Diffusion-controlled reaction

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width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Diffusion-controlled_reaction" title="Special:EditPage/Diffusion-controlled reaction">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Diffusion-controlled+reaction%22">"Diffusion-controlled reaction"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Diffusion-controlled+reaction%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Diffusion-controlled+reaction%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Diffusion-controlled+reaction%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Diffusion-controlled+reaction%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Diffusion-controlled+reaction%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">January 2020</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><b>Diffusion-controlled</b> (or diffusion-limited) <a href="/wiki/Chemical_reaction" title="Chemical reaction">reactions</a> are reactions in which the reaction rate is equal to the rate of transport of the reactants through the reaction medium (usually a solution).<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> The process of chemical reaction can be considered as involving the diffusion of reactants until they encounter each other in the right stoichiometry and form an activated complex which can form the product species. The observed rate of chemical reactions is, generally speaking, the rate of the slowest or "rate determining" step. In diffusion controlled reactions the formation of products from the <a href="/wiki/Activated_complex" title="Activated complex">activated complex</a> is much faster than the diffusion of reactants and thus the rate is governed by <a href="/wiki/Collision_frequency" title="Collision frequency">collision frequency</a>. </p><p>Diffusion control is rare in the gas phase, where rates of <a href="/wiki/Diffusion" title="Diffusion">diffusion</a> of molecules are generally very high. Diffusion control is more likely in solution where diffusion of reactants is slower due to the greater number of collisions with solvent molecules. Reactions where the <a href="/wiki/Activated_complex" title="Activated complex">activated complex</a> forms easily and the products form rapidly are most likely to be limited by diffusion control. Examples are those involving <a href="/wiki/Catalysis" title="Catalysis">catalysis</a> and <a href="/wiki/Enzyme" title="Enzyme">enzymatic</a> reactions. <a href="/wiki/Homogeneous_and_heterogeneous_reactions" class="mw-redirect" title="Homogeneous and heterogeneous reactions">Heterogeneous reactions</a> where reactants are in different phases are also candidates for diffusion control. </p><p>One classical test for diffusion control of a heterogeneous reaction is to observe whether the rate of reaction is affected by stirring or agitation; if so then the reaction is almost certainly diffusion controlled under those conditions. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Derivation">Derivation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion-controlled_reaction&action=edit&section=1" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following derivation is adapted from <i>Foundations of Chemical Kinetics</i>.<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> This derivation assumes the reaction <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 A+B\rightarrow C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+B\rightarrow C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3624c88b316f4df8fb9ddf959df68e998541b3a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.728ex; height:2.343ex;" alt="{\displaystyle A+B\rightarrow C}"></span>. Consider a sphere of radius <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_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b096f1c60d7fdc543f3bc583fe32601f1c2f0cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.229ex; height:2.509ex;" alt="{\displaystyle R_{A}}"></span>, centered at a spherical molecule A, with reactant B flowing in and out of it. A reaction is considered to occur if molecules A and B touch, that is, when the distance between the two molecules is <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_{AB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa22b9617fe3558b77fc9e242bb5e851f495c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.476ex; height:2.509ex;" alt="{\displaystyle R_{AB}}"></span> apart. </p><p>If we assume a local steady state, then the rate at which B reaches <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_{AB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa22b9617fe3558b77fc9e242bb5e851f495c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.476ex; height:2.509ex;" alt="{\displaystyle R_{AB}}"></span> is the limiting factor and balances the reaction. </p><p>Therefore, the steady state condition becomes </p><p><b>1.