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1.9.2: Entropy- Dependence on Temperature and Pressure

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    The volume of a given closed system at equilibrium prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of solute-\(j\) is defined by the set of independent variables shown in equation (a).

    \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0, \xi^{\mathrm{eq}}\right] \nonumber \]

    The same set of independent variables defines the entropy \(\mathrm{S}\).

    \[\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0, \xi^{\mathrm{eq}}\right] \nonumber \]

    We envisage that the system is displaced by a change in pressure along a path where the system remains at equilibrium (i.e. \(\mathrm{A} = 0\)) and the volume remains the same as defined by equation (a). In a plot of entropy against \(\mathrm{p}\), the gradient of the plot at the point defined by the independent variables, \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0, \boldsymbol{\xi}^{\mathrm{eq}}\right]\) is given by equation (c).

    Isochoric

    \[\left(\frac{\partial S}{\partial p}\right)_{V, A=0} \nonumber \]

    The set of derivatives is completed by the following partial derivatives.

    Isothermal

    \[\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \nonumber \]

    Isobaric

    \[\left(\frac{\partial S}{\partial T}\right)_{p, A=0} \nonumber \]

    This page titled 1.9.2: Entropy- Dependence on Temperature and Pressure was last modified on Mon, 20 May 2024 07:03:08 GMT and is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer and Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform.