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1.7.14: Compressions- Ratio- Isentropic and Isothermal

  • Page ID
    374372
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    Using a calculus operation, we obtain equations relating isothermal and isentropic dependencies of volume on pressure.

    Thus,

    \[\begin{aligned}
    (\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}} &=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}} \\ &=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}
    \end{aligned}\]

    and,

    \[\begin{aligned}
    (\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=&-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{v}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \\
    &=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}
    \end{aligned}\]

    Then,

    \[(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}} /(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}} /(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}\]

    The Gibbs-Helmholtz equation requires that

    \[\mathrm{H}=\mathrm{G}-\mathrm{T} \,(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}}\]

    Also

    \[(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}}=\mathrm{C}_{\mathrm{p}}=-\mathrm{T} \,\left(\partial^{2} \mathrm{G} / \partial \mathrm{T}^{2}\right)_{\mathrm{p}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}\]

    Similarly,

    \[(\partial \mathrm{U} / \partial \mathrm{T})_{\mathrm{V}}=\mathrm{C}_{\mathrm{V}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}\]

    Hence,

    \[\mathrm{K}_{\mathrm{T}} / \mathrm{K}_{\mathrm{s}}=\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{V}}\]


    This page titled 1.7.14: Compressions- Ratio- Isentropic and Isothermal is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.