# Helmholtz energy of Simple solid

1. Dec 1, 2013

The problem is :

a) Find Helmholtz free energy F(V, T) of a simple solid.
b) Use the result of part a) to verify that (∂F/∂T)v and (∂F/∂V)T are consistent with S(T, V) and P(V, T) in equation P=a0T-b0ln(V/V0)

I know:
Helmholtz free energy is F=U-TS
and dF=-SdT-PdV
S=-((∂F/∂T)v)
P=-(∂F/∂V)T
Maxwell relation: (∂S/∂V)T=(∂P/∂T)V

My problem is that the only examples I have here of Helmholtz free energy is for an ideal gas, NOT a simple solid. Is this correct to say internal energy of simple solid is U=ncvT+nu0 ?
And S=ncvln(T/Tr)+nRln(V/Vr+S(Tr, Vr) ?
Where you could just substitute the equations for U and S into F and simplify?

I found the above equations on a power point from another classes slides so I'm not sure on the background if they're accurate or not...
Any help would be appreciated to get me on the right track!! Thanks!

2. Dec 30, 2013

### shockwaver

the trick is to specify second derivatives of F. they are the physical observables. i.e., bulk modulus, KT=-v(dp/dv)v can be chosen as murnaghan's =KTo(v0/v)^n. specific heat, CV=T(ds/dt)v can be 3R and (dp/dt)v=gamma/v*Cv, gamma being the gruneisen's ratio. you can integrate twice to get F, closed form and you can find constants v0,kt0,n,and gamma for many materials in tables. p.s., often gamma/v is assumed constant and experiments bear this out.

3. Dec 30, 2013

### shockwaver

by the way, integration is much easier if you just call the bulk modulus constant. with gamma/v*cv also constant, integration should be a snap.

4. Dec 31, 2013

### shockwaver

typo correction: KT=-v(dp/dv)t
also, n=1 for linear compression solid