Show that cv=-T((∂v)/(∂T))s((∂P)/(∂T))v and cp=T((∂P)/(∂T))s((∂v)/(∂T))

Name: Problem 82, Chapter 12: Thermodynamics: An Engineering Approach
Uploaded: 2020-09-27T14:56:22-08:00
Duration: 2 min 26 s
Channel: Sheh Lit Chang
Description: Show that cv=-T((∂v)/(∂T))s((∂P)/(∂T))v and cp=T((∂P)/(∂T))s((∂v)/(∂T))

step 1 In this question, we need to show these two relationships, this CV and CP. Okay, so solution, ok

Show that cv=-T((∂v)/(∂T))s((∂P)/(∂T))v and ...

Show that $$c_{v}=-T\left(\frac{\partial v}{\partial T}\right)_{s}\left(\frac{\partial P}{\partial T}\right)_{v} \text { and } c_{p}=T\left(\frac{\partial P}{\partial T}\right)_{s}\left(\frac{\partial v}{\partial T}\right)$$

Key Concepts

Heat Capacities

Heat capacities, specifically at constant volume (c_v) and constant pressure (c_p), measure a system’s ability to absorb heat with changes in temperature under constant volume or pressure conditions. They are defined in terms of partial derivatives of internal energy (for c_v) and enthalpy (for c_p) with respect to temperature, and are fundamental in understanding energy storage and transfer in thermodynamic systems.

Partial Derivatives in Thermodynamics

Partial derivatives in thermodynamics express how one state variable changes with respect to another while holding a third variable constant. They are crucial for formulating relations among pressure, volume, temperature, and entropy, and are used to connect various thermodynamic properties through mathematical manipulation and chain rules.

Maxwell Relations

Maxwell relations arise from the equality of mixed second derivatives of thermodynamic potentials. These relations allow the conversion of difficult-to-measure derivatives, like the derivative of pressure with respect to temperature at constant volume, into more accessible forms, and play a key role in linking the expressions for heat capacities to other measurable thermodynamic quantities.

Thermodynamic Identities

Thermodynamic identities are fundamental equations that relate different thermodynamic variables. They form the backbone of deriving relationships among properties like temperature, pressure, volume, and entropy, and allow one to express complex quantities such as heat capacities in terms of partial derivatives and measurable parameters.

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