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Kinetic diameter

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Kinetic diameter is a measure applied to <a href="Atom" title="Atom">atoms</a> and <a href="Molecule" title="Molecule">molecules</a> that expresses the likelihood that a molecule in a gas will collide with another molecule. It is an indication of the size of the molecule as a target. The kinetic diameter is not the same as <a href="Atomic_radius" title="Atomic radius">atomic diameter</a> defined in terms of the size of the atom's <a href="Electron_shell" title="Electron shell">electron shell</a>, which is generally a lot smaller, depending on the exact definition used. Rather, it is the size of the sphere of influence that can lead to a <a href="Scattering" title="Scattering">scattering</a> event.<a href="#cite_note-1">[1]</a>

Kinetic diameter is related to the <a href="Mean_free_path" title="Mean free path">mean free path</a> of molecules in a gas. Mean free path is the average distance that a particle will travel without collision. For a fast moving particle (that is, one moving much faster than the particles it is moving through) the kinetic diameter is given by,<a href="#cite_note-Ismail14-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle d^{2}={1 \over \pi ln}}</annotation>

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</math><img src="./b7853d07bde342971374aeeda6fc06cc59941b2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.627ex; height:5.343ex;" alt="{\displaystyle d^{2}={1 \over \pi ln}}" loading="lazy"></dd>

<dd>where,</dd>

<dd>d is the kinetic diameter,</dd>

<dd>r is the kinetic radius, r = d/2,</dd>

<dd>n is the <a href="Number_density" title="Number density">number density</a> of particles</dd></dl>

However, a more usual situation is that the colliding particle being considered is indistinguishable from the population of particles in general. Here, the <a href="Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann distribution</a> of energies must be considered, which leads to the modified expression,<a href="#cite_note-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle d^{2}={1 \over {\sqrt {2}}\pi ln}}</annotation>

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</math><img src="./c3b6a1cf4bf6bde6737bc90b813a8a67052bd42e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.725ex; height:6.176ex;" alt="{\displaystyle d^{2}={1 \over {\sqrt {2}}\pi ln}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="List_of_diameters">List of diameters</h2></div>

The following table lists the kinetic diameters of some common molecules;

<th colspan="2">Molecule</th>

<th rowspan="2">Molecular mass</th>

<th rowspan="2">Kinetic diameter (<a href="Picometre" title="Picometre">pm</a>)</th>

<th rowspan="2">ref

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<th>Formula

</th></tr>

<tr>

<td><a href="Hydrogen" title="Hydrogen">Hydrogen</a></td>

</td></tr>

<tr>

<td><a href="Helium" title="Helium">Helium</a></td>

</td></tr>

<tr>

<td><a href="Methane" title="Methane">Methane</a></td>

</td></tr>

<tr>

<td><a href="Ammonia" title="Ammonia">Ammonia</a></td>

</td></tr>

<tr>

<td><a href="Water" title="Water">Water</a></td>

</td></tr>

<tr>

</td></tr>

<tr>

<td><a href="Acetylene" title="Acetylene">Acetylene</a></td>

</td></tr>

<tr>

<td><a href="Nitrogen" title="Nitrogen">Nitrogen</a></td>

</td></tr>

<tr>

<td><a href="Carbon_monoxide" title="Carbon monoxide">Carbon monoxide</a></td>

</td></tr>

<tr>

<td><a href="Ethylene" title="Ethylene">Ethylene</a></td>

</td></tr>

<tr>

<td><a href="Nitric_oxide" title="Nitric oxide">Nitric oxide</a></td>

</td></tr>

<tr>

<td><a href="Oxygen" title="Oxygen">Oxygen</a></td>

</td></tr>

<tr>

<td><a href="Hydrogen_sulfide" title="Hydrogen sulfide">Hydrogen sulfide</a></td>

</td></tr>

<tr>

<td><a href="Hydrogen_chloride" title="Hydrogen chloride">Hydrogen chloride</a></td>

