# 14.13: Relating the Differentials of Chemical Potential and Activity


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.