Isotope dilution - Wikipedia

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dilution method subsection</span> </button> <ul id="toc-Single_dilution_method-sublist" class="vector-toc-list"> <li id="toc-Optimum_composition_of_the_blend" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optimum_composition_of_the_blend"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Optimum composition of the blend</span> </div> </a> <ul id="toc-Optimum_composition_of_the_blend-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Double_dilution_method" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Double_dilution_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Double dilution method</span> </div> </a> <ul id="toc-Double_dilution_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Triple_dilution_method" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Triple_dilution_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Triple dilution method</span> </div> </a> <ul id="toc-Triple_dilution_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calculations_using_calibration_curve" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Calculations_using_calibration_curve"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Calculations using calibration curve</span> </div> </a> <ul id="toc-Calculations_using_calibration_curve-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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" 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href="https://de.wikipedia.org/wiki/Isotopenverd%C3%BCnnungsanalyse" title="Isotopenverdünnungsanalyse – German" lang="de" hreflang="de" data-title="Isotopenverdünnungsanalyse" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Diluci%C3%B3n_isot%C3%B3pica" title="Dilución isotópica – Spanish" lang="es" hreflang="es" data-title="Dilución isotópica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Dilution_isotopique" title="Dilution isotopique – French" lang="fr" hreflang="fr" data-title="Dilution isotopique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pengenceran_isotop" title="Pengenceran isotop – Indonesian" lang="id" hreflang="id" data-title="Pengenceran isotop" 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/Diluizione_isotopica" title="Diluizione isotopica – Italian" lang="it" hreflang="it" data-title="Diluizione isotopica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Metoda_rozcie%C5%84cze%C5%84_izotopowych" title="Metoda rozcieńczeń izotopowych – Polish" lang="pl" hreflang="pl" data-title="Metoda rozcieńczeń izotopowych" data-language-autonym="Polski" 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src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Principle_of_isotope_dilution.svg/330px-Principle_of_isotope_dilution.svg.png" decoding="async" width="330" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Principle_of_isotope_dilution.svg/495px-Principle_of_isotope_dilution.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Principle_of_isotope_dilution.svg/660px-Principle_of_isotope_dilution.svg.png 2x" data-file-width="1052" data-file-height="744" /></a><figcaption><b>Basic principle of isotope dilution</b><br /> Adding of an isotopically altered standard to the sample changes the natural isotopic composition of the analyte. By measuring the resulting isotopic composition, it is possible to calculate the amount of the analyte present in the sample.</figcaption></figure> <p><b>Isotope dilution analysis</b> is a method of determining the quantity of chemical substances. In its most simple conception, the method of isotope dilution comprises the addition of known amounts of isotopically enriched substance to the analyzed sample. Mixing of the isotopic standard with the sample effectively "dilutes" the isotopic enrichment of the standard and this forms the basis for the isotope dilution method. Isotope dilution is classified as a method of <a href="/wiki/Internal_standard" title="Internal standard">internal standardisation</a>, because the standard (isotopically enriched form of analyte) is added directly to the sample. In addition, unlike traditional analytical methods which rely on signal intensity, isotope dilution employs signal ratios. Owing to both of these advantages, the method of isotope dilution is regarded among chemistry measurement methods of the highest metrological standing.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p><b>Isotopes</b> are variants of a particular <a href="/wiki/Chemical_element" title="Chemical element">chemical element</a> which differ in <a href="/wiki/Neutron_number" title="Neutron number">neutron number</a>. All isotopes of a given element have the same number of <a href="/wiki/Proton" title="Proton">protons</a> in each <a href="/wiki/Atom" title="Atom">atom</a>. The term <i>isotope</i> is formed from the Greek roots <i>isos</i> (<a href="https://en.wiktionary.org/wiki/%E1%BC%B4%CF%83%CE%BF%CF%82" class="extiw" title="wikt:ἴσος">ἴσος</a> "equal") and <i>topos</i> (<a href="https://en.wiktionary.org/wiki/%CF%84%CF%8C%CF%80%CE%BF%CF%82" class="extiw" title="wikt:τόπος">τόπος</a> "place"), meaning "the same place"; thus, the meaning behind the name is that different isotopes of a single element occupy the same position on the <a href="/wiki/Periodic_table" title="Periodic table">periodic table</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Early_history">Early history</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isotope_dilution&action=edit&section=1" title="Edit section: Early history"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:George_de_Hevesy.