</td></tr>

<tr>

<td><a href="Argon" title="Argon">Argon</a></td>

</td></tr>

<tr>

<td><a href="Propylene" title="Propylene">Propylene</a></td>

</td></tr>

<tr>

<td><a href="Carbon_dioxide" title="Carbon dioxide">Carbon dioxide</a></td>

</td></tr>

<tr>

<td><a href="Nitrous_oxide" title="Nitrous oxide">Nitrous oxide</a></td>

</td></tr>

<tr>

<td><a href="Propane" title="Propane">Propane</a></td>

</td></tr>

<tr>

<td><a href="Sulfur_dioxide" title="Sulfur dioxide">Sulfur dioxide</a></td>

</td></tr>

<tr>

<td><a href="Chlorine" title="Chlorine">Chlorine</a></td>

</td></tr>

<tr>

<td><a href="Benzene" title="Benzene">Benzene</a></td>

</td></tr>

<tr>

<td><a href="Hydrogen_bromide" title="Hydrogen bromide">Hydrogen bromide</a></td>

</td></tr>

<tr>

<td><a href="Krypton" title="Krypton">Krypton</a></td>

</td></tr>

<tr>

<td><a href="Xenon" title="Xenon">Xenon</a></td>

</td></tr>

<tr>

<td><a href="Sulfur_hexafluoride" title="Sulfur hexafluoride">Sulfur hexafluoride</a></td>

</td></tr>

<tr>

<td><a href="Carbon_tetrachloride" title="Carbon tetrachloride">Carbon tetrachloride</a></td>

</td></tr>

<tr>

<td><a href="Bromine" title="Bromine">Bromine</a></td>

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Dissimilar_particles">Dissimilar particles</h2></div>

Collisions between two dissimilar particles occur when a beam of fast particles is fired into a gas consisting of another type of particle, or two dissimilar molecules randomly collide in a gas mixture. For such cases, the above formula for scattering cross section has to be modified.

The scattering cross section, σ, in a collision between two dissimilar particles or molecules is defined by the sum of the kinetic diameters of the two particles,

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<annotation encoding="application/x-tex">{\displaystyle \sigma =\pi (r_{1}+r_{2})^{2}}</annotation>

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</math><img src="./9ec4009f55f6d575ca3042368bd93045d6305064.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.67ex; height:3.176ex;" alt="{\displaystyle \sigma =\pi (r_{1}+r_{2})^{2}}" loading="lazy"></dd>

<dd>where.</dd>

<dd>r1, r2 are, half the kinetic diameter (ie, the kinetic radii) of the two particles, respectively.</dd></dl>

We define an <a href="Intensive_quantity" class="mw-redirect" title="Intensive quantity">intensive quantity</a>, the scattering coefficient α, as the product of the gas number density and the scattering cross section,

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<annotation encoding="application/x-tex">{\displaystyle \alpha \equiv n\sigma }</annotation>

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</math><img src="./4e43ed1e3def22eb3a64015a5abd1eeecaef62aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.31ex; height:1.676ex;" alt="{\displaystyle \alpha \equiv n\sigma }" loading="lazy"></dd></dl>

The mean free path is the inverse of the scattering coefficient,

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<annotation encoding="application/x-tex">{\displaystyle l={1 \over \alpha }={1 \over \sigma n}}</annotation>

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</math><img src="./f70b6b15b47df0aa04deaaf194416d2a8face994.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.774ex; height:5.176ex;" alt="{\displaystyle l={1 \over \alpha }={1 \over \sigma n}}" loading="lazy"></dd></dl>

For similar particles, r1 = r2 and,

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<annotation encoding="application/x-tex">{\displaystyle l={1 \over \sigma n}={1 \over 4\pi r^{2}n}={1 \over \pi d^{2}n}}</annotation>