</b> <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 k[B]=-4\pi r^{2}J_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo>−</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k[B]=-4\pi r^{2}J_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7be18a656772541afa906b238a42cb70907838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.543ex; height:3.176ex;" alt="{\displaystyle k[B]=-4\pi r^{2}J_{B}}"></span> </p><p>where </p><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 J_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b58f5463d5483d92900f34f76dad9a1d27f4454b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.77ex; height:2.509ex;" alt="{\displaystyle J_{B}}"></span> is the flux of B, as given by <a href="/wiki/Fick%27s_law_of_diffusion" class="mw-redirect" title="Fick's law of diffusion">Fick's law of diffusion</a>, </p><p><b>2. </b><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 J_{B}=-D_{AB}({\frac {dB(r)}{dr}}+{\frac {[B]}{k_{B}T}}{\frac {dU}{dr}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>U</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{B}=-D_{AB}({\frac {dB(r)}{dr}}+{\frac {[B]}{k_{B}T}}{\frac {dU}{dr}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbd96d640797fccc87c696f7ff5586b293700293" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.634ex; height:6.176ex;" alt="{\displaystyle J_{B}=-D_{AB}({\frac {dB(r)}{dr}}+{\frac {[B]}{k_{B}T}}{\frac {dU}{dr}})}"></span>, </p><p>where <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 D_{AB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{AB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71f1d6ad0071554a7c23f9cd15831136cafdd143" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.636ex; height:2.509ex;" alt="{\displaystyle D_{AB}}"></span> is the diffusion coefficient and can be obtained by the <a href="/wiki/Stokes-Einstein_equation" class="mw-redirect" title="Stokes-Einstein equation">Stokes-Einstein equation</a>, and the second term is the gradient of the chemical potential with respect to position. Note that [B] refers to the average concentration of B in the solution, while [B](r) is the "local concentration" of B at position r. </p><p><br /> Inserting <b>2 </b> into <b>1</b> results in </p><p><b>3.</b> <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 k[B]=4\pi r^{2}D_{AB}({\frac {dB(r)}{dr}}+{\frac {[B](r)}{k_{B}T}}{\frac {dU}{dr}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>U</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k[B]=4\pi r^{2}D_{AB}({\frac {dB(r)}{dr}}+{\frac {[B](r)}{k_{B}T}}{\frac {dU}{dr}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df8d075b645d6254bc14da6413a3d7a8f8584e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.511ex; height:6.176ex;" alt="{\displaystyle k[B]=4\pi r^{2}D_{AB}({\frac {dB(r)}{dr}}+{\frac {[B](r)}{k_{B}T}}{\frac {dU}{dr}})}"></span>. </p><p>It is convenient at this point to use the identity <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 \exp(-U(r)/k_{B}T)\cdot {\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)=({\frac {dB(r)}{dr}}+{\frac {[B](r)}{k_{B}T}}{\frac {dU}{dr}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>U</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(-U(r)/k_{B}T)\cdot {\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)=({\frac {dB(r)}{dr}}+{\frac {[B](r)}{k_{B}T}}{\frac {dU}{dr}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36e510b669c593a9202c452879b8b179ba9968b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:69.787ex; height:6.176ex;" alt="{\displaystyle \exp(-U(r)/k_{B}T)\cdot {\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)=({\frac {dB(r)}{dr}}+{\frac {[B](r)}{k_{B}T}}{\frac {dU}{dr}})}"></span> allowing us to rewrite <b>3</b> as </p><p><br /> <b>4. </b><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 k[B]=4\pi r^{2}D_{AB}\exp(-U(r)/k_{B}T)\cdot {\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k[B]=4\pi r^{2}D_{AB}\exp(-U(r)/k_{B}T)\cdot {\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0458e569ab5726427dc277cb6ba67b20ae9de6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:61.767ex; height:5.509ex;" alt="{\displaystyle k[B]=4\pi r^{2}D_{AB}\exp(-U(r)/k_{B}T)\cdot {\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)}"></span>. </p><p>Rearranging <b>4</b> allows us to write </p><p><b>5. </b><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 {\frac {k[B]\exp(U(r)/k_{B}T)}{4\pi r^{2}D_{AB}}}={\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {k[B]\exp(U(r)/k_{B}T)}{4\pi r^{2}D_{AB}}}={\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8da22ef05a62efd4008b56592bccb4011108e68e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.882ex; height:6.343ex;" alt="{\displaystyle {\frac {k[B]\exp(U(r)/k_{B}T)}{4\pi r^{2}D_{AB}}}={\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)}"></span> </p><p>Using the boundary conditions that <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 [B](r)\rightarrow [B]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [B](r)\rightarrow [B]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae7bef7e16f193b7ecfac86a612e4617eaf3f6f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.587ex; height:2.843ex;" alt="{\displaystyle [B](r)\rightarrow [B]}"></span>, ie the local concentration of B approaches that of the solution at large distances, and consequently <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(r)\rightarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(r)\rightarrow 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55949dc1c7d26637a2fce36b63bd06ad19b2ddbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.417ex; height:2.843ex;" alt="{\displaystyle U(r)\rightarrow 0}"></span>, as <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\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/170e3065b668161cd003c9fd57119e51d3424ce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.986ex; height:1.843ex;" alt="{\displaystyle r\rightarrow \infty }"></span>, we can solve <b>5</b> by separation of variables, we get </p><p><b>6. </b><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 \int _{R_{AB}}^{\infty }dr{\frac {k[B]\exp(U(r)/k_{B}T)}{4\pi r^{2}D_{AB}}}=\int _{R_{AB}}^{\infty }d([B](r)\exp(U(r)/k_{B}T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>d</mi> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>d</mi> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{R_{AB}}^{\infty }dr{\frac {k[B]\exp(U(r)/k_{B}T)}{4\pi r^{2}D_{AB}}}=\int _{R_{AB}}^{\infty }d([B](r)\exp(U(r)/k_{B}T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea37c85df2e5b84ce1a94fba2eb04df6e9effe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:60.936ex; height:6.509ex;" alt="{\displaystyle \int _{R_{AB}}^{\infty }dr{\frac {k[B]\exp(U(r)/k_{B}T)}{4\pi r^{2}D_{AB}}}=\int _{R_{AB}}^{\infty }d([B](r)\exp(U(r)/k_{B}T)}"></span> or </p><p><b>7. </b><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 {\frac {k[B]}{4\pi D_{AB}\beta }}=[B]-[B](R_{AB})\exp(U(R_{AB})/k_{B}T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo>−</mo> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {k[B]}{4\pi D_{AB}\beta }}=[B]-[B](R_{AB})\exp(U(R_{AB})/k_{B}T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acd8b6d7ac45dfd9ccb4b295a57ee74d8cddc507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.945ex; height:6.176ex;" alt="{\displaystyle {\frac {k[B]}{4\pi D_{AB}\beta }}=[B]-[B](R_{AB})\exp(U(R_{AB})/k_{B}T)}"></span> (where : <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 \beta ^{-1}=\int _{R_{AB}}^{\infty }{\frac {1}{r^{2}}}\exp({\frac {U(r)}{k_{B}T}}dr)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{-1}=\int _{R_{AB}}^{\infty }{\frac {1}{r^{2}}}\exp({\frac {U(r)}{k_{B}T}}dr)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39ed7f3cad4bdd02065435adc1e4af526d7c81c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.534ex; height:6.509ex;" alt="{\displaystyle \beta ^{-1}=\int _{R_{AB}}^{\infty }{\frac {1}{r^{2}}}\exp({\frac {U(r)}{k_{B}T}}dr)}"></span>) </p><p>For the reaction between A and B, there is an inherent reaction constant <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 k_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7e5bb94a98612867e7ad219e44b5499bb7ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.185ex; height:2.509ex;" alt="{\displaystyle k_{r}}"></span>, so <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 [B](R_{AB})=k[B]/k_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [B](R_{AB})=k[B]/k_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dab7b195bc9b7585cb38a0219a0cfb73dcaaa001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.058ex; height:2.843ex;" alt="{\displaystyle [B](R_{AB})=k[B]/k_{r}}"></span>. Substituting this into <b>7</b> and rearranging yields </p><p><b>8.