical equation then becomes the problem of determining the <var style="padding-right: 1px;">J</var><sub>N</sub>-dimensional kernel of the composition matrix. It is important to note that only for <var style="padding-right: 1px;">J</var><sub>N</sub> = 1 will there be a unique preferred solution to the balancing problem. For <var style="padding-right: 1px;">J</var><sub>N</sub> > 1 there will be an infinite number of preferred solutions with <var style="padding-right: 1px;">J</var><sub>N</sub> of them linearly independent. If <var style="padding-right: 1px;">J</var><sub>N</sub> = 0, there will be only the unusable trivial solution, the zero vector. </p><p>Techniques have been developed<sup id="cite_ref-Thorne2010_15-0" class="reference"><a href="#cite_note-Thorne2010-15"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Holmes2015_16-0" class="reference"><a href="#cite_note-Holmes2015-16"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> to quickly calculate a set of <var style="padding-right: 1px;">J</var><sub>N</sub> independent solutions to the balancing problem, which are superior to the inspection and <span class="citation-needed-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">algebraic method</span><sup class="noprint Inline-Template Template-Fact" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This method has those properties too. (July 2022)">citation needed</span></a></i>]</sup> in that they are determinative and yield all solutions to the balancing problem. </p> <dl><dt>Example</dt></dl> <p>Using the same chemical equation again, write the corresponding matrix equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {{\mathit {s}}_{1}{CH4}+{\mathit {s}}_{2}{O2}->{\mathit {s}}_{3}{CO2}+{\mathit {s}}_{4}{H2O}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext class="MJX-tex-mathit" mathvariant="italic">s</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>CH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext class="MJX-tex-mathit" mathvariant="italic">s</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>O</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo stretchy="false">⟶</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext class="MJX-tex-mathit" mathvariant="italic">s</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>CO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext class="MJX-tex-mathit" mathvariant="italic">s</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>O</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {{\mathit {s}}_{1}{CH4}+{\mathit {s}}_{2}{O2}->{\mathit {s}}_{3}{CO2}+{\mathit {s}}_{4}{H2O}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07f85bf087629c1dc5eb9a008bf0d368c8c27bb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.282ex; height:2.843ex;" alt="{\displaystyle {\ce {{\mathit {s}}_{1}{CH4}+{\mathit {s}}_{2}{O2}->{\mathit {s}}_{3}{CO2}+{\mathit {s}}_{4}{H2O}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{C:}}\\{\text{H:}}\\{\text{O:}}\end{matrix}}\quad {\begin{bmatrix}1&0&1&0\\4&0&0&2\\0&2&2&1\end{bmatrix}}{\begin{bmatrix}\nu _{1}\\\nu _{2}\\\nu _{3}\\\nu _{4}\end{bmatrix}}=\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>C:</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>H:</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>O:</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{C:}}\\{\text{H:}}\\{\text{O:}}\end{matrix}}\quad {\begin{bmatrix}1&0&1&0\\4&0&0&2\\0&2&2&1\end{bmatrix}}{\begin{bmatrix}\nu _{1}\\\nu _{2}\\\nu _{3}\\\nu _{4}\end{bmatrix}}=\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a853eea2e8e6fe0008447b012df038f3d4a1693" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:31.489ex; height:12.509ex;" alt="{\displaystyle {\begin{matrix}{\text{C:}}\\{\text{H:}}\\{\text{O:}}\end{matrix}}\quad {\begin{bmatrix}1&0&1&0\\4&0&0&2\\0&2&2&1\end{bmatrix}}{\begin{bmatrix}\nu _{1}\\\nu _{2}\\\nu _{3}\\\nu _{4}\end{bmatrix}}=\mathbf {0} }"></span></dd></dl> <p><br /> Its solutions are of the following form, where <span class="texhtml mvar" style="font-style:italic;">r</span> is any real number: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\nu _{1}\\\nu _{2}\\\nu _{3}\\\nu _{4}\end{bmatrix}}={\begin{bmatrix}-s_{1}\\-s_{2}\\s_{3}\\s_{4}\end{bmatrix}}=r{\begin{bmatrix}-1\\-2\\1\\2\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\nu _{1}\\\nu _{2}\\\nu _{3}\\\nu _{4}\end{bmatrix}}={\begin{bmatrix}-s_{1}\\-s_{2}\\s_{3}\\s_{4}\end{bmatrix}}=r{\begin{bmatrix}-1\\-2\\1\\2\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfc63742eec84c4d3eeefdfaee138245886ae50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:27.928ex; height:12.509ex;" alt="{\displaystyle {\begin{bmatrix}\nu _{1}\\\nu _{2}\\\nu _{3}\\\nu _{4}\end{bmatrix}}={\begin{bmatrix}-s_{1}\\-s_{2}\\s_{3}\\s_{4}\end{bmatrix}}=r{\begin{bmatrix}-1\\-2\\1\\2\end{bmatrix}}}"></span></dd></dl> <p>The choice of <span class="nowrap"><span class="texhtml mvar" style="font-style:italic;">r</span> = 1</span> and a <a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">sign</a>-flip of the first two rows yields the preferred solution to the balancing problem: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}-\nu _{1}\\-\nu _{2}\\\nu _{3}\\\nu _{4}\end{bmatrix}}={\begin{bmatrix}s_{1}\\s_{2}\\s_{3}\\s_{4}\end{bmatrix}}={\begin{bmatrix}1\\2\\1\\2\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}-\nu _{1}\\-\nu _{2}\\\nu _{3}\\\nu _{4}\end{bmatrix}}={\begin{bmatrix}s_{1}\\s_{2}\\s_{3}\\s_{4}\end{bmatrix}}={\begin{bmatrix}1\\2\\1\\2\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3b1269921ef5cd7621126363ba32844923bc42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:25.071ex; height:12.509ex;" alt="{\displaystyle {\begin{bmatrix}-\nu _{1}\\-\nu _{2}\\\nu _{3}\\\nu _{4}\end{bmatrix}}={\begin{bmatrix}s_{1}\\s_{2}\\s_{3}\\s_{4}\end{bmatrix}}={\begin{bmatrix}1\\2\\1\\2\end{bmatrix}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Ionic_equations">Ionic equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=11" title="Edit section: Ionic equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An ionic equation is a chemical equation in which <a href="/wiki/Electrolyte" title="Electrolyte">electrolytes</a> are written as dissociated <a href="/wiki/Ion" title="Ion">ions</a>. Ionic equations are used for <a href="/wiki/Single_displacement_reaction" title="Single displacement reaction">single</a> and <a href="/wiki/Double_displacement_reaction" class="mw-redirect" title="Double displacement reaction">double displacement reactions</a> that occur in <a href="/wiki/Aqueous_solution" title="Aqueous solution">aqueous solutions</a>. </p><p>For example, in the following precipitation reaction: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {CaCl2 + 2AgNO3 -> Ca(NO3)2 + 2 AgCl(v)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>CaCl</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>AgNO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo stretchy="false">⟶</mo> <mtext>Ca</mtext> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msubsup> <mtext>NO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mtext>AgCl</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {CaCl2 + 2AgNO3 -> Ca(NO3)2 + 2 AgCl(v)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a60caffe451e908b0e37044b6d780b454afc6154" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.309ex; height:3.009ex;" alt="{\displaystyle {\ce {CaCl2 + 2AgNO3 -> Ca(NO3)2 + 2 AgCl(v)}}}"></span></dd></dl> <p>the full ionic equation is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {Ca^2+ + 2Cl^- + 2Ag+ + 2NO3^- -> Ca^2+ + 2NO3^- + 2AgCl(v)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>Ca</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msup> <mtext>Cl</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msup> <mtext>Ag</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>NO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mo stretchy="false">⟶</mo> <msup> <mtext>Ca</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>NO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mtext>AgCl</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {Ca^2+ + 2Cl^- + 2Ag+ + 2NO3^- -> Ca^2+ + 2NO3^- + 2AgCl(v)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c766d96acad66802d2ed0d4da1b68a0f7b7aee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:62.161ex; height:3.343ex;" alt="{\displaystyle {\ce {Ca^2+ + 2Cl^- + 2Ag+ + 2NO3^- -> Ca^2+ + 2NO3^- + 2AgCl(v)}}}"></span></dd></dl> <p>or, with all physical states included: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {Ca^2+(aq) + 2Cl^{-}(aq) + 2Ag+(aq) + 2NO3^{-}(aq) -> Ca^2+(aq) + 2NO3^{-}(aq) + 