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</math><img src="./3702320bd594f572fdcda03ccf62f96b341e97eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.213ex; height:5.509ex;" alt="{\displaystyle l={1 \over \sigma n}={1 \over 4\pi r^{2}n}={1 \over \pi d^{2}n}}" loading="lazy"></dd></dl>

as before.<a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> Joos & Freeman, p. 573

</li>

<li id="cite_note-Ismail14-2">^ <a href="#cite_ref-Ismail14_2-0">a</a> <a href="#cite_ref-Ismail14_2-1">b</a> <a href="#cite_ref-Ismail14_2-2">c</a> <a href="#cite_ref-Ismail14_2-3">d</a> <a href="#cite_ref-Ismail14_2-4">e</a> <a href="#cite_ref-Ismail14_2-5">f</a> <a href="#cite_ref-Ismail14_2-6">g</a> Ismail et al., p. 14

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Freude, p. 4

</li>

<li id="cite_note-Matteucci6-4">^ <a href="#cite_ref-Matteucci6_4-0">a</a> <a href="#cite_ref-Matteucci6_4-1">b</a> <a href="#cite_ref-Matteucci6_4-2">c</a> <a href="#cite_ref-Matteucci6_4-3">d</a> <a href="#cite_ref-Matteucci6_4-4">e</a> <a href="#cite_ref-Matteucci6_4-5">f</a> <a href="#cite_ref-Matteucci6_4-6">g</a> <a href="#cite_ref-Matteucci6_4-7">h</a> Matteucci et al., p. 6

</li>

<li id="cite_note-Breck-5">^ <a href="#cite_ref-Breck_5-0">a</a> <a href="#cite_ref-Breck_5-1">b</a> <a href="#cite_ref-Breck_5-2">c</a> <a href="#cite_ref-Breck_5-3">d</a> <a href="#cite_ref-Breck_5-4">e</a> <a href="#cite_ref-Breck_5-5">f</a> <a href="#cite_ref-Breck_5-6">g</a> <a href="#cite_ref-Breck_5-7">h</a> <a href="#cite_ref-Breck_5-8">i</a> <a href="#cite_ref-Breck_5-9">j</a> <a href="#cite_ref-Breck_5-10">k</a> <a href="#cite_ref-Breck_5-11">l</a> <a href="#cite_ref-Breck_5-12">m</a> Breck

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> Li & Talu, p. 373

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> Freude, pp. 3-4

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</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li>Breck, Donald W., "Zeolite Molecular Sieves: Structure, Chemistry, and Use", New York: Wiley, 1974 <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0471099856</bdi>.</li>

<li>Freude, D., Molecular Physics, <a rel="nofollow" class="external text" href="http://home.uni-leipzig.de/energy/pdf/freume2.pdf">chapter 2</a>, 2004 unpublished draft, retrieved and <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160909124658/http://home.uni-leipzig.de/energy/pdf/freume2.pdf">archived</a> 18 October 2015.</li>

<li>Ismail, Ahmad Fauzi; Khulbe, Kailash; Matsuura, Takeshi, Gas Separation Membranes: Polymeric and Inorganic, Springer, 2015 <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3319010956</bdi>.</li>

<li>Joos, Georg; Freeman, Ira Maximilian, Theoretical Physics, Courier Corporation, 1958 <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0486652270</bdi>.</li>

<li>Li, Jian-Min; Talu, Orhan, "Effect of structural heterogeneity on multicomponent adsorption: benzene and p-xylene mixture on silicalite", in Suzuki, Motoyuki (ed), Fundamentals of Adsorption, pp. 373-380, Elsevier, 1993 <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0080887724</bdi>.</li>

<li>Matteucci, Scott; Yampolskii, Yuri; Freeman, Benny D.; Pinnau, Ingo, "Transport of gases and vapors in glassy and rubbery polymers" in, Yampolskii, Yuri; Freeman, Benny D.; Pinnau, Ingo, Materials Science of Membranes for Gas and Vapor Separation, pp. 1-47, John Wiley & Sons, 2006 <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0470029048</bdi>. ئئ</li></ul></div><div><div class="zim-footer">

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