</b> <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 k={\frac {4\pi D_{AB}\beta k_{r}}{k_{r}+4\pi D_{AB}\beta \exp({\frac {U(R_{AB})}{k_{B}T}})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mi>β</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mi>β</mi> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {4\pi D_{AB}\beta k_{r}}{k_{r}+4\pi D_{AB}\beta \exp({\frac {U(R_{AB})}{k_{B}T}})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a43a926df29bdf1c56c4adf1e4daed39f70749b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:31.184ex; height:7.843ex;" alt="{\displaystyle k={\frac {4\pi D_{AB}\beta k_{r}}{k_{r}+4\pi D_{AB}\beta \exp({\frac {U(R_{AB})}{k_{B}T}})}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Limiting_conditions">Limiting conditions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion-controlled_reaction&action=edit&section=2" title="Edit section: Limiting conditions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Very_fast_intrinsic_reaction">Very fast intrinsic reaction</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion-controlled_reaction&action=edit&section=3" title="Edit section: Very fast intrinsic reaction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose <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 k_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7e5bb94a98612867e7ad219e44b5499bb7ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.185ex; height:2.509ex;" alt="{\displaystyle k_{r}}"></span> is very large compared to the diffusion process, so A and B react immediately. This is the classic diffusion limited reaction, and the corresponding diffusion limited rate constant, can be obtained from <b>8</b> as <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 k_{D}=4\pi D_{AB}\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{D}=4\pi D_{AB}\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/896c61dc700bac0b9be508658dfbfa5f52aed1cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.366ex; height:2.509ex;" alt="{\displaystyle k_{D}=4\pi D_{AB}\beta }"></span>. <b>8</b> can then be re-written as the "diffusion influenced rate constant" as </p><p><b>9.</b> <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 k={\frac {k_{D}k_{r}}{k_{r}+k_{D}\exp({\frac {U(R_{AB})}{k_{B}T}})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {k_{D}k_{r}}{k_{r}+k_{D}\exp({\frac {U(R_{AB})}{k_{B}T}})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c299318f9b7f5c855a2aa93a7bde872b2125a06e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:25.525ex; height:7.843ex;" alt="{\displaystyle k={\frac {k_{D}k_{r}}{k_{r}+k_{D}\exp({\frac {U(R_{AB})}{k_{B}T}})}}}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Weak_intermolecular_forces">Weak intermolecular forces</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion-controlled_reaction&action=edit&section=4" title="Edit section: Weak intermolecular forces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the forces that bind A and B together are weak, ie <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(r)\approx 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(r)\approx 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7167c6dca963e60a7af687df0526778de684a273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.901ex; height:2.843ex;" alt="{\displaystyle U(r)\approx 0}"></span> for all r except very small r, <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 \beta ^{-1}\approx {\frac {1}{R_{AB}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{-1}\approx {\frac {1}{R_{AB}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ff1cf48413810b37950e7c187f432a1abc4f3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.081ex; height:5.676ex;" alt="{\displaystyle \beta ^{-1}\approx {\frac {1}{R_{AB}}}}"></span>. The reaction rate <b>9</b> simplifies even further to </p><p><b>10. </b> <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 k={\frac {k_{D}k_{r}}{k_{r}+k_{D}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {k_{D}k_{r}}{k_{r}+k_{D}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e437e00fca9edc89151cb00a12ef469d04723ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.975ex; height:5.843ex;" alt="{\displaystyle k={\frac {k_{D}k_{r}}{k_{r}+k_{D}}}}"></span> This equation is true for a very large proportion of industrially relevant reactions in solution. </p> <div class="mw-heading mw-heading3"><h3 id="Viscosity_dependence">Viscosity dependence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion-controlled_reaction&action=edit&section=5" title="Edit section: Viscosity dependence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Stokes-Einstein" class="mw-redirect" title="Stokes-Einstein">Stokes-Einstein</a> equation describes a frictional force on a sphere of diameter <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_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b096f1c60d7fdc543f3bc583fe32601f1c2f0cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.229ex; height:2.509ex;" alt="{\displaystyle R_{A}}"></span> as <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 D_{A}={\frac {k_{B}T}{3\pi R_{A}\eta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> <mrow> <mn>3</mn> <mi>π</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>η</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{A}={\frac {k_{B}T}{3\pi R_{A}\eta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0e03c98be6d1ec5bd597905867162091ed7357a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.216ex; height:5.843ex;" alt="{\displaystyle D_{A}={\frac {k_{B}T}{3\pi R_{A}\eta }}}"></span> where <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 \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span> is the viscosity of the solution. Inserting this into <b>9</b> gives an estimate for <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 k_{D}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{D}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/162142306eaa8b5b78330b50a0a1569a05a63ee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.804ex; height:2.509ex;" alt="{\displaystyle k_{D}}"></span> as <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 {\frac {8RT}{3\eta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>R</mi> <mi>T</mi> </mrow> <mrow> <mn>3</mn> <mi>η</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {8RT}{3\eta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b47b2dea8f3acd52d679871b46f13b356489bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:5.399ex; height:5.676ex;" alt="{\displaystyle {\frac {8RT}{3\eta }}}"></span>, where R is the gas constant, and <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 \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span> is given in centipoise. For the following molecules, an estimate for <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 k_{D}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{D}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/162142306eaa8b5b78330b50a0a1569a05a63ee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.804ex; height:2.509ex;" alt="{\displaystyle k_{D}}"></span> is given: </p> <table class="wikitable sortable"> <caption>Solvents and <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 k_{D}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{D}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/162142306eaa8b5b78330b50a0a1569a05a63ee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.804ex; height:2.509ex;" alt="{\displaystyle k_{D}}"></span> </caption> <tbody><tr> <th>Solvent</th> <th>Viscosity (<a href="/wiki/Centipoise" class="mw-redirect" title="Centipoise">centipoise</a>)</th> <th><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 k_{D}({\frac {\times 1e9}{M\cdot s}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>×</mo> <mn>1</mn> <mi>e</mi> <mn>9</mn> </mrow> <mrow> <mi>M</mi> <mo>⋅</mo> <mi>s</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{D}({\frac {\times 1e9}{M\cdot s}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8560d425a865428133a2649d378240d3c7ed3786" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.666ex; height:5.176ex;" alt="{\displaystyle k_{D}({\frac {\times 1e9}{M\cdot s}})}"></span> </th></tr> <tr> <td><a href="/wiki/N-Pentane" class="mw-redirect" title="N-Pentane">n-Pentane</a></td> <td>0.24</td> <td>27 </td></tr> <tr> <td>Hexadecane</td> <td>3.34</td> <td>1.9 </td></tr> <tr> <td><a href="/wiki/Methanol" title="Methanol">Methanol</a></td> <td>0.55</td> <td>11.8 </td></tr> <tr> <td><a href="/wiki/Water" title="Water">Water</a></td> <td>0.89</td> <td>7.42 </td></tr> <tr> <td><a href="/wiki/Toluene" title="Toluene">Toluene</a></td> <td>0.59</td> <td>11 </td></tr></tbody></table><p><sup id="cite_ref-NIST_3-0" class="reference"><a href="#cite_note-NIST-3"><span class="cite-bracket">[</span>3<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=Diffusion-controlled_reaction&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Diffusion_limited_enzyme" class="mw-redirect" title="Diffusion limited enzyme">Diffusion limited enzyme</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=Diffusion-controlled_reaction&action=edit&section=7" 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"> <div class="mw-references-wrap"><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="CITEREFAtkins1998" class="citation book cs1">Atkins, Peter (1998). <i>Physical Chemistry</i> (6th ed.). New York: Freeman. pp. <span class="nowrap">825–</span>8.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+Chemistry&rft.place=New+York&rft.pages=%3Cspan+class%3D%22nowrap%22%3E825-%3C%2Fspan%3E8&rft.edition=6th&rft.pub=Freeman&rft.date=1998&rft.aulast=Atkins&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffusion-controlled+reaction" class="Z3988"></span></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="CITEREFRoussel" class="citation web cs1">Roussel, Marc R. <a rel="nofollow" class="external text" href="http://people.uleth.ca/~roussel/C4000foundations/slides/28diffusion_influencedI.pdf">"Lecture 28:Diffusion-influenced reactions, Part I"</a> <span class="cs1-format">(PDF)</span>. <i>Foundations of Chemical Kinetics</i>. University of Lethbridge (Canada)<span class="reference-accessdate">. Retrieved <span class="nowrap">19 February</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Foundations+of+Chemical+Kinetics&rft.atitle=Lecture+28%3ADiffusion-influenced+reactions%2C+Part+I&rft.aulast=Roussel&rft.aufirst=Marc+R.&rft_id=http%3A%2F%2Fpeople.uleth.ca%2F~roussel%2FC4000foundations%2Fslides%2F28diffusion_influencedI.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffusion-controlled+reaction" class="Z3988"></span></span> </li> <li id="cite_note-NIST-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-NIST_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerg" class="citation book cs1">Berg, Howard, C. <i>Random Walks in Biology</i>. pp. <span class="nowrap">145–</span>148.</cite><span 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href="/wiki/SN1_reaction" title="SN1 reaction">Unimolecular nucleophilic substitution</a> (S<sub>N</sub>1)</li> <li><a href="/wiki/SN2_reaction" title="SN2 reaction">Bimolecular nucleophilic substitution</a> (S<sub>N</sub>2)</li> <li><a href="/wiki/Nucleophilic_aromatic_substitution" title="Nucleophilic aromatic substitution">Nucleophilic aromatic substitution</a> (S<sub>N</sub>Ar)</li> <li><a href="/wiki/SNi" title="SNi">Nucleophilic internal substitution</a> (S<sub>N</sub>i)</li> <li><a href="/wiki/Nucleophilic_acyl_substitution" class="mw-redirect" title="Nucleophilic acyl substitution">Nucleophilic acyl substitution</a> (S<sub>N</sub>Acyl)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Electrophilic_substitution" title="Electrophilic substitution">Electrophilic substitutions</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Electrophilic_aromatic_substitution" title="Electrophilic aromatic substitution">Electrophilic aromatic substitution</a> (S<sub>E</sub>Ar)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Elimination_reaction" title="Elimination reaction">Elimination reactions</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/E1_elimination" class="mw-redirect" title="E1 elimination">Unimolecular elimination</a> (E1)</li> <li><a href="/wiki/E1cB-elimination_reaction" title="E1cB-elimination reaction">E1cB-elimination</a></li> <li><a href="/wiki/E2_elimination" class="mw-redirect" title="E2 elimination">Bimolecular elimination</a> (E2)</li> <li><a href="/wiki/Ei_mechanism" title="Ei mechanism">E<sub>i</sub> elimination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Addition_reaction" title="Addition reaction">Addition reactions</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Electrophilic_addition" title="Electrophilic addition">Electrophilic addition</a> (A<sub>E</sub>)</li> <li><a href="/wiki/Nucleophilic_addition" title="Nucleophilic addition">Nucleophilic addition</a> (A<sub>N</sub>)</li> <li><a