How to Derive the Equation for Internal Energy in Thermodynamics Homework?

[arpon]

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.

 

[DrDu]

I think your solution is correct while the problem as stated is wrong.