Statistical definition of entropy

[rbwang1225]

In the book of Pathria (p.15), Cp =: T (ds/dt)N,P = (d(E+PV)/dT)N,P

S=S(N,V,E)

I don't know how it comes the 2nd equal sigh.

Does anybody can help me? Thanks in advance!

 

[vanhees71]

Within classical thermodynamics everything can be deduced from the first and second law of thermodynamics, which reads

$$\mathrm{d} U=T \mathrm{d} S-p \mathrm{d}V+\mu \mathrm{d} N.$$

The "natural independent" variables for the internal energy, $$U$$, are thus $$S$$, $$V$$, $$N$$.

Now, you like to calculate the specific heat at constant pressure and particle number. With $$U$$ this is not so simple to achieve, but you can go to another thermodynamical potential, the enthalpy, $$H$$ via the Legendre transform

$$H=U+ p V.$$

Then we get

$$\mathrm{d} H = \mathrm{U}+\mathrm{d} p V + \mathrm{d}V p = T \mathrm{d} S + V \mathrm{d} p + \mu \mathrm{d} N.$$

That means that the change of the enthalpy at constant pressure and constant particle number is identical with the change of heat $$\mathrm{d} Q=T \mathrm{d} S$$ and thus

$$c_p=\left( \frac{\partial H}{\partial T} \right )_{p,N} = T \left (\frac{\partial S}{\partial T} \right )_{p,N}.$$