href="/wiki/Free-radical_addition" title="Free-radical addition">Free-radical addition</a></li> <li><a href="/wiki/Cycloaddition" title="Cycloaddition">Cycloaddition</a></li> <li><a href="/wiki/Oxidative_addition" title="Oxidative addition">Oxidative addition</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Unimolecular reactions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Intramolecular_reaction" title="Intramolecular reaction">Intramolecular reaction</a></li> <li><a href="/wiki/Isomerization" title="Isomerization">Isomerization</a></li> <li><a href="/wiki/Photodissociation" title="Photodissociation">Photodissociation</a></li> <li><a href="/wiki/Lindemann%E2%80%93Hinshelwood_mechanism" class="mw-redirect" title="Lindemann–Hinshelwood mechanism">Lindemann–Hinshelwood mechanism</a></li> <li><a href="/wiki/RRKM_theory" title="RRKM theory">RRKM theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Electron_transfer" title="Electron transfer">Electron/Proton transfer</a> reactions</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Redox" title="Redox">Redox</a></li> <li><a href="/wiki/Harpoon_reaction" title="Harpoon reaction">Harpoon reaction</a></li> <li><a href="/wiki/Grotthuss_mechanism" title="Grotthuss mechanism">Grotthuss mechanism</a></li> <li><a href="/wiki/Marcus_theory" title="Marcus theory">Marcus theory</a></li> <li><a href="/wiki/Inner_sphere_electron_transfer" title="Inner sphere electron transfer">Inner sphere electron transfer</a></li> <li><a href="/wiki/Outer_sphere_electron_transfer" title="Outer sphere electron transfer">Outer sphere electron transfer</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Medium effects</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Solvent_effects" title="Solvent effects">Solvent effects</a></li> <li><a href="/wiki/Cage_effect" title="Cage effect">Cage effect</a></li> <li><a href="/wiki/Matrix_isolation" title="Matrix isolation">Matrix isolation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Elementary_reaction" title="Elementary reaction">Elementary reaction</a></li> <li><a href="/wiki/Reaction_dynamics" title="Reaction dynamics">Reaction dynamics</a></li> <li><a href="/wiki/Reactive_intermediate" title="Reactive intermediate">Reactive intermediate</a></li> <li><a href="/wiki/Radical_(chemistry)" title="Radical (chemistry)">Radical (chemistry)</a></li> <li><a href="/wiki/Molecularity" title="Molecularity">Molecularity</a></li> <li><a href="/wiki/Stereochemistry" title="Stereochemistry">Stereochemistry</a></li> <li><a href="/wiki/Catalysis" title="Catalysis">Catalysis</a></li> <li><a href="/wiki/Collision_theory" title="Collision theory">Collision theory</a></li> <li><a href="/wiki/Arrow_pushing" title="Arrow pushing">Arrow pushing</a></li> <li><a href="/wiki/Potential_energy_surface" title="Potential energy surface">Potential energy surface</a></li> <li><a href="/wiki/More_O%27Ferrall%E2%80%93Jencks_plot" title="More O'Ferrall–Jencks plot">More O'Ferrall–Jencks plot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Chemical_kinetics" title="Chemical kinetics">Chemical kinetics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Rate_equation" title="Rate equation">Rate equation</a></li> <li><a href="/wiki/Equilibrium_constant" title="Equilibrium constant">Equilibrium constant</a></li> <li><a href="/wiki/Rate-determining_step" title="Rate-determining step">Rate-determining step</a></li> <li><a href="/wiki/Reaction_coordinate" title="Reaction coordinate">Reaction coordinate</a></li> <li><a href="/wiki/Energy_profile_(chemistry)" title="Energy profile (chemistry)">Energy profile (chemistry)</a></li> <li><a href="/wiki/Transition_state_theory" title="Transition state theory">Transition state theory</a></li> <li><a href="/wiki/Activation_energy" title="Activation energy">Activation energy</a></li> <li><a href="/wiki/Activated_complex" title="Activated complex">Activated complex</a></li> <li><a href="/wiki/Arrhenius_equation" title="Arrhenius equation">Arrhenius equation</a></li> <li><a href="/wiki/Eyring_equation" title="Eyring equation">Eyring equation</a></li> <li><a href="/wiki/Michaelis%E2%80%93Menten_kinetics" title="Michaelis–Menten kinetics">Michaelis–Menten kinetics</a></li> <li><a class="mw-selflink selflink">Diffusion-controlled reaction</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" 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Diffusion-controlled reaction - Wikipedia