In the previous module, we saw how the chemical potential changes with pressure. However, the most important application of chemical potential is understanding how it changes with composition.
Consider a pure substance A at temperature \(T\) and pressure \(p\). Its chemical potential is
\[ \mu_A^\circ \]
where the superscript "\(^{\circ}\)" indicates the pure substance at the specified temperature and pressure.
Now suppose that A is mixed with another substance B. How does the chemical potential of A change?
For an ideal mixture,
\[ \Delta G_{\mathrm{mix}} = RT \sum_i n_i\ln x_i \]
where \(x_i\) is the mole fraction of component \(i\).
The total Gibbs function of the mixture is therefore
\[ G = \sum_i n_i\mu_i^\circ + RT \sum_i n_i\ln x_i \]
The first term represents the Gibbs function of the unmixed pure substances. The second term is the contribution arising from mixing.
The chemical potential is defined as the partial molar Gibbs function:
\[ \mu_A = \left( \frac{\partial G} {\partial n_A} \right)_{T,p,n_B} \]
Applying this derivative to the expression for the Gibbs function of an ideal mixture yields
\[ \boxed{ \mu_A = \mu_A^\circ + RT\ln x_A } \]
and similarly
\[ \boxed{ \mu_B = \mu_B^\circ + RT\ln x_B } \]
These equations describe the composition dependence of the chemical potentials of the components of an ideal mixture.
Since the mole fraction must satisfy
\[ 0 < x_i \le 1 \]
it follows that
\[ \ln(x_i)\le 0 \]
Therefore,
\[ \mu_i \le \mu_i^\circ \]
for every component in an ideal mixture.
Mixing always lowers the chemical potential of each component.
This result explains why mixing is spontaneous. The system lowers its total Gibbs function by lowering the chemical potentials of the components.
If component A is pure,
\[ x_A = 1 \]
and therefore
\[ \ln(1)=0 \]
giving
\[ \mu_A = \mu_A^\circ \]
as expected.
As the mixture becomes more dilute in A,
\[ x_A \rightarrow 0 \]
and
\[ \mu_A \rightarrow -\infty \]
according to the ideal-mixture model.
Combining the pressure dependence of an ideal gas with the composition dependence of an ideal mixture gives
\[ \boxed{ \mu_i = \mu_i^\circ + RT\ln \left( \frac{p_i}{p^\circ} \right) } \]
where
\[ p_i = x_i p \]
is the partial pressure of component \(i\).
This equation is one of the most important relationships in the thermodynamics of mixtures and will be used repeatedly throughout the remainder of the chapter.
Big picture: Mixing lowers the chemical potential of every component in an ideal mixture. The relationship \(\mu_i=\mu_i^\circ+RT\ln x_i\) provides a direct connection between composition and chemical potential and explains why mixing is a spontaneous process.
Benzene and toluene form an approximately ideal liquid mixture. At \(298\ \mathrm{K}\), the chemical potential of pure benzene is
\[ \mu_{\mathrm{benzene}}^\circ = -12.5\ \mathrm{kJ\,mol^{-1}} \]
Calculate the chemical potential of benzene in a mixture where the mole fraction of benzene is
\[ x_{\mathrm{benzene}} = 0.350 \]
For an ideal mixture,
\[ \mu_i = \mu_i^\circ + RT\ln x_i \]
Substitute the values:
\[ \mu_{\mathrm{benzene}} = -12.5\ \mathrm{kJ\,mol^{-1}} + (8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}) (298\ \mathrm{K}) \ln(0.350) \]
Convert the \(RT\ln x_i\) term to \(\mathrm{kJ\,mol^{-1}}\):
\[ RT\ln(0.350) = -2.60\ \mathrm{kJ\,mol^{-1}} \]
Therefore,
\[ \mu_{\mathrm{benzene}} = -12.5 - 2.60 \]
\[ \boxed{ \mu_{\mathrm{benzene}} = -15.1\ \mathrm{kJ\,mol^{-1}} } \]
Interpretation: Mixing lowers the chemical potential of benzene because \(x_{\mathrm{benzene}}<1\), making \(\ln x_{\mathrm{benzene}}\) negative. The benzene is therefore more stable in the mixture than it is in the pure liquid at the same temperature and pressure.
A component in an ideal mixture will be generated at random. Use \(\mu_i=\mu_i^\circ+RT\ln x_i\) to calculate its chemical potential in the mixture.
Big picture: The chemical potential of a component depends not only on temperature and pressure, but also on composition. In an ideal mixture, the relationship \[ \mu_i = \mu_i^\circ + RT\ln x_i \] quantifies how mixing lowers the chemical potential of each component, providing the thermodynamic driving force for spontaneous mixing.