Helmholtz energy of Simple solid

[Kelsi_Jade]

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!

 

[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.

 

[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.

 

[shockwaver]

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

 

[paranormal]

First of all, it is important to note that the Helmholtz free energy is a thermodynamic potential that describes the maximum amount of work that a system can perform at constant temperature and volume. It is often denoted as F and is defined as F = U - TS, where U is the internal energy of the system, T is the temperature, and S is the entropy. However, for a simple solid, the internal energy U is not simply given by U = ncvT, where n is the number of moles, cv is the molar heat capacity at constant volume, and T is the temperature. In fact, the internal energy of a simple solid is more accurately described by the expression U = ncVT, where V is the volume of the solid. This is because in a solid, the atoms or molecules are tightly packed and do not have the same degree of freedom as in a gas, so the molar heat capacity at constant volume is not a constant and depends on the volume.

To find the Helmholtz free energy F(V, T) of a simple solid, we can use the definition F = U - TS and substitute the expression for U mentioned above. This gives us F = ncVT - TS. We can then use the Maxwell relation (∂S/∂V)T = (∂P/∂T)V to find (∂F/∂V)T, which gives us (∂F/∂V)T = ncT - (∂S/∂V)T. Similarly, we can use the definition P = -(∂F/∂V)T to find (∂F/∂T)V, which gives us (∂F/∂T)V = -ncV + P. These expressions are consistent with S(T, V) and P(V, T) in the equation P = a0T - b0ln(V/V0), as can be seen by substituting the expressions for (∂F/∂V)T and (∂F/∂T)V into the equation and comparing it to the given equation.

In conclusion, the Helmholtz free energy of a simple solid can be found by using the definition F = U - TS and substituting the appropriate expression for the internal energy of the solid. The resulting expressions for (∂F/∂V)T and (∂F/∂T)V are consistent with