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Formula for grand potential is:

$蠒=U-TS-渭N$

$蠒=U-TS-渭N$

Use thermodynamic equation:

$dU=TdS-PdV+渭dN$

Equation for an infinitesimal change in grand potential is:

$$\begin{gathered} d蠒=d(U-TS-渭N) \\ =dU-d\left(TS\right)-d\left(渭N\right) \\ =dU-TdS-SdT-渭dN-Nd渭 \end{gathered}$$

Substitute $dU=TdS-PdV+渭dN$

localid="1647239388505" $$\begin{gathered} d蠒=TdS-PdV+渭dN-TdS-SdT-渭dN-Nd渭 \\ =-SdT-PdV-Nd渭 \end{gathered}$$

Differentiate equation $d蠒=-SdT-PdV-Nd渭$with respect to $渭$

$\frac{鈭�蠒}{鈭�{渭}_{TV}}=-N$

We differentiate function $\widetilde{蠒}=-kT脳\ln (\widehat{Z)}$ with respect to $渭$.

${\left(\frac{鈭�\widetilde{蠒}}{鈭�渭}\right)}_{TV}=-\frac{kT}{\hat{Z}}\left(\frac{鈭�\hat{Z}}{鈭�渭}\right)$

Here, $\hat{Z}$ is grand partition function.

As, average number of particles is given by:

localid="1647240173822" $\stackrel{炉}{N}=\frac{kT}{\hat{Z}}\left(\frac{鈭�\hat{Z}}{鈭�渭}\right)$

Substitute $\frac{kT}{\hat{Z}}\left(\frac{鈭�\hat{Z}}{鈭�渭}\right)=\stackrel{炉}{N}$ in above differential equation,

${\left(\frac{鈭�\widetilde{蠒}}{鈭�渭}\right)}_{TV}=-\stackrel{炉}{N}$

For grand canonical function $蠒and\widetilde{蠒}$at $渭=0$we get,

$$\begin{gathered} \widetilde{蠒}=-kT脳\ln \left(Z\right) \\ =F \end{gathered}$$

Here, $F=-kT脳\ln \left(Z\right)$is called Helmholtz free energy.

Substitute $渭=0$in equation $蠒=U-TS-渭N$

$蠒=U-TS$

As, Helmholtz free energy is $F=U-TS$

Substitute $F=U-TS$in equation $蠒=U-TS$we get,

$蠒=F$

So, $蠒$and $\widetilde{蠒}$at same initial conditions have same values. Therefore, they are the same functions. Hence, the grand canonical function is $蠒=-kT脳\ln \left(\hat{Z}\right)$.