2AgCl(v)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>Ca</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mtext>aq</mtext> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msup> <mtext>Cl</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mtext>aq</mtext> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msup> <mtext>Ag</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mtext>aq</mtext> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>NO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mtext>aq</mtext> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">⟶</mo> <msup> <mtext>Ca</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mtext>aq</mtext> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>NO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mtext>aq</mtext> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mtext>AgCl</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {Ca^2+(aq) + 2Cl^{-}(aq) + 2Ag+(aq) + 2NO3^{-}(aq) -> Ca^2+(aq) + 2NO3^{-}(aq) + 2AgCl(v)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/607e8960ed2262e138b2d020e1a038b1700efc68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:87.357ex; height:3.343ex;" alt="{\displaystyle {\ce {Ca^2+(aq) + 2Cl^{-}(aq) + 2Ag+(aq) + 2NO3^{-}(aq) -> Ca^2+(aq) + 2NO3^{-}(aq) + 2AgCl(v)}}}"></span></dd></dl> <p>In this reaction, the Ca<sup>2+</sup> and the NO<sub>3</sub><sup>−</sup> ions remain in solution and are not part of the reaction. That is, these ions are identical on both the reactant and product side of the chemical equation. Because such ions do not participate in the reaction, they are called <a href="/wiki/Spectator_ion" title="Spectator ion">spectator ions</a>. A <i>net ionic</i> equation is the full ionic equation from which the spectator ions have been removed.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The net ionic equation of the proceeding reactions is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {2Cl^- + 2Ag+ -> 2AgCl(v)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mspace width="thinmathspace" /> <msup> <mtext>Cl</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msup> <mtext>Ag</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">⟶</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mtext>AgCl</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {2Cl^- + 2Ag+ -> 2AgCl(v)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/247f6a5347181c97980e0562b43f07423252964c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.23ex; height:2.843ex;" alt="{\displaystyle {\ce {2Cl^- + 2Ag+ -> 2AgCl(v)}}}"></span></dd></dl> <p>or, in <i>reduced</i> balanced form, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {Ag+ + Cl^- -> AgCl(v)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>Ag</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>+</mo> <msup> <mtext>Cl</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">⟶</mo> <mtext>AgCl</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {Ag+ + Cl^- -> AgCl(v)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c7171a8a1b43a85ae9c98a01350f3424dc41a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.581ex; height:2.843ex;" alt="{\displaystyle {\ce {Ag+ + Cl^- -> AgCl(v)}}}"></span></dd></dl> <p>In a <a href="/wiki/Neutralization_(chemistry)" title="Neutralization (chemistry)">neutralization</a> or <a href="/wiki/Acid" title="Acid">acid</a>/<a href="/wiki/Base_(chemistry)" title="Base (chemistry)">base</a> reaction, the net ionic equation will usually be: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {H+ (aq) + OH^{-}(aq) -> H2O(l)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mtext>aq</mtext> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msup> <mtext>OH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mtext>aq</mtext> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">⟶</mo> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>O</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mtext>l</mtext> <mo stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {H+ (aq) + OH^{-}(aq) -> H2O(l)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba86d546c335f4424cde3ca3e6d4c6b4fde47ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.712ex; height:3.176ex;" alt="{\displaystyle {\ce {H+ (aq) + OH^{-}(aq) -> H2O(l)}}}"></span></dd></dl> <p>There are a few acid/base reactions that produce a precipitate in addition to the water molecule shown above. An example is the reaction of <a href="/wiki/Barium_hydroxide" title="Barium hydroxide">barium hydroxide</a> with <a href="/wiki/Phosphoric_acid" title="Phosphoric acid">phosphoric acid</a>, which produces not only water but also the insoluble salt <a href="/w/index.php?title=Barium_phosphate&action=edit&redlink=1" class="new" title="Barium phosphate (page does not exist)">barium phosphate</a>. In this reaction, there are no spectator ions, so the net ionic equation is the same as the full ionic equation. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {3Ba(OH)2 + 2H3PO4 -> 6H2O + Ba3(PO4)2(v)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mspace width="thinmathspace" /> <mtext>Ba</mtext> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mtext>OH</mtext> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <msubsup> <mtext>PO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo stretchy="false">⟶</mo> <mn>6</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>O</mtext> <mo>+</mo> <msubsup> <mtext>Ba</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msubsup> <mtext>PO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {3Ba(OH)2 + 2H3PO4 -> 6H2O + Ba3(PO4)2(v)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac8ce09f4f27e1cb96c3cffbaa3d09323f244b88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.83ex; height:3.009ex;" alt="{\displaystyle {\ce {3Ba(OH)2 + 2H3PO4 -> 6H2O + Ba3(PO4)2(v)}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {{3Ba^{2}+}+{6OH^{-}}+{6H+}}}+\underbrace {\ce {2PO4^{3}-}} _{\ce {phosphate}}{\ce {->{6H2O}+\underbrace {Ba3(PO4)2(v)} _{barium~phosphate}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mspace width="thinmathspace" /> <msup> <mtext>Ba</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> </mrow> </msup> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mspace width="thinmathspace" /> <msup> <mtext>OH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mspace width="thinmathspace" /> <msup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </mrow> <mo>+</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mn>2</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>PO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> </mrow> </msubsup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>phosphate</mtext> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">⟶</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mspace width="thinmathspace" /> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>O</mtext> </mrow> <mo>+</mo> <munderover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <msubsup> <mtext>Ba</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msubsup> <mtext>PO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>barium</mtext> <mtext> </mtext> <mtext>phosphate</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </munderover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {{3Ba^{2}+}+{6OH^{-}}+{6H+}}}+\underbrace {\ce {2PO4^{3}-}} _{\ce {phosphate}}{\ce {->{6H2O}+\underbrace {Ba3(PO4)2(v)} _{barium~phosphate}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30887789728a5f55e509321b6c3f24fe7ea19107" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:60.409ex; height:7.843ex;" alt="{\displaystyle {\ce {{3Ba^{2}+}+{6OH^{-}}+{6H+}}}+\underbrace {\ce {2PO4^{3}-}} _{\ce {phosphate}}{\ce {->{6H2O}+\underbrace {Ba3(PO4)2(v)} _{barium~phosphate}}}}"></span></dd></dl> <p>Double displacement reactions that feature a <a href="/wiki/Carbonate" title="Carbonate">carbonate</a> reacting with an acid have the net ionic equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {2H+}}+\underbrace {{\ce {CO3^2-}}} _{{\ce {carbonate}}}{\ce {-> H2O + CO2 (^)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mspace width="thinmathspace" /> <msup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> <mo>+</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>CO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> </mrow> </msubsup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>carbonate</mtext> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">⟶</mo> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>O</mtext> <mo>+</mo> <msubsup> <mtext>CO</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↑</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {2H+}}+\underbrace {{\ce {CO3^2-}}} _{{\ce {carbonate}}}{\ce {-> H2O + CO2 (^)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f31bfcd08a92ebd1de3c052683e394fbb81cade" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:32.282ex; height:6.676ex;" alt="{\displaystyle {\ce {2H+}}+\underbrace {{\ce {CO3^2-}}} _{{\ce {carbonate}}}{\ce {-> H2O + CO2 (^)}}}"></span></dd></dl> <p>If every ion is a "spectator ion" then there was no reaction, and the net ionic equation is null. </p><p>Generally, if <i>z<sub>j</sub></i> is the multiple of elementary charge on the <i>j-th</i> molecule, charge neutrality may be written as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j=1}^{J}z_{j}\nu _{j}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j=1}^{J}z_{j}\nu _{j}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd130dcbb3fc5b2dc725076cd97f9d0620bd235d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:12.052ex; height:7.676ex;" alt="{\displaystyle \sum _{j=1}^{J}z_{j}\nu _{j}=0}"></span></dd></dl> <p>where the <i>ν<sub>j</sub></i> are the stoichiometric coefficients described above. The <i>z<sub>j</sub></i> may be incorporated<sup id="cite_ref-Thorne2010_15-1" class="reference"><a href="#cite_note-Thorne2010-15"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Holmes2015_16-1" class="reference"><a href="#cite_note-Holmes2015-16"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> as an additional row in the <i>a<sub>ij</sub></i> matrix described above, and a properly balanced ionic equation will then also obey: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j=1}^{J}a_{ij}\nu _{j}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j=1}^{J}a_{ij}\nu _{j}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb0c3c8854b7cf17e9b32282edcd25d1f941648f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:12.768ex; height:7.676ex;" alt="{\displaystyle \sum _{j=1}^{J}a_{ij}\nu _{j}=0}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=12" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link 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href="https://en.wikipedia.org/w/index.php?title=Chemical_equation&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">June 2022</span>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Typesetting">Typesetting</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=13" title="Edit section: Typesetting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Empty_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span"><b>This section is empty.</b> You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Chemical_equation&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">June 2022</span>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=14" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Not to be confused with a related quantity called <a href="/wiki/Stoichiometric_number" class="mw-redirect" title="Stoichiometric number">stoichiometric number</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Not to be confused with <a href="/wiki/Yield_(chemistry)" title="Yield (chemistry)">yield (chemistry)</a>, a quantification of synthesis efficiency.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">The notation ⇄ was proposed in 1884 by the Dutch chemist <a href="/wiki/Jacobus_Henricus_van_%27t_Hoff" title="Jacobus Henricus van 't Hoff">Jacobus Henricus van 't Hoff</a>. Van 't Hoff called reactions that didn't proceed to completion "limited reactions". He wrote (translation from French):<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>Now Mr. Pfaundler has joined these two phenomena in a single concept by considering the observed limit as the result of two opposing reactions, driving the one in the example cited to the formation of sea salt [i.e., NaCl] and nitric acid, [and] the other to hydrochloric acid and sodium nitrate. This consideration, which experiment validates, justifies the expression "chemical equilibrium", which is used to characterize the final state of limited reactions. I would propose to translate this expression by the following symbol: </p><dl><dd>HCl + NO<sub>3</sub> Na ⇄ NO<sub>3</sub> H + Cl Na.</dd></dl><p> I thus replace, in this case, the = sign in the chemical equation by the sign ⇄, which in reality doesn't express just equality but shows also the direction of the reaction. This clearly expresses that a chemical action occurs simultaneously in two opposing directions.</p></blockquote></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">The notation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {<=>}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded height="0" depth="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↽</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">⇀</mo> </mrow> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {<=>}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961c77b79c6c5277df106f207db09fb691d7bf72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0; margin-bottom: -0.505ex; width:3.358ex; height:2.843ex;" alt="{\displaystyle {\ce {<=>}}}"></span> was suggested by <a href="/wiki/Hugh_Marshall" title="Hugh Marshall">Hugh Marshall</a> in 1902.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Triangle (△) was originally the alchemical symbol for fire.