Grand partition function Z of a system

[sm09]

The grand partition function Z of a system is given by formula:

Z = Ʃ exp ((-Ei/KbT) + (μni/KbT))

where , 1, 2... i E i= are permitted energy levels, μ is the chemical
potential, , 1,2... i n i= are number of particles of different types.
Taking into account that averaged internal energy

U = Ʃ Pi(Ei-μni) show that

U = Kb(T^2)(d(lnZ)/dT)

any help?

 

[dydxforsn]

The grand partition function is as follows:

$\Large{Z = \sum_{i}{e^{(\frac{-\epsilon_i}{kT}) + (\frac{{\mu}n_i}{kT})}}}$

Remembering that the expectation value of the energy is the following:

$\large{<U> = \sum_{i}{P(i)({\epsilon}_i-{\mu}n_i)}}$

(where P(i) is the probability of finding the system in the i^th state..)
Show that:

$\large{<U> = kT^2\frac{d(ln(Z))}{dT}}$

Just dressing up your equations in Latex for the practice. This proof should in a standard Thermal Physics text, but unfortunately I am without mine at the moment :(

 

[jfy4]

Note:
$$\frac{d}{dT}\ln(Z)=\frac{1}{Z}\frac{dZ}{dT}$$
also, you should look up the definition of the canonical probability distribution $P(\sigma_i)$. With those definitions, you should be on your way. Also remember that the temperature derivative can pass through the sum in the first equation.