From QED

The entropy of a system is found to be:

<math>S=Nk_{B}\left(\ln U^{\alpha}+\ln V^{\beta}+\ln N^{\gamma}+\frac{7}{2}\right)</math> 

a. In order for the entropy to remain extensive (i.e. be twice as big for two identical systems as for one), what must be the relationship among α, β, and γ?

b. Find the temperature T in terms of U, V, N.

c. Find the Helmholtz free energy F(T, V, N) as a function of the indicated quantities.

d. Find the equation of state P(T, V, N)

e. Find the specific heat at constant pressure; at constant volume.

We must have extensivity: 2S(U,V,N) = S(2U,2V,2N)

Dividing, etc:

α + β + γ = αln2 + βln2 + γln2

implying:

<span class="texhtml">α + β + γ = 0</span>

We use the definition of the temperature:

From our definition of the entropy:

so:

<math>T=\frac{U}{Nk_{B}\alpha}</math>

The Helmholtz free energy is given by:

F = U − ST

Plugging in from part b for U in terms of T, V, and N, and for S from the problem:

<math>F=Nk_{B}T\left(\alpha-\ln U^{\alpha}-\ln V^{\beta}-\ln N^{\gamma}-\frac{7}{2}\right)</math>

The pressure is given by:

<math>P=-\frac{\partial F}{\partial V}=\frac{\beta Nk_{B}T}{V}</math>

Specific heat at constant pressure, using U = TNk_Bα:

<math>C_{P}=\left(\frac{\partial U}{\partial T}\right)_{P,N}=\alpha Nk_{B}</math> 

We can find the entropy as a function of T with U = TNk_Bα:

so:

<math>C_{V}=T\left(\frac{\partial S}{\partial T}\right)_{V,N}=T\frac{Nk_{B}\alpha}{T}=Nk_{B}\alpha</math>

This page was recovered in October 2009 from the Plasmagicians page on Prelim_J97_SMT2 dated 21:40, 5 July 2007.