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A van der Waal's gas obeys the equation of state $\left(\mathrm{P}+\frac{\mathrm{n}^{2} \mathrm{a}}{\mathrm{V}^{2}}\right)(\mathrm{V}-\mathrm{nb})=\mathrm{nRT}$. Its internal energy is given by $\mathrm{U}=\mathrm{CT}-\frac{\mathrm{n}^{2} \mathrm{a}}{\mathrm{V}} .$ The equation of a quasistatic ad iabat for this gas is given by-
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$\mathrm{T}^{\mathrm{ChR}}(\mathrm{V}-\mathrm{nb})=$ constant
For adiabatic process
$\mathrm{dQ}=0$ and $-\mathrm{dU}=\mathrm{dW} \Rightarrow-\mathrm{nC}_{\mathrm{V}} \Delta \mathrm{T}=\mathrm{P} \Delta \mathrm{V}$ or $-\mathrm{nC}_{\mathrm{V}} \mathrm{dT}=\mathrm{PdV}$
when change is very small
now given $\mathrm{U}=\mathrm{CT}-\frac{\mathrm{n}^{2} \mathrm{a}}{\mathrm{V}} \quad \therefore \mathrm{d} \mathrm{U}=\mathrm{CdT}+\frac{\mathrm{n}^{2} \mathrm{a}}{\mathrm{V}^{2}} \mathrm{~d} \mathrm{~V}$
put this value of $\mathrm{dU} \mathrm{in}-\mathrm{dU}=\mathrm{d} \mathrm{W}$
$$
\therefore-\left(\mathrm{CdT}+\frac{\mathrm{n}^{2} \mathrm{a}}{\mathrm{V}^{2}} \mathrm{~d} \mathrm{~V}\right)=\mathrm{PdV}
$$
also $\mathrm{P}=\left(\frac{\mathrm{nRT}}{\mathrm{V}-\mathrm{nb}}\right)-\frac{\mathrm{n}^{2} \mathrm{a}}{\mathrm{V}^{2}}$ replace it in (1)
$$
-\left(C d T+\frac{n^{2} a}{V^{2}} d V\right)=\left(\left(\frac{n R T}{V-n b}\right)-\frac{n^{2} a}{V^{2}}\right) d V
$$
$$
\begin{array}{l}
\therefore-\mathrm{CdT}=\left(\frac{\mathrm{nRT}}{\mathrm{V}-\mathrm{nb}}\right) \mathrm{dV} \\
\therefore-\frac{\mathrm{C}}{\mathrm{nR}} \frac{\mathrm{dT}}{\mathrm{T}}=\frac{\mathrm{dV}}{\mathrm{V}-\mathrm{nb}}
\end{array}
$$
Integrating we get
$-\ell \mathrm{n} \mathrm{T}^{\mathrm{C} \mathrm{n} \mathrm{R}}=\ell \mathrm{n}(\mathrm{V}-\mathrm{nb})+\mathrm{k} \quad(\mathrm{k} \rightarrow$ constant of integration)
$\therefore \ln \left(\mathrm{T}^{\mathrm{C} / \mathrm{nR}}\right)(\mathrm{V}-\mathrm{nb})=-\mathrm{k}$
or $\left(\mathrm{T}^{\mathrm{C}^{\mathrm{Tn} \mathrm{R}}}\right)(\mathrm{V}-\mathrm{nb})=$ constant
$\mathrm{dQ}=0$ and $-\mathrm{dU}=\mathrm{dW} \Rightarrow-\mathrm{nC}_{\mathrm{V}} \Delta \mathrm{T}=\mathrm{P} \Delta \mathrm{V}$ or $-\mathrm{nC}_{\mathrm{V}} \mathrm{dT}=\mathrm{PdV}$
when change is very small
now given $\mathrm{U}=\mathrm{CT}-\frac{\mathrm{n}^{2} \mathrm{a}}{\mathrm{V}} \quad \therefore \mathrm{d} \mathrm{U}=\mathrm{CdT}+\frac{\mathrm{n}^{2} \mathrm{a}}{\mathrm{V}^{2}} \mathrm{~d} \mathrm{~V}$
put this value of $\mathrm{dU} \mathrm{in}-\mathrm{dU}=\mathrm{d} \mathrm{W}$
$$
\therefore-\left(\mathrm{CdT}+\frac{\mathrm{n}^{2} \mathrm{a}}{\mathrm{V}^{2}} \mathrm{~d} \mathrm{~V}\right)=\mathrm{PdV}
$$
also $\mathrm{P}=\left(\frac{\mathrm{nRT}}{\mathrm{V}-\mathrm{nb}}\right)-\frac{\mathrm{n}^{2} \mathrm{a}}{\mathrm{V}^{2}}$ replace it in (1)
$$
-\left(C d T+\frac{n^{2} a}{V^{2}} d V\right)=\left(\left(\frac{n R T}{V-n b}\right)-\frac{n^{2} a}{V^{2}}\right) d V
$$
$$
\begin{array}{l}
\therefore-\mathrm{CdT}=\left(\frac{\mathrm{nRT}}{\mathrm{V}-\mathrm{nb}}\right) \mathrm{dV} \\
\therefore-\frac{\mathrm{C}}{\mathrm{nR}} \frac{\mathrm{dT}}{\mathrm{T}}=\frac{\mathrm{dV}}{\mathrm{V}-\mathrm{nb}}
\end{array}
$$
Integrating we get
$-\ell \mathrm{n} \mathrm{T}^{\mathrm{C} \mathrm{n} \mathrm{R}}=\ell \mathrm{n}(\mathrm{V}-\mathrm{nb})+\mathrm{k} \quad(\mathrm{k} \rightarrow$ constant of integration)
$\therefore \ln \left(\mathrm{T}^{\mathrm{C} / \mathrm{nR}}\right)(\mathrm{V}-\mathrm{nb})=-\mathrm{k}$
or $\left(\mathrm{T}^{\mathrm{C}^{\mathrm{Tn} \mathrm{R}}}\right)(\mathrm{V}-\mathrm{nb})=$ constant
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