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/George_de_Hevesy.jpg/220px-George_de_Hevesy.jpg" decoding="async" width="220" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/George_de_Hevesy.jpg/330px-George_de_Hevesy.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/George_de_Hevesy.jpg/440px-George_de_Hevesy.jpg 2x" data-file-width="816" data-file-height="1113" /></a><figcaption>The Hungarian chemist George de Hevesy was awarded the Nobel Prize in Chemistry for development of radiotracer method, which is a forerunner of isotope dilution</figcaption></figure> <p>Analytical application of the radiotracer method is a forerunner of isotope dilution. This method was developed in the early 20th century by <a href="/wiki/George_de_Hevesy" title="George de Hevesy">George de Hevesy</a> for which he was awarded the <a href="/wiki/Nobel_Prize_in_Chemistry" title="Nobel Prize in Chemistry">Nobel Prize in Chemistry</a> for 1943. </p><p>An early application of isotope dilution in the form of radiotracer method was determination of the solubility of lead sulphide and lead chromate in 1913 by <a href="/wiki/George_de_Hevesy" title="George de Hevesy">George de Hevesy</a> and <a href="/wiki/Friedrich_Adolf_Paneth" class="mw-redirect" title="Friedrich Adolf Paneth">Friedrich Adolf Paneth</a>.<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> In the 1930s, US biochemist <a href="/wiki/David_Rittenberg" title="David Rittenberg">David Rittenberg</a> pioneered the use of isotope dilution in biochemistry enabling detailed studies of cell metabolism.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Tutorial_example">Tutorial example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isotope_dilution&action=edit&section=2" title="Edit section: Tutorial example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Illustration_of_isotope_dilution_analysis_with_fish_counting_in_lakes.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Illustration_of_isotope_dilution_analysis_with_fish_counting_in_lakes.jpg/330px-Illustration_of_isotope_dilution_analysis_with_fish_counting_in_lakes.jpg" decoding="async" width="330" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/c2/Illustration_of_isotope_dilution_analysis_with_fish_counting_in_lakes.jpg 1.5x" data-file-width="385" data-file-height="195" /></a><figcaption>Tutorial illustration of isotope dilution analysis with fish counting in lakes</figcaption></figure> <p>Isotope dilution is analogous to the <a href="/wiki/Mark_and_recapture" title="Mark and recapture">mark and recapture</a> method, commonly used in ecology to estimate population size. </p><p>For instance, consider the determination of the number of fish (<i>n</i><sub>A</sub>) in a lake. For the purpose of this example, assume all fish native to the lake are blue. On their first visit to the lake, an ecologist adds five yellow fish (<i>n</i><sub>B</sub> = 5). On their second visit, the ecologist captures a number of fish according to a <a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">sampling</a> plan and observes that the ratio of blue-to-yellow (i.e. native-to-marked) fish is 10:1. The number of fish native to the lake can be calculated using the following 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 n_{\mathrm {A} }=n_{\mathrm {B} }\times {\frac {10}{1}}=50}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>1</mn> </mfrac> </mrow> <mo>=</mo> <mn>50</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\mathrm {A} }=n_{\mathrm {B} }\times {\frac {10}{1}}=50}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78be2d4387d7cd7e503697384c61b06a75f3bb58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.173ex; height:5.176ex;" alt="{\displaystyle n_{\mathrm {A} }=n_{\mathrm {B} }\times {\frac {10}{1}}=50}"></span></dd></dl> <p>This is a simplified view of isotope dilution but it illustrates the method's salient features. A more complex situation arises when the distinction between marked and unmarked fish becomes fuzzy. This can occur, for example, when the lake already contains a small number of marked fish from previous field experiments; and vice versa, where the amount of marked fish added contains a small number of unmarked fish. In a laboratory setting, an unknown (the "lake") may contain a quantity of a compound that is naturally present in major ("blue") and minor ("yellow") isotopic forms. A standard that is enriched in the minor isotopic form may then be added to the unknown, which can be subsequently analyzed. Keeping to the fish analogy, the following expression can be employed: </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 n_{\mathrm {A} }=n_{\mathrm {B} }\times {\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}\times {\frac {1+R_{\mathrm {A} }}{1+R_{\mathrm {B} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\mathrm {A} }=n_{\mathrm {B} }\times {\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}\times {\frac {1+R_{\mathrm {A} }}{1+R_{\mathrm {B} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86bf95b607dea3d0c5e27026b7b151e86d148cd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.795ex; height:5.843ex;" alt="{\displaystyle n_{\mathrm {A} }=n_{\mathrm {B} }\times {\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}\times {\frac {1+R_{\mathrm {A} }}{1+R_{\mathrm {B} }}}}"></span></dd></dl> <p>where, as indicated above, <i>n</i><sub>A</sub> and n<sub>B</sub> represent the number of fish in the lake and the number of fish added to the lake, respectively; <i>R</i><sub>A</sub> is the ratio of the native-to-marked fish in the lake prior to the addition of marked fish; <i>R</i><sub>B</sub> is the ratio of the native-to-marked fish in the amount of marked fish added to the lake; finally, <i>R</i><sub>AB</sub> is the ratio of the native-to-marked fish captured during the second visit. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isotope_dilution&action=edit&section=3" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Isotope dilution is almost exclusively employed with mass spectrometry in applications where high-accuracy is demanded. For example, all National Metrology Institutes rely significantly on isotope dilution when producing certified reference materials. In addition to high-precision analysis, isotope dilution is applied when low recovery of the analyte is encountered. In addition to the use of stable isotopes, radioactive isotopes can be employed in isotope dilution which is often encountered in biomedical applications, for example, in estimating the <a href="/wiki/Blood_volume" title="Blood volume">volume of blood</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Single_dilution_method">Single dilution method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isotope_dilution&action=edit&section=4" title="Edit section: Single dilution method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="infobox" width="200px"> <tbody><tr> <th colspan="2">Isotope dilution notation </th></tr> <tr> <td></td> <td> </td></tr> <tr> <th>Name</th> <th>Symbol </th></tr> <tr> <td>Analyte</td> <td><strong>A</strong> </td></tr> <tr> <td>Isotopic standard (Spike)</td> <td><strong>B</strong> </td></tr> <tr> <td></td> <td> </td></tr> <tr> <td>Analyte + Spike</td> <td><strong>AB</strong> </td></tr></tbody></table> <p>Consider a natural analyte rich in isotope <sup><i>i</i></sup>A (denoted as A), and the same analyte, enriched in isotope <sup><i>j</i></sup>A (denoted as B). Then, the obtained mixture is analyzed for the isotopic composition of the analyte, <i>R</i><sub>AB</sub> = <i>n</i>(<sup><i>i</i></sup>A)<sub>AB</sub>/<i>n</i>(<sup><i>j</i></sup>A)<sub>AB</sub>. If the amount of the isotopically enriched substance (<i>n</i><sub>B</sub>) is known, the amount of substance in the sample (<i>n</i><sub>A</sub>) can be obtained:<sup id="cite_ref-Debus_4-0" class="reference"><a href="#cite_note-Debus-4"><span class="cite-bracket">[</span>4<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 n_{\mathrm {A} }=n_{\mathrm {B} }{\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}\times {\frac {x(^{j}\mathrm {A} )_{\mathrm {B} }}{x(^{j}\mathrm {A} )_{\mathrm {A} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>x</mi> <msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\mathrm {A} }=n_{\mathrm {B} }{\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}\times {\frac {x(^{j}\mathrm {A} )_{\mathrm {B} }}{x(^{j}\mathrm {A} )_{\mathrm {A} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66737c109590750f6dc3dbb4cd6b9205caa515bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.979ex; height:6.676ex;" alt="{\displaystyle n_{\mathrm {A} }=n_{\mathrm {B} }{\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}\times {\frac {x(^{j}\mathrm {A} )_{\mathrm {B} }}{x(^{j}\mathrm {A} )_{\mathrm {A} }}}}"></span></dd></dl> <p>Here, <i>R</i><sub>A</sub> is the isotope amount ratio of the natural analyte, <i>R</i><sub>A</sub> = <i>n</i>(<sup><i>i</i></sup>A)<sub>A</sub>/<i>n</i>(<sup><i>j</i></sup>A)<sub>A</sub>, <i>R</i><sub>B</sub> is the isotope amount ratio of the isotopically enriched analyte, <i>R</i><sub>B</sub> = <i>n</i>(<sup><i>i</i></sup>A)<sub>B</sub>/<i>n</i>(<sup><i>j</i></sup>A)<sub>B</sub>, <i>R</i><sub>AB</sub> is the isotope amount ratio of the resulting mixture, <i>x</i>(<sup><i>j</i></sup>A)<sub>A</sub> is the isotopic abundance of the minor isotope in the natural analyte, and <i>x</i>(<sup><i>j</i></sup>A)<sub>B</sub> is the isotopic abundance of the major isotope in the isotopically enriched analyte. </p><p>For elements with only two stable isotopes, such as boron, chlorine, or silver, the above single dilution equation simplifies to the following: </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 n_{\mathrm {A} }=n_{\mathrm {B} }{\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}\times {\frac {1+R_{\mathrm {A} }}{1+R_{\mathrm {B} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\mathrm {A} }=n_{\mathrm {B} }{\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}\times {\frac {1+R_{\mathrm {A} }}{1+R_{\mathrm {B} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ea97919f820c71413b3ef05f4822d0b7fc47488" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.955ex; height:5.843ex;" alt="{\displaystyle n_{\mathrm {A} }=n_{\mathrm {B} }{\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}\times {\frac {1+R_{\mathrm {A} }}{1+R_{\mathrm {B} }}}}"></span></dd></dl> <p>In a typical <a href="/wiki/Gas_chromatography" title="Gas chromatography">gas chromatography</a> analysis, isotopic dilution can decrease the uncertainty of the measurement results from 5% to 1%. It can also be used in <a href="/wiki/Mass_spectrometry" title="Mass spectrometry">mass spectrometry</a> (commonly referred to as isotopic dilution mass spectrometry or IDMS), in which the isotopic ratio can be determined with precision typically better than 0.25%.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Optimum_composition_of_the_blend">Optimum composition of the blend</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isotope_dilution&action=edit&section=5" title="Edit section: Optimum composition of the blend"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a simplified manner, the uncertainty of the measurement results is largely determined from the measurement of <i>R</i><sub>AB</sub>: </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 u(n_{\mathrm {A} })^{2}\propto \left({\frac {\partial {n_{\mathrm {A} }}}{\partial R_{\mathrm {AB} }}}\right)^{2}u(R_{\mathrm {AB} })^{2}=n_{\mathrm {A} }^{2}{\frac {(R_{\mathrm {A} }-R_{\mathrm {B} })^{2}}{(R_{\mathrm {A} }-R_{\mathrm {AB} })^{2}(R_{\mathrm {AB} }-R_{\mathrm {B} })^{2}}}u(R_{\mathrm {AB} })^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∝</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>u</mi> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(n_{\mathrm {A} })^{2}\propto \left({\frac {\partial {n_{\mathrm {A} }}}{\partial R_{\mathrm {AB} }}}\right)^{2}u(R_{\mathrm {AB} })^{2}=n_{\mathrm {A} }^{2}{\frac {(R_{\mathrm {A} }-R_{\mathrm {B} })^{2}}{(R_{\mathrm {A} }-R_{\mathrm {AB} })^{2}(R_{\mathrm {AB} }-R_{\mathrm {B} })^{2}}}u(R_{\mathrm {AB} })^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ed7cd4ed7e45483dd4ecaea9115a99890446bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:71.72ex; height:6.676ex;" alt="{\displaystyle u(n_{\mathrm {A} })^{2}\propto \left({\frac {\partial {n_{\mathrm {A} }}}{\partial R_{\mathrm {AB} }}}\right)^{2}u(R_{\mathrm {AB} })^{2}=n_{\mathrm {A} }^{2}{\frac {(R_{\mathrm {A} }-R_{\mathrm {B} })^{2}}{(R_{\mathrm {A} }-R_{\mathrm {AB} })^{2}(R_{\mathrm {AB} }-R_{\mathrm {B} })^{2}}}u(R_{\mathrm {AB} })^{2}}"></span></dd></dl> <p>From here, we obtain the relative uncertainty of <i>n</i><sub>A</sub>, <i>u</i><sub>r</sub>(<i>n</i><sub>A</sub>) = <i>u</i>(<i>n</i><sub>A</sub>)/<i>n</i><sub>A</sub>: </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 u_{\mathrm {r} }(n_{\mathrm {A} })^{2}\propto {\frac {(R_{\mathrm {A} }-R_{\mathrm {B} })^{2}}{(R_{\mathrm {A} }-R_{\mathrm {AB} })^{2}(R_{\mathrm {AB} }-R_{\mathrm {B} })^{2}}}u(R_{\mathrm {AB} })^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>u</mi> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{\mathrm {r} }(n_{\mathrm {A} })^{2}\propto {\frac {(R_{\mathrm {A} }-R_{\mathrm {B} })^{2}}{(R_{\mathrm {A} }-R_{\mathrm {AB} })^{2}(R_{\mathrm {AB} }-R_{\mathrm {B} })^{2}}}u(R_{\mathrm {AB} })^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d4bebc7d0aed17c07ce90a4f99a9688978fb5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:47.031ex; height:6.676ex;" alt="{\displaystyle u_{\mathrm {r} }(n_{\mathrm {A} })^{2}\propto {\frac {(R_{\mathrm {A} }-R_{\mathrm {B} })^{2}}{(R_{\mathrm {A} }-R_{\mathrm {AB} })^{2}(R_{\mathrm {AB} }-R_{\mathrm {B} })^{2}}}u(R_{\mathrm {AB} })^{2}}"></span></dd></dl> <p>The lowest relative uncertainty of <i>n</i><sub>A</sub> corresponds to the condition when the first derivative with respect to <i>R</i><sub>AB</sub> equals zero. In addition, it is common in mass spectrometry that <i>u</i>(<i>R</i><sub>AB</sub>)/<i>R</i><sub>AB</sub> is constant and therefore we can replace <i>u</i>(<i>R</i><sub>AB</sub>) with <i>R</i><sub>AB</sub>. These ideas combine to give </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 u_{\mathrm {r} }(n_{\mathrm {A} })_{\mathrm {min} }\mapsto \partial \left({\frac {(R_{\mathrm {A} }-R_{\mathrm {B} })}{(R_{\mathrm {A} }-R_{\mathrm {AB} })(R_{\mathrm {AB} }-R_{\mathrm {B} })}}R_{\mathrm {AB} }\right)/\partial R_{\mathrm {AB} }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">↦</mo> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{\mathrm {r} }(n_{\mathrm {A} })_{\mathrm {min} }\mapsto \partial \left({\frac {(R_{\mathrm {A} }-R_{\mathrm {B} })}{(R_{\mathrm {A} }-R_{\mathrm {AB} })(R_{\mathrm {AB} }-R_{\mathrm {B} })}}R_{\mathrm {AB} }\right)/\partial R_{\mathrm {AB} }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e93bfc55a72d25530756c99762b1ad95cea2c261" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.811ex; height:6.509ex;" alt="{\displaystyle u_{\mathrm {r} }(n_{\mathrm {A} })_{\mathrm {min} }\mapsto \partial \left({\frac {(R_{\mathrm {A} }-R_{\mathrm {B} })}{(R_{\mathrm {A} }-R_{\mathrm {AB} })(R_{\mathrm {AB} }-R_{\mathrm {B} })}}R_{\mathrm {AB} }\right)/\partial R_{\mathrm {AB} }=0}"></span></dd></dl> <p>Solving this equation leads to the optimum composition of the blend AB, i.e., the <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a> between the isotopic compositions of standard (A) and spike (B): </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 R_{\mathrm {AB} }={\sqrt {R_{\mathrm {A} }R_{\mathrm {B} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {AB} }={\sqrt {R_{\mathrm {A} }R_{\mathrm {B} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e47f4d5e6f16cc294f2d4efb747b2ad3614abd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.203ex; height:3.509ex;" alt="{\displaystyle R_{\mathrm {AB} }={\sqrt {R_{\mathrm {A} }R_{\mathrm {B} }}}}"></span></dd></dl> <p>This simplified equation was first proposed by De Bievre and Debus numerically<sup id="cite_ref-Debus_4-1" class="reference"><a href="#cite_note-Debus-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and later by Komori et al.