State equation in grand canonical ensemble

Question

My teacher told us that $$ \ln Z = \frac{PV}{kT} $$ is the equation of state for an ideal gas, being $Z$ the grand canonical partition function and $k$ the Boltzmann constant. Where does this formula come from? (we then used this formula for studying Bose-Einstein and Fermi-Dirac systems).

I have tried the following: in the grand canonical ensemble I know that $\Xi = -kT \ln Z $ is the grand canonical potential. Also, from thermodynamics, I know that the grand canonical potential differential is $ d\Xi = -S~dT - P~dV - N~d\mu $ so, integrating, $ \Xi = -PV + f(\mu,T) $ where $f$ is some unknown function. Equating this two facts about $\Xi$ gives $$ \ln Z = \frac{PV - f(\mu,T)}{kT} $$ but how can be shown that $f(\mu,T)=0$?

Answer 1

I have just found a way to prove that $\ln Z = \frac{PV}{kT}$ as follows: the grand canonical potential is defined, in thermodynamics, to be $ \Xi = U - TS - \mu N $. Using Euler relation (Callen, eq. (3.6), also known as Euler integrals here in Wikipedia) $ U = TS - PV + \mu N $ then $$ \Xi = -PV. $$

On the other hand the grand canonical potential can be obtained from the grand canonical partition function as $$ \Xi = -kT \ln Z .$$

Now it is trivial that $$ \ln Z = \frac{PV}{kT} $$

Answer 2

Let us consider a grand canonical ensemble of $M$ identical systems such that the total number of particles in the ensemble is $M\bar{N}$ and a total energy of $M\bar{E}$. Let $n_{r,s}$ be the number of systems having energy $E_{s}$ and number of particles $N_{r}$. Putting these statements mathematically, we have

$\sum_{r,s} n_{r,s}= M$

$\sum_{r,s} n_{r,s} N_{r}=M\bar{N}$

$\sum_{r,s} n_{r,s} E_{s}=M \bar{E}$

The above part lays the groundwork. The solutions begins here.

Let $P_{r,s}$ be the probability that a given member of the ensemble has an energy $E_{s}$ and number of particles $n_{r}$. From basic considerations of the grand canonical ensemble, we have

$P_{r,s}=\frac{exp(-\alpha N_{r}-\beta E_{s})}{\sum_{r,s}exp(-\alpha N_{r}-\beta E_{s})}$ .............. (1)

Now, the denominator is the grand canonical partition function $Z_{G}$

$Z_{G}=\sum_{r,s}exp(-\alpha N_{r}-\beta E_{s})$ .............. (2)

Therefore,

$P_{r,s}=\frac{exp(-\alpha N_{r}-\beta E_{s})}{Z_{G}}$ .............. (3)

Now, from the definition of entropy, we have

$S=-k_{B}<ln(P_{r,s})>$ .............. (4)

$=> S=-k_{B}\sum_{r,s} P_{r,s} ln(P_{r,s}) $ .............. (5)

Now, we substitute the expression of $P_{r,s}$ obtained in equation (3) inside the $ln$. We leave the $P_{r,s}$ outside the $ln$ unchanged. (This makes the calculations easier as we will see shortly).

$S=-k_{B} \sum_{r,s} P_{r,s} ln[\frac{exp(-\alpha N_{r}-\beta E_{s})}{Z_G}]$ .............. (6)

$=>S=-k_{B} \sum_{r,s} P_{r,s}[-\alpha N_{r}-\beta E_{s}-ln(Z_{G})]$ .............. (7)

$=>S=k_{B} \alpha \sum_{r,s} P_{r,s}N_{r}+k_{B} \beta \sum_{r,s} P_{r,s} E_{s}+k_{B}ln(Z_{G}) \sum_{r,s} P_{r,s} $ .............. (8)

From normalization of probabilities,

$\sum_{r,s} P_{r,s}=1$.

From the basic definition of mean, we have

$\sum_{r,s} P_{r,s}N_{r}=\bar{N}$

and

$\sum_{r,s} P_{r,s}E_{s}=\bar{E}$

Therefore, equation (8) can be written as

$S=k_{B} \alpha \bar{N}+k_{B} \beta \bar{E}+k_{B}ln(Z_{G}) $ .............. (9)

Now, using the first law and second law of thermodynamics,

$d\bar{E}=TdS-PdV+\mu d\bar{N}$ .............. (10)

Since $S$, $V$, and $N$ are extensive quantities, using Euler's homogeneous function theorem, we get

$\bar{E}=TS-PV+\mu\bar{N}$ .............. (11)

Rearranging the terms of (11) and using the fact that $\beta=\frac{1}{k_{B}T}$, we get

$S=k_{B}\beta\bar{E}+k_{B}\beta PV - k_{B}\beta \mu \bar{N} $ .............. (12)

Comparing (9) and (12), we get

$k_{B}\beta PV =k_{B}ln(Z_{G})$

$=>\beta PV=ln(Z_{G})$

$=>\frac{PV}{k_{B}T}=ln(Z_{G})$ .............. (13)

[Hence proved]

Note : The key here is to bridge statistical mechanics, and thermodynamics. Entropy is often a useful tool for this purpose as we saw here.