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">This expression comes from the <a href="/wiki/Planck_relation" title="Planck relation">Planck equation for the energy of a photon</a>, <span class="nowrap"><span class="texhtml mvar" style="font-style:italic;">E</span> = <span class="texhtml mvar" style="font-style:italic;">hν</span></span>. The Greek letter <a href="/wiki/Nu_(letter)" title="Nu (letter)"><span class="texhtml mvar" style="font-style:italic;">ν</span> ("nu")</a> is sometimes mistakenly replaced with a Latin letter <a href="/wiki/V" title="V"><span class="texhtml mvar" style="font-style:italic;">v</span> ("vee")</a>.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">A negative stoichiometric coefficient signifies a substance placed on the incorrect side of the chemical equation.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">An equivalent approach is flipping the signs of <span class="texhtml mvar" style="font-style:italic;">a<sub>ij</sub></span> for reactants instead of replacing the stoichiometric coefficients <span class="texhtml mvar" style="font-style:italic;">s<sub>j</sub></span> with stoichiometric numbers <span class="texhtml mvar" style="font-style:italic;">ν<sub>j</sub></span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chemical_equation&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-goldbook-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-goldbook_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-goldbook_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/International_Union_of_Pure_and_Applied_Chemistry" title="International Union of Pure and Applied Chemistry">IUPAC</a>, <i><a href="/wiki/IUPAC_books#Gold_Book" class="mw-redirect" title="IUPAC books">Compendium of Chemical Terminology</a></i>, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "<a rel="nofollow" class="external text" href="https://goldbook.iupac.org/terms/view/C01034.html">chemical reaction equation</a>". <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1351%2Fgoldbook.C01034">10.1351/goldbook.C01034</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrosland,_M.P.1959" class="citation journal cs1">Crosland, M.P. (1959). "The use of diagrams as chemical 'equations' in the lectures of William Cullen and Joseph Black". <i>Annals of Science</i>. <b>15</b> (2): 75–90. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00033795900200088">10.1080/00033795900200088</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Science&rft.atitle=The+use+of+diagrams+as+chemical+%27equations%27+in+the+lectures+of+William+Cullen+and+Joseph+Black&rft.volume=15&rft.issue=2&rft.pages=75-90&rft.date=1959&rft_id=info%3Adoi%2F10.1080%2F00033795900200088&rft.au=Crosland%2C+M.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChemical+equation" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_'t_Hoff1884" class="citation book cs1 cs1-prop-foreign-lang-source">van 't Hoff, J.H. (1884). <a rel="nofollow" class="external text" href="https://archive.org/stream/etudesdedynamiqu00hoff#page/4/mode/2up"><i>Études de Dynamique Chemique</i></a> [<i>Studies of chemical dynamics</i>] (in French). Amsterdam, Netherlands: Frederik Muller & Co. pp. 4–5. <q>Or M. Pfaundler a relié ces deux phénomênes ... s'accomplit en même temps dans deux sens opposés.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C3%89tudes+de+Dynamique+Chemique&rft.place=Amsterdam%2C+Netherlands&rft.pages=4-5&rft.pub=Frederik+Muller+%26+Co.&rft.date=1884&rft.aulast=van+%27t+Hoff&rft.aufirst=J.H.&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fetudesdedynamiqu00hoff%23page%2F4%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChemical+equation" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarshall1902" class="citation journal cs1">Marshall, Hugh (1902). "Suggested Modifications of the Sign of Equality for Use in Chemical Notation". <i>Proceedings of the Royal Society of Edinburgh</i>. <b>24</b>: 85–87. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0370164600007720">10.1017/S0370164600007720</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+Edinburgh&rft.atitle=Suggested+Modifications+of+the+Sign+of+Equality+for+Use+in+Chemical+Notation&rft.volume=24&rft.pages=85-87&rft.date=1902&rft_id=info%3Adoi%2F10.1017%2FS0370164600007720&rft.aulast=Marshall&rft.aufirst=Hugh&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChemical+equation" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcLean" class="citation web cs1">McLean, Les. <a rel="nofollow" class="external text" href="https://www.quora.com/Why-cant-we-write-the-symbol-in-chemical-equations">"Why can't we write the symbol "-" in chemical equations?"