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> and by Riepe and Kaiser analytically.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> It has been noted that this simple expression is only a general approximation and it does not hold, for example, in the presence of Poisson statistics<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> or in the presence of strong isotope signal ratio correlation.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Double_dilution_method">Double dilution method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isotope_dilution&action=edit&section=6" title="Edit section: Double dilution method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The single dilution method requires the knowledge of the isotopic composition of the isotopically enriched analyte (<i>R</i><sub>B</sub>) and the amount of the enriched analyte added (<i>n</i><sub>B</sub>). Both of these variables are hard to establish since isotopically enriched substances are generally available in small quantities of questionable purity. As a result, before isotope dilution is performed on the sample, the amount of the enriched analyte is ascertained beforehand using isotope dilution. This preparatory step is called the <i>reverse isotope dilution</i> and it involves a standard of natural isotopic-composition analyte (denoted as A*). First proposed in the 1940s<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and further developed in the 1950s,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> reverse isotope dilution remains an effective means of characterizing a labeled material. </p> <table class="infobox" width="200px"> <tbody><tr> <th colspan="2">Isotope dilution notation </th></tr> <tr> <td></td> <td> </td></tr> <tr> <th>Name</th> <th>Symbol </th></tr> <tr> <td>Analyte</td> <td><strong>A</strong> </td></tr> <tr> <td>Natural standard</td> <td><strong>A*</strong> </td></tr> <tr> <td>Isotopic standard (Spike)</td> <td><strong>B</strong> </td></tr> <tr> <td></td> <td> </td></tr> <tr> <td>Analyte + Spike</td> <td><strong>AB</strong> </td></tr> <tr> <td>Standard + Spike</td> <td><strong>A*B</strong> </td></tr></tbody></table> <p>Reverse isotope dilution analysis of the enriched analyte: </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 n_{\mathrm {B} }=n_{\mathrm {A*} }{\frac {R_{\mathrm {A*} }-R_{\mathrm {A*B} }}{R_{\mathrm {A*B} }-R_{\mathrm {B} }}}\times {\frac {x(^{j}\mathrm {A} )_{\mathrm {A*} }}{x(^{j}\mathrm {A} )_{\mathrm {B} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mo>∗</mo> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mo>∗</mo> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mo>∗</mo> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mo>∗</mo> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mo>∗</mo> </mrow> </mrow> </msub> </mrow> <mrow> <mi>x</mi> <msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\mathrm {B} }=n_{\mathrm {A*} }{\frac {R_{\mathrm {A*} }-R_{\mathrm {A*B} }}{R_{\mathrm {A*B} }-R_{\mathrm {B} }}}\times {\frac {x(^{j}\mathrm {A} )_{\mathrm {A*} }}{x(^{j}\mathrm {A} )_{\mathrm {B} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7966cd4c18444061337fa39ab7c84d4051c4710d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.267ex; height:6.676ex;" alt="{\displaystyle n_{\mathrm {B} }=n_{\mathrm {A*} }{\frac {R_{\mathrm {A*} }-R_{\mathrm {A*B} }}{R_{\mathrm {A*B} }-R_{\mathrm {B} }}}\times {\frac {x(^{j}\mathrm {A} )_{\mathrm {A*} }}{x(^{j}\mathrm {A} )_{\mathrm {B} }}}}"></span></dd></dl> <p>Isotope dilution analysis of the analyte: </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 n_{\mathrm {A} }=n_{\mathrm {B} }{\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}\times {\frac {x(^{j}\mathrm {A} )_{\mathrm {B} }}{x(^{j}\mathrm {A} )_{\mathrm {A} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>x</mi> <msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\mathrm {A} }=n_{\mathrm {B} }{\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}\times {\frac {x(^{j}\mathrm {A} )_{\mathrm {B} }}{x(^{j}\mathrm {A} )_{\mathrm {A} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66737c109590750f6dc3dbb4cd6b9205caa515bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.979ex; height:6.676ex;" alt="{\displaystyle n_{\mathrm {A} }=n_{\mathrm {B} }{\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}\times {\frac {x(^{j}\mathrm {A} )_{\mathrm {B} }}{x(^{j}\mathrm {A} )_{\mathrm {A} }}}}"></span></dd></dl> <p>Since isotopic composition of A and A* are identical, combining these two expressions eliminates the need to measure the amount of the added enriched standard (<i>n</i><sub>B</sub>): </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 n_{\mathrm {A} }=n_{\mathrm {A*} }{\frac {R_{\mathrm {A*} }-R_{\mathrm {A*B} }}{R_{\mathrm {A*B} }-R_{\mathrm {B} }}}\times {\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mo>∗</mo> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mo>∗</mo> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mo>∗</mo> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mo>∗</mo> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\mathrm {A} }=n_{\mathrm {A*} }{\frac {R_{\mathrm {A*} }-R_{\mathrm {A*B} }}{R_{\mathrm {A*B} }-R_{\mathrm {B} }}}\times {\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e40915b665d6fd84ee097add6d92ef44ba9feb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.719ex; height:5.843ex;" alt="{\displaystyle n_{\mathrm {A} }=n_{\mathrm {A*} }{\frac {R_{\mathrm {A*} }-R_{\mathrm {A*B} }}{R_{\mathrm {A*B} }-R_{\mathrm {B} }}}\times {\frac {R_{\mathrm {B} }-R_{\mathrm {AB} }}{R_{\mathrm {AB} }-R_{\mathrm {A} }}}}"></span></dd></dl> <p>Double dilution method can be designed such that the isotopic composition of the two blends, A+B and A*+B, is identical, <i>i.e.</i>, <i>R</i><sub>AB</sub> = <i>R</i><sub>A*B</sub>. This condition of <i>exact-matching</i> double isotope dilution simplifies the above equation significantly:<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<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 n_{\mathrm {A} }=n_{\mathrm {A*} }\;(R_{\mathrm {A*B} }=R_{\mathrm {AB} }\land R_{\mathrm {A*} }=R_{\mathrm {A} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mo>∗</mo> </mrow> </mrow> </msub> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mo>∗</mo> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>∧</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mo>∗</mo> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\mathrm {A} }=n_{\mathrm {A*} }\;(R_{\mathrm {A*B} }=R_{\mathrm {AB} }\land R_{\mathrm {A*} }=R_{\mathrm {A} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f41439f4af5053ea5e114be29513c65a70472b40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.76ex; height:2.843ex;" alt="{\displaystyle n_{\mathrm {A} }=n_{\mathrm {A*} }\;(R_{\mathrm {A*B} }=R_{\mathrm {AB} }\land R_{\mathrm {A*} }=R_{\mathrm {A} })}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Triple_dilution_method">Triple dilution method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isotope_dilution&action=edit&section=7" title="Edit section: Triple dilution method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To avoid contamination of the mass spectrometer with the isotopically enriched spike, an additional blend of the primary standard (A*) and the spike (B) can be measured instead of measuring the enriched spike (B) directly. This approach was first put forward in the 1970s and developed in 2002.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Calculations_using_calibration_curve">Calculations using calibration curve</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isotope_dilution&action=edit&section=8" title="Edit section: Calculations using calibration curve"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many analysts do not employ analytical equations for isotope dilution analysis. Instead, they rely on building a calibration curve from mixtures of the natural primary standard (A*) and the isotopically enriched standard (the spike, B). Calibration curves are obtained by plotting measured isotope ratios in the prepared blends against the known ratio of the sample mass to the mass of the spike solution in each blend. Isotope dilution calibration plots sometimes show nonlinear relationships and in practice polynomial fitting is often performed to empirically describe such curves.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>When calibration plots are markedly nonlinear, one can bypass the empirical polynomial fitting and employ the ratio of two linear functions (known as <a href="/wiki/Pad%C3%A9_approximant" title="Padé approximant">Padé approximant</a>) which is shown to describe the curvature of isotope dilution curves exactly.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isotope_dilution&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Standard_addition" title="Standard addition">Standard addition</a></li> <li><a href="/wiki/Internal_standard" title="Internal standard">Internal standard</a></li> <li><a href="/wiki/Mass_spectrometry" title="Mass spectrometry">Mass spectrometry</a></li> <li><a href="/wiki/Mark_and_recapture" title="Mark and recapture">Mark and recapture</a></li> <li><a href="/wiki/Lincoln_index" title="Lincoln index">Lincoln index</a></li></ul> <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=Isotope_dilution&action=edit&section=10" title="Edit section: References"><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-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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><cite id="CITEREFM.J.T._MiltonR._I._Wielgosz2000" class="citation journal cs1">M.J.T. 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V. Hevesy; F. Paneth (1913). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1428142">"Die Löslichkeit des Bleisulfids und Bleichromats"</a>. <i><a href="/wiki/Z._Anorg._Allg._Chem." class="mw-redirect" title="Z. Anorg. Allg. Chem.">Z. Anorg. Allg. Chem.