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Homework Statement
Derive the equation
##U=-T(\frac{\partial A}{\partial T})_V##
where ##U## is the internal energy, ##T## is the temperature, ##A## is the Helmholtz function.
Reference: Heat and Thermodynamics, Zemansky, Dittman, Page 272, Problem 10.4 (a)
Homework Equations
##dA=-PdV-SdT## ... (i) [##S## is the entropy]
##A=U-TS## ... (ii)
The Attempt at a Solution
As ##A(V,T)## is a state function,
##dA = (\frac{\partial A}{\partial V})_T dV + (\frac{\partial A}{\partial T})_V dT## ... (iii)
Comparing (i) and (iii),
##(\frac{\partial A}{\partial V})_T = -P## ... (iv)
and ##(\frac{\partial A}{\partial T})_V = -S## ... (v)
Using (ii) and (v),
##U = A +TS = A- T(\frac{\partial A}{\partial T})_V##
Any help would be appreciated.
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