Second order relation of entropy

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thonwer
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Homework Statement


Find:
Second partial derivative of entropy S with respect to volume V, taken at constant U: (∂²S/∂V²) with subscript U,n.


Homework Equations


Equation dS = (1/T) dU + (P/T) dV - (mu/T) dN, the fundamental thermodynamic differential relation for entropy S in terms of internal energy U, volume V and particle number N.

Equation: alpha = (1/V)(dV/dT) at constant pressure P, the isobaric thermal expansion coefficient.

Equation for isothermal compressibility: kappa_T = -(1/V)(dV/dP) at constant temperature T.

The Attempt at a Solution


Partial derivative chain for second derivative of entropy: (∂²S/∂V²) at constant U, n equals (∂(P/T)/∂V) at constant U, N, which equals (1/T)(∂P/∂V) minus (P/T²)(∂T/∂V), equals -1/(V κT) minus P/(T² α V).
 

Discussion

Have I done it in the right way? (I forgot to put the question :oops:)
 

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