</a></i> <b>82</b> (1): <span class="nowrap">323–</span>328. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fzaac.19130820125">10.1002/zaac.19130820125</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Z.+Anorg.+Allg.+Chem.&rft.atitle=Die+L%C3%B6slichkeit+des+Bleisulfids+und+Bleichromats&rft.volume=82&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E323-%3C%2Fspan%3E328&rft.date=1913&rft_id=info%3Adoi%2F10.1002%2Fzaac.19130820125&rft.au=G.+V.+Hevesy&rft.au=F.+Paneth&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1428142&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotope+dilution" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/rittenberg-david.pdf">Isotope dilution</a> — <i><a href="/wiki/Biographical_Memoirs_of_the_National_Academy_of_Sciences" title="Biographical Memoirs of the National Academy of Sciences">Biographical Memoirs</a></i> of the <a href="/wiki/National_Academy_of_Sciences" title="National Academy of Sciences">National Academy of Sciences</a></span> </li> <li id="cite_note-Debus-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Debus_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Debus_4-1"><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="CITEREFP._J._De_BievreG._H._Debus1965" class="citation journal cs1">P. J. De Bievre; G. H. Debus (1965). "Precision mass spectrometric isotope dilution analysis". <i><a href="/wiki/Nuclear_Instruments_and_Methods" class="mw-redirect" title="Nuclear Instruments and Methods">Nucl. Instrum. Methods</a></i>. <b>32</b> (2): <span class="nowrap">224–</span>228. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1965NucIM..32..224D">1965NucIM..32..224D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0029-554X%2865%2990516-1">10.1016/0029-554X(65)90516-1</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nucl.+Instrum.+Methods&rft.atitle=Precision+mass+spectrometric+isotope+dilution+analysis&rft.volume=32&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E224-%3C%2Fspan%3E228&rft.date=1965&rft_id=info%3Adoi%2F10.1016%2F0029-554X%2865%2990516-1&rft_id=info%3Abibcode%2F1965NucIM..32..224D&rft.au=P.+J.+De+Bievre&rft.au=G.+H.+Debus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotope+dilution" 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">EPA publication SW-846, "Test Methods for Evaluating Solid Waste, Physical/Chemical Methods", available at <a rel="nofollow" class="external free" href="http://www.epa.gov/epaoswer/hazwaste/test/sw846.htm">http://www.epa.gov/epaoswer/hazwaste/test/sw846.htm</a>. See Method 6800, "Elemental and Speciated Isotope Dilution Mass Spectrometry", available at <a rel="nofollow" class="external free" href="http://www.epa.gov/epaoswer/hazwaste/test/pdfs/6800.pdf">http://www.epa.gov/epaoswer/hazwaste/test/pdfs/6800.pdf</a>.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFT._Komori1966" class="citation journal cs1">T. Komori; et al. 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Milton; J.A. Wang (2002). "High Accuracy Method for Isotope Dilution Mass Spectrometry with Application to the Measurement of Carbon Dioxide". <i><a href="/wiki/International_Journal_of_Mass_Spectrometry" title="International Journal of Mass Spectrometry">Int. J. Mass Spectrom.</a></i> <b>218</b> (1): <span class="nowrap">63–</span>73. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002IJMSp.218...63M">2002IJMSp.218...63M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS1387-3806%2802%2900663-2">10.1016/S1387-3806(02)00663-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Int.+J.+Mass+Spectrom.&rft.atitle=High+Accuracy+Method+for+Isotope+Dilution+Mass+Spectrometry+with+Application+to+the+Measurement+of+Carbon+Dioxide&rft.volume=218&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E63-%3C%2Fspan%3E73&rft.date=2002&rft_id=info%3Adoi%2F10.1016%2FS1387-3806%2802%2900663-2&rft_id=info%3Abibcode%2F2002IJMSp.218...63M&rft.au=M.J.T.+Milton&rft.au=J.A.+Wang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotope+dilution" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJ.A._JonckheereA.P._De_LeenheerH.L._Steyaert1983" class="citation journal cs1">J.A. Jonckheere; A.P. De Leenheer; H.L. Steyaert (1983). "Statistical evaluation of calibration curve nonlinearity in isotope dilution gas chromatography/mass spectrometry". <i>Anal. Chem</i>. <b>55</b>: <span class="nowrap">153–</span>155. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fac00252a042">10.1021/ac00252a042</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Anal.+Chem.&rft.atitle=Statistical+evaluation+of+calibration+curve+nonlinearity+in+isotope+dilution+gas+chromatography%2Fmass+spectrometry&rft.volume=55&rft.pages=%3Cspan+class%3D%22nowrap%22%3E153-%3C%2Fspan%3E155&rft.date=1983&rft_id=info%3Adoi%2F10.1021%2Fac00252a042&rft.au=J.A.+Jonckheere&rft.au=A.P.+De+Leenheer&rft.au=H.L.+Steyaert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotope+dilution" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaglianoMesterMeija2015" class="citation journal cs1">Pagliano, E.; Mester, Zoltan; Meija, Juris (2015). <a rel="nofollow" class="external text" href="https://nrc-publications.canada.ca/eng/view/accepted/?id=c963874f-036a-4da0-9def-eead416e46d9">"Calibration graphs in isotope dilution mass spectrometry"</a>. <i><a href="/wiki/Analytica_Chimica_Acta" title="Analytica Chimica Acta">Analytica Chimica Acta</a></i>. <b>896</b>: <span class="nowrap">63–</span>67. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015AcAC..896...63P">2015AcAC..896...63P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.aca.2015.09.020">10.1016/j.aca.2015.09.020</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26481988">26481988</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7543394">7543394</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Analytica+Chimica+Acta&rft.atitle=Calibration+graphs+in+isotope+dilution+mass+spectrometry&rft.volume=896&rft.pages=%3Cspan+class%3D%22nowrap%22%3E63-%3C%2Fspan%3E67&rft.date=2015&rft_id=info%3Adoi%2F10.1016%2Fj.aca.2015.09.020&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7543394%23id-name%3DS2CID&rft_id=info%3Apmid%2F26481988&rft_id=info%3Abibcode%2F2015AcAC..896...63P&rft.aulast=Pagliano&rft.aufirst=E.&rft.au=Mester%2C+Zoltan&rft.au=Meija%2C+Juris&rft_id=https%3A%2F%2Fnrc-publications.canada.ca%2Feng%2Fview%2Faccepted%2F%3Fid%3Dc963874f-036a-4da0-9def-eead416e46d9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotope+dilution" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isotope_dilution&action=edit&section=11" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSargent_(ed.)Harte_(ed.)Harrington_(ed.)2002" class="citation book cs1">Sargent (ed.), Mike; Harte (ed.), Rita; Harrington (ed.), Chris (2002). <a rel="nofollow" class="external text" href="http://www.rsc.org/shop/books/2002/9780854044184.asp"><i>Guidelines for Achieving High Accuracy in Isotope Dilution Mass Spectrometry (IDMS)</i></a>. Royal Society of Chemistry. p. 58. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-85404-418-4" title="Special:BookSources/978-0-85404-418-4"><bdi>978-0-85404-418-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Guidelines+for+Achieving+High+Accuracy+in+Isotope+Dilution+Mass+Spectrometry+%28IDMS%29&rft.pages=58&rft.pub=Royal+Society+of+Chemistry&rft.date=2002&rft.isbn=978-0-85404-418-4&rft.aulast=Sargent+%28ed.%29&rft.aufirst=Mike&rft.au=Harte+%28ed.%29%2C+Rita&rft.au=Harrington+%28ed.%29%2C+Chris&rft_id=http%3A%2F%2Fwww.rsc.org%2Fshop%2Fbooks%2F2002%2F9780854044184.asp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotope+dilution" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment"><code class="cs1-code">|last1=</code> has generic name (<a href="/wiki/Help:CS1_errors#generic_name" title="Help:CS1 errors">help</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGarcia-AlonsoRodriguez-Gonzalez2013" class="citation book cs1">Garcia-Alonso, J. Ignacio; Rodriguez-Gonzalez, Pablo (2013). <a rel="nofollow" class="external text" href="http://www.rsc.org/shop/books/2013/9781849733335.asp"><i>Isotope Dilution Mass Spectrometry</i></a>. Royal Society of Chemistry. p. 453. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84973-333-5" title="Special:BookSources/978-1-84973-333-5"><bdi>978-1-84973-333-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Isotope+Dilution+Mass+Spectrometry&rft.pages=453&rft.pub=Royal+Society+of+Chemistry&rft.date=2013&rft.isbn=978-1-84973-333-5&rft.aulast=Garcia-Alonso&rft.aufirst=J.+Ignacio&rft.au=Rodriguez-Gonzalez%2C+Pablo&rft_id=http%3A%2F%2Fwww.rsc.org%2Fshop%2Fbooks%2F2013%2F9781849733335.asp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotope+dilution" class="Z3988"></span></li></ul> </div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist 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spectrometer</a></li> <li><a href="/wiki/Spectrophotometry" title="Spectrophotometry">Spectrophotometer</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Techniques</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calorimetry" title="Calorimetry">Calorimetry</a></li> <li><a href="/wiki/Chromatography" title="Chromatography">Chromatography</a></li> <li><a href="/wiki/Electroanalytical_methods" title="Electroanalytical methods">Electroanalytical methods</a></li> <li><a href="/wiki/Gravimetric_analysis" title="Gravimetric analysis">Gravimetric analysis</a></li> <li><a href="/wiki/Ion_mobility_spectrometry" title="Ion mobility spectrometry">Ion mobility spectrometry</a></li> <li><a href="/wiki/Mass_spectrometry" title="Mass spectrometry">Mass spectrometry</a></li> <li><a href="/wiki/Spectroscopy" title="Spectroscopy">Spectroscopy</a></li> <li><a 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title="Separation process">Separation process</a></li> <li><a href="/wiki/Sub-sampling_(chemistry)" title="Sub-sampling (chemistry)">Sub-sampling</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Calibration</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chemometrics" title="Chemometrics">Chemometrics</a></li> <li><a href="/wiki/Calibration_curve" title="Calibration curve">Calibration curve</a></li> <li><a href="/wiki/Matrix_(chemical_analysis)" title="Matrix (chemical analysis)">Matrix effect</a></li> <li><a href="/wiki/Internal_standard" title="Internal standard">Internal standard</a></li> <li><a href="/wiki/Standard_addition" title="Standard addition">Standard addition</a></li> <li><a class="mw-selflink selflink">Isotope dilution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Prominent <a 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Isotope dilution - Wikipedia