</a>. <i>Quora</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Quora&rft.atitle=Why+can%27t+we+write+the+symbol+%22-%22+in+chemical+equations%3F&rft.aulast=McLean&rft.aufirst=Les&rft_id=https%3A%2F%2Fwww.quora.com%2FWhy-cant-we-write-the-symbol-in-chemical-equations&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChemical+equation" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20210615072545/https://chemistry.stackexchange.com/questions/42075/why-is-the-minus-sign-not-allowed-in-reaction-equations">"Why is the minus sign (-) not allowed in reaction equations?"</a>. <i><a href="/wiki/Stack_Exchange" title="Stack Exchange">Stack Exchange</a></i>. 2017-09-20. Answer by Nicolau Saker Neto. Archived from <a rel="nofollow" class="external text" href="https://chemistry.stackexchange.com/questions/42075/why-is-the-minus-sign-not-allowed-in-reaction-equations">the original</a> on 2021-06-15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stack+Exchange&rft.atitle=Why+is+the+minus+sign+%28-%29+not+allowed+in+reaction+equations%3F&rft.pages=Answer+by+Nicolau+Saker+Neto&rft.date=2017-09-20&rft_id=https%3A%2F%2Fchemistry.stackexchange.com%2Fquestions%2F42075%2Fwhy-is-the-minus-sign-not-allowed-in-reaction-equations&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChemical+equation" class="Z3988"></span></span> </li> <li id="cite_note-Thorne2010-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-Thorne2010_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Thorne2010_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThorne2010" class="citation journal cs1">Thorne, Lawrence R. (2010). "An Innovative Approach to Balancing Chemical-Reaction Equations: A Simplified Matrix-Inversion Technique for Determining the Matrix Null Space". <i>Chem. Educator</i>. <b>15</b>: 304–308. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1110.4321">1110.4321</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Chem.+Educator&rft.atitle=An+Innovative+Approach+to+Balancing+Chemical-Reaction+Equations%3A+A+Simplified+Matrix-Inversion+Technique+for+Determining+the+Matrix+Null+Space&rft.volume=15&rft.pages=304-308&rft.date=2010&rft_id=info%3Aarxiv%2F1110.4321&rft.aulast=Thorne&rft.aufirst=Lawrence+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChemical+equation" class="Z3988"></span></span> </li> <li id="cite_note-Holmes2015-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-Holmes2015_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Holmes2015_16-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHolmes2015" class="citation web cs1">Holmes, Dylan (2015). <a rel="nofollow" class="external text" href="http://www.logical.ai/chemistry/html/chem-nullspace.html">"The null space's insight into chemical balance"</a>. Dylan Holmes<span class="reference-accessdate">. Retrieved <span class="nowrap">Oct 10,</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+null+space%27s+insight+into+chemical+balance&rft.pub=Dylan+Holmes&rft.date=2015&rft.aulast=Holmes&rft.aufirst=Dylan&rft_id=http%3A%2F%2Fwww.logical.ai%2Fchemistry%2Fhtml%2Fchem-nullspace.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChemical+equation" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames_E._BradyFrederick_SeneseNeil_D._Jespersen2007" class="citation book cs1">James E. Brady; Frederick Senese; Neil D. Jespersen (December 14, 2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=A1UvAQAAIAAJ&q=%22a+net+ionic+equation+is+obtained+by+eliminating+spectator+ions+from+the+ionic+equation%22"><i>Chemistry: matter and its changes</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780470120941" title="Special:BookSources/9780470120941"><bdi>9780470120941</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2007033355">2007033355</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chemistry%3A+matter+and+its+changes&rft.pub=John+Wiley+%26+Sons&rft.date=2007-12-14&rft_id=info%3Alccn%2F2007033355&rft.isbn=9780470120941&rft.au=James+E.+Brady&rft.au=Frederick+Senese&rft.au=Neil+D.+Jespersen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DA1UvAQAAIAAJ%26q%3D%2522a%2Bnet%2Bionic%2Bequation%2Bis%2Bobtained%2Bby%2Beliminating%2Bspectator%2Bions%2Bfrom%2Bthe%2Bionic%2Bequation%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChemical+equation" 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