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Physics LibreTexts

17.7: Chemical Potential, Pressure, Fugacity

  • Page ID
    8667
  • 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 CP 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 µi0 (T), or as RTφi. The constant µi0 (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 ni 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 fi is the fugacity of component i.