{"id":44,"date":"2021-10-02T23:21:34","date_gmt":"2021-10-02T23:21:34","guid":{"rendered":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/ficks-law\/"},"modified":"2022-01-10T18:24:37","modified_gmt":"2022-01-10T18:24:37","slug":"ficks-law","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/ficks-law\/","title":{"raw":"2.4 Fick\u2019s Law","rendered":"2.4 Fick\u2019s Law"},"content":{"raw":"
\r\n

Fick\u2019s first law of diffusion (Fick, 1855) is a central feature of practically all discussions of diffusion. A variety of mathematical expressions for Fick\u2019s law that calculate different fluxes using different forms of concentration gradient are present in the literature. Not all of these expressions are equivalent to one another. We have elected in this book to carry out all of our developments in terms of molar fluxes and concentrations. In that notation, only Equation\u00a011 is referred to as Fick\u2019s law from this point forward.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle J_{i}=-DC\\frac{dx_{i}}{dl}[\/latex] , \u00a0\u00a0\u00a0\u00a0 [latex]\\displaystyle i=A,B[\/latex]<\/td>\r\n(11)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The flux calculated by Equation 11 is the equimolar flux defined by Equation 10<\/a>. The parameter D<\/em> is the effective diffusion coefficient (L2<\/sup>\/T), a modification of the molecular diffusion coefficient, D<\/em>m<\/em><\/sub>\u00a0(L2<\/sup>\/T), to account for the reduction of cross-sectional area available for gas diffusion and the increase in diffusion path length caused by the presence of solids and liquids (see Section\u00a06<\/a>). If there are no obstructions then the effective diffusion coefficient is equal to the molecular diffusion coefficient available in handbooks. We set aside for the time being any further discussion of the physical ingredients of D<\/em> and D<\/em>m<\/em><\/sub> except to note that the coefficient pertaining to diffusion of A<\/em> into B<\/em> is the same as for B<\/em> into A<\/em>. Further, kinetic theory predicts that this binary molecular diffusion coefficient is inversely proportional to the gas pressure. It is clear from Equation\u00a033<\/a> in Section\u00a06 that these characteristics of the molecular diffusion coefficient are true of the effective diffusion coefficient as well.<\/p>\r\n

Importantly, Equation\u00a011 satisfies Equation\u00a07<\/a>, a condition that is not restricted to isobaric diffusion. That is, Fick\u2019s law in this book calculates the equimolar fluxes in either an isobaric or non-isobaric binary system. Many authors assume J<\/em>i<\/em><\/sub>\u00a0=\u00a0<\/em>-<\/em>D\u00a0<\/em>dC<\/em>i<\/em><\/sub>\/dl<\/em> as the form for Fick\u2019s law (or the equivalent form on a mass basis). However, this form is not consistent with the flux definitions presented herein; in particular Equation 10<\/a> is not satisfied by this alternate form when the diffusion is influenced by a pressure gradient. We regard Equation\u00a011 as the more general form, applicable for both liquids and gases under either isobaric or non-isobaric conditions.<\/p>\r\n\r\n<\/div>","rendered":"

\n

Fick\u2019s first law of diffusion (Fick, 1855) is a central feature of practically all discussions of diffusion. A variety of mathematical expressions for Fick\u2019s law that calculate different fluxes using different forms of concentration gradient are present in the literature. Not all of these expressions are equivalent to one another. We have elected in this book to carry out all of our developments in terms of molar fluxes and concentrations. In that notation, only Equation\u00a011 is referred to as Fick\u2019s law from this point forward.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle J_{i}=-DC\frac{dx_{i}}{dl}\" , \u00a0\u00a0\u00a0\u00a0 \"\displaystyle i=A,B\"<\/td>\n(11)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The flux calculated by Equation 11 is the equimolar flux defined by Equation 10<\/a>. The parameter D<\/em> is the effective diffusion coefficient (L2<\/sup>\/T), a modification of the molecular diffusion coefficient, D<\/em>m<\/em><\/sub>\u00a0(L2<\/sup>\/T), to account for the reduction of cross-sectional area available for gas diffusion and the increase in diffusion path length caused by the presence of solids and liquids (see Section\u00a06<\/a>). If there are no obstructions then the effective diffusion coefficient is equal to the molecular diffusion coefficient available in handbooks. We set aside for the time being any further discussion of the physical ingredients of D<\/em> and D<\/em>m<\/em><\/sub> except to note that the coefficient pertaining to diffusion of A<\/em> into B<\/em> is the same as for B<\/em> into A<\/em>. Further, kinetic theory predicts that this binary molecular diffusion coefficient is inversely proportional to the gas pressure. It is clear from Equation\u00a033<\/a> in Section\u00a06 that these characteristics of the molecular diffusion coefficient are true of the effective diffusion coefficient as well.<\/p>\n

Importantly, Equation\u00a011 satisfies Equation\u00a07<\/a>, a condition that is not restricted to isobaric diffusion. That is, Fick\u2019s law in this book calculates the equimolar fluxes in either an isobaric or non-isobaric binary system. Many authors assume J<\/em>i<\/em><\/sub>\u00a0=\u00a0<\/em>–<\/em>D\u00a0<\/em>dC<\/em>i<\/em><\/sub>\/dl<\/em> as the form for Fick\u2019s law (or the equivalent form on a mass basis). However, this form is not consistent with the flux definitions presented herein; in particular Equation 10<\/a> is not satisfied by this alternate form when the diffusion is influenced by a pressure gradient. We regard Equation\u00a011 as the more general form, applicable for both liquids and gases under either isobaric or non-isobaric conditions.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":9,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-44","chapter","type-chapter","status-publish","hentry"],"part":86,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/44","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":9,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/44\/revisions"}],"predecessor-version":[{"id":398,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/44\/revisions\/398"}],"part":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/parts\/86"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/44\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/media?parent=44"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapter-type?post=44"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/contributor?post=44"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/license?post=44"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}