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13.1: The Gibbs Free Energy of an Ideal Gas

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    151739
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    In Chapter 11, we find a general equation for the molar Gibbs free energy of a pure gas. We adopt the Gibbs free energy of formation of the hypothetical ideal gas, in its standard state at 1 bar, \(P^o\), as the reference state for the Gibbs free energy of the gas at other pressures and the same temperature. Then, the molar Gibbs free energy of pure gas \(A\), at pressure \(P\), is

    \[{\overline{G}}_A\left(P\right)={\Delta }_fG^o\left(A,{HIG}^o\right)+RT{ \ln \left(\frac{P}{P^o}\right)\ }+RT\int^P_0{\left(\frac{\overline{V}}{RT}-\frac{1}{P}\right)dP}\] (any pure gas)

    \({\overline{G}}_A\left(P\right)\) is the difference between the Gibbs free energy of the gas at pressure \(P\) and that of its constituent elements at 1 bar and the same temperature. If gas \(A\) is an ideal gas, the integral is zero, and the standard-state Gibbs free energy of formation is that of an “actual” ideal gas, not a “hypothetical state” of a real gas. To recognize this distinction, let us write \({\Delta }_fG^o\left(A,P^o\right)\), rather than \({\Delta }_fG^o\left(A,{HIG}^o\right)\), when the gas behaves ideally. In a mixture of ideal gases, the partial pressure of gas \(A\) is given by \(P_A=x_AP\), where \(x_A\) is the mole fraction of \(A\) and \(P\) is the pressure of the mixture. In §3, we find that the Gibbs free energy of one mole of pure ideal gas \(A\) at pressure \(P_A\) has the same Gibbs free energy as one mole of gas \(A\) in a gaseous mixture in which the partial pressure of \(A\) is \(P_A=x_AP\). Recognizing these properties of an ideal gas, we can express the molar Gibbs free energy of an ideal gas—pure or in a mixture—as

    \[\overline{G}_A\left(P_A\right)=\Delta_fG^o\left(A,P^o\right)+RT \ln \left(\frac{P_A}{P^o}\right)\ \]

    (ideal gas)

    Note that we can obtain this result for pure gas \(A\) directly from \({\left({\partial \overline{G}}/{\partial P}\right)}_T=\overline{V}={RT}/{P}\) by evaluating the definite integrals

    \[\int^{\overline{G}_A\left(P_A\right)}_{\Delta_fG^o\left(A,P^o\right)}{d\overline{G}}=\int^{P_A}_{P^o}{\frac{RT}{P}dP}\]

    Including the constant, \(P^o\), in these relationships is a useful reminder that \(RT{ \ln \left({P_A}/{P^o}\right)\ }\) represents a Gibbs free energy difference. Including \(P^o\) makes the argument of the natural-log function dimensionless; if we express \(P\) in bars, including\({\ P}^o=1\ \mathrm{bar}\) leaves the numerical value of the argument unchanged. If we express \(P\) in other units, \(P^o\) becomes the conversion factor for converting those units to bars; if we express \(P\) in atmospheres, we have \({\ P}^o=1\ \mathrm{bar}=0.986923\ \mathrm{atm}\).

    However, including the “\(P^o\)” is frequently a typographical nuisance. Therefore, let us introduce another bit of notation; we use a lower-case “\(p\)” to denote the ratio “\({P}/{P^0}\)”. That is, \(p_A\) is a dimensionless quantity whose numerical value is that of the partial pressure of \(A\), expressed in bars. The molar Gibbs free energy becomes

    \[{\overline{G}}_A\left(P_A\right)={\Delta }_fG^o\left(A,P^o\right)+RT{ \ln p_A\ }\] (ideal gas)


    13.1: The Gibbs Free Energy of an Ideal Gas 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.