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14.13: Relating the Differentials of Chemical Potential and Activity

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    Let us write \({\left(d{\mu }_A\right)}_{PT}\) to represent the differential of \({\mu }_A\) at constant pressure and temperature. From the general expression for \(d{\mu }_A\) and the definition of activity, we can write the total differential of the chemical potential of substance \(A\) in a particular system in several equivalent ways

    \[\begin{aligned} d{\mu }_A & = \left(\frac{\partial {\mu }_A}{\partial P}\right)_TdP+ \left(\frac{\partial {\mu }_A}{\partial T}\right)_PdT+ \left(d{\mu }_A\right)_{PT} \\ ~ & = \left(\frac{\partial {\mu }_A}{\partial P}\right)_TdP+ \left(\frac{\partial {\mu }_A}{\partial T}\right)_PdT+\sum^{\omega }_{j=1} \left(\frac{\partial {\mu }_A}{\partial n_j}\right)_{PT}dn_j \\ ~ & =\overline{V}_AdP-\overline{S}_AdT+RT \left(d \ln \tilde{a}_A \right)_{PT} \\ ~ & =RT \left(\frac{\partial \ln \tilde{a}_A}{\partial P}\right)_TdP+ \left(\frac{\partial {\mu }_A}{\partial T}\right)_PdT + RT \left(d \ln \tilde{a}_A \right)_{PT} \end{aligned}\]

    In short, we have developed several alternative notations for the same physical quantities. From the dependence of chemical potential on pressure, and because \({\widetilde{\mu }}^o_A\) is not a function of pressure, we have a very useful relationship:

    \[{\left(\frac{\partial {\mu }_A}{\partial P}\right)}_T=RT{\left(\frac{\partial { \ln {\tilde{a}}_A\ }}{\partial P}\right)}_T={\overline{V}}_A\]

    From the definition of activity and the dependence of chemical potential on temperature, we have: \[{\left(\frac{\partial {\mu }_A}{\partial T}\right)}_P={\left(\frac{\partial {\widetilde{\mu }}^o_A}{\partial T}\right)}_P+R{ \ln {\tilde{a}}_A\ }+RT{\left(\frac{\partial { \ln {\tilde{a}}_A\ }}{\partial T}\right)}_P=-{\overline{S}}_A\] From the dependence of chemical potential on the composition of the system, we have

    \[\sum^{\omega }_{j=1}{{\left(\frac{\partial {\mu }_A}{\partial n_j}\right)}_{PT}dn_j}={\left(d{\mu }_A\right)}_{PT}=RT{\left(d{ \ln {\tilde{a}}_A\ }\right)}_{PT}\]

    This last equation shows explicitly that the activity of component \(A\) depends on all of the species present. The effects of interactions between \(A\) molecules and \(B\) molecules are represented in this sum by the term \({\left({\partial {\mu }_A}/{\partial n_B}\right)}_{PT}\). When the effects of intermolecular interactions on the chemical potential are independent of the component concentrations, \({\left({\partial {\mu }_A}/{\partial n_B}\right)}_{PT}=0\), and the only surviving term is \({\left({\partial {\mu }_A}/{\partial n_A}\right)}_{PT}\). If the interactions between \(A\) molecules and the rest of the system are constant over a range of concentrations of \(A\), \({\gamma }_A\) is constant over this range.

    14.13: Relating the Differentials of Chemical Potential and Activity is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.