Partial derivatives of enthelpy and Maxwell relations

AI Thread Summary
The discussion focuses on the application of Maxwell relations and the definitions of thermal coefficients to derive partial derivatives of enthalpy. The user has made progress by expressing enthalpy in terms of internal energy and pressure-volume work, leading to the equation for the partial derivative of enthalpy with respect to volume at constant internal energy. Further, the conversation explores how to express pressure as a function of entropy and volume, allowing for the decomposition of the partial derivative of pressure. The user is guided to utilize the relationship between temperature, entropy, and volume to continue their derivation. The discussion emphasizes the importance of Maxwell relations in simplifying the expressions for thermodynamic properties.
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Homework Statement
Write the following partial derivatives in terms of heat capacity (##c##), compressibility (##\kappa##) and coefficient of thermal expansion (##\alpha##).

##(\frac{\partial H}{\partial V})_U##
##(\frac{\partial H}{\partial V})_S##

With ##H##: enthalpy, ##U##: internal energy, ##V##: volume, ##S##: entropy and the subscript denotes a magnitud held constant in differentiation.
Relevant Equations
##dH=dU+Pdv+vdP##
##dU=TdS-PdV##
I've attached images showing my progress. I have used Maxwell relations and the definitions of ##\alpha##, ##\kappa## and ##c##, but I don't know how to continue. Can you help me?
 

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You have that ##H = U + PV##. And you also have that ##dU = T dS - P dV##. So when ##U## is constant, you have: ##dH = d(PV) = PdV + V dP##. And you also have when ##U## is constant, ##TdS - PdV = 0##. So we conclude:

##\frac{\partial H}{\partial V}|_U = P + V \frac{\partial P}{\partial V}|_U##

Now, if you think of ##P## as a function of ##S## and ##V##, then

##\frac{\partial P}{\partial V}|_U = \frac{\partial P}{\partial V}|_S + \frac{\partial P}{\partial S}|_V \frac{\partial S}{\partial V}|_U##

At this point, you can use ##TdS - PdV = 0## (when ##dU = 0##) to rewrite ##\frac{\partial S}{\partial V}|_U##, and you can use one of the Maxwell relations to rewrite ##\frac{\partial P}{\partial S}|_V##.
 
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