17.7: Chemical Potential, Pressure, Fugacity

Equation 12.9.11 told us how to calculate the change in the Gibbs function of a mole of an ideal gas going from one state to another. For N moles it would be

\[ \Delta G=N \int C_{P} d T-N T_{2} \int C_{P} d(\ln T)+N R T_{2} \ln \left(\dfrac{P_{2}}{P_{1}}\right)-N S\left(T_{2}-T_{1}\right),\]

where C_P and S are molar, and G is total.

Since we know now how to calculate the absolute entropy and also know that the entropy at T = 0 is zero, this can be written

\[ G(T, P)=N(R T \ln P+\text { constant }) \label{17.7.2}\]

The “constant” here depends on the temperature, but is not a function of the pressure, being in fact the value of the molar Gibbs function extrapolated to the limit of zero pressure. Sometimes it is convenient to write Equation \ref{17.7.2} in the form

\[ G=N R T(\ln P+\phi)\]

where \(φ\) is a function of temperature.

If we have a mixture of several components, the total Gibbs function is

\[ G(T, P)=\sum_{i} N_{i}\left(R T \ln p_{i}+\text { constant }\right)\]

We can now write this in terms of the partial molar Gibbs function of the component i – that is to say, the chemical potential of the component i, which is given by \( \mu_{i}=\left(\partial G / \partial N_{i}\right)_{P, T, N_{j \neq 1}}\), and the partial pressure of component i. Thus we obtain

\[ \mu_{i} =\mu_{i}^{0}(T)+R T \ln p_{i}\]

and

\[ \mu_{i} =R T\left(\ln p_{i}+\phi_{i}\right) \]

Here I have written the “constant” as 0 µ_i⁰ (T), or as RTφ_i. The constant µ_i⁰ (T) is the value of the chemical potential at temperature \(T\) extrapolated to the limit of zero pressure. If the system consists of a mixture of ideal gases, the partial pressure of the ith component is related to the total pressure simply by Dalton’s law of partial pressures:

\[ p_{i}=n_{i} P,\]

where n_i is the mole fraction of the ith component. In that case, equation 17.7.4 becomes

\[ \mu_{i}=\mu_{i}^{0}(T)+R T \ln n_{i}+R T \ln P.\]

and equation 17.7.5 becomes 

\[ \mu_{i}=R T\left(\ln n_{i}+\ln P+\phi_{i}\right).\]

However, in a common deviation from ideality, volumes in a mixture are not simply additive, and we write equation 17.7.4 in the form

\[ \mu_{i}=\mu_{i}^{0}(T)+R T \ln f_{i},\]

or equation 17.7.5 in the form

\[ \mu_{i}=R T\left(\ln f_{i}+\phi_{i}\right).\]

where f_i is the fugacity of component i.