- #1
Dario SLC
Homework Statement
This is a state ecuation of a gas:
PV=AT+B/V, where A and B there are constants.
First: Demonstrate that ##c_V## depends only of T
Second: Find U(T,V) and S(T,V)
Homework Equations
##\left(\frac{\partial U}{\partial S}\right)_V=T\text{ (1)}##
##\left(\frac{\partial U}{\partial V}\right)_S=-p\text{ (2)}##
##F=U-TS\text{ (3)}##
##\left(\frac{\partial F}{\partial V}\right)_T=-p\text{ (4)}##
##\left(\frac{\partial F}{\partial T}\right)_V=-S\text{ (5)}##
The Attempt at a Solution
For second item, I think use first (2) and then integrate respect to V using the state ecuation, I got:
##U(S,V)=-AT\ln V+B/V+u(S)##
when u(S) is a constant integration, then I use the (1) for find u(S) and the complete expression for U(S,V):
##U(S,V)=-AT\ln V+B/V+TS+U_0##
now using (4) I would like to obtain the free energy of Helmholtz:
\begin{equation}
F(T,V)=-ATlnV+B/V+f(T)
\end{equation}
when f(T) it is a new constant integration, then I use the (5) I got:
-S(T,V)=-AlnV+f'(T) then S(T,V)=AlnV-f'(T)
if F=U(S,V)-TS(T,V) and derivate respect to T, this it is the same
-AlnV=-AlnV+f'(T)
therefore f(T)=c with c constant, then
S(T,V)=AlnV-c
and
U(T,V)=-ATlnV+B/V-cT+U0
Well, I doubt that the entropy don't depends of T, I haven't see the error, it should be something like to
##S(T,V)=S_0+c_V\ln\frac{T}{T_0}+R\ln\frac{V}{V_0}## for a van der Waals gases
(Sears, Thermodynamics, chapter 9)