Question

A gas obeying the equation of state $PV=RT$ undergoes a hypothetical reversible process described by the equation, $P{V}^{5/3}exp(-\frac{PV}{E_0})=C_1$ where $C_1$ and $E_0$ are dimensioned constants. Then, for this process, the thermal compressibility at high temperature.

• A. Approaches a constant value
• B. Is proportional to $T$
• C. Is proportional to $T^{1/2}$
• D. Is proportional to $T^2$

Answer 1

The correct option is A Approaches a constant value

$p{v}^{\frac{5}{3}}=C_{1}e\times p\left(\frac{PV}{E_0}\right)$

$dp{v}^{\frac{5}{3}}+p^{\frac{5}{3}}{v}^{\frac{2}{3}}dv=\frac{C_1}{E_0}e\times p\left(\frac{pv}{E_0}\right)$ $[pdv+vdp]$

$\Rightarrow dp[v^{\frac{5}{3}}+\frac{C_1}{E_0}\{e\times p\left(\frac{pv}{E_0}\right)\}v]$

$=[\frac{C_1}{E_0}\{e\times p\left(\frac{pv}{E_0}\right)\}p-\frac{5}{3}{v}^{\frac{2}{3}}p]dv$

$\Rightarrow Vdp[v^{\frac{2}{3}}+\frac{C_1}{E_0}e\times p\left(\frac{pv}{E_0}\right)]$

$=-dv[p^{\frac{5}{3}}{v}^{\frac{2}{3}}-\frac{GP}{E_0}e\times p\left(\frac{pv}{E_0}\right)]$

$\Rightarrow \propto =-\frac{dv}{vdp}=\frac{v^{\frac{2}{3}}+\frac{C_1}{E_0}e\times p\left(\frac{pv}{e_0}\right)}{p^{\frac{5}{3}}{v}^{\frac{2}{3}}-+\frac{C_{1}p}{E_0}e\times p\left(\frac{pv}{E_0}\right)}$

$\propto =-\frac{1}{v}\frac{dv}{dp}=\frac{1}{p}\left\{\frac{v^{\frac{2}{3}}+\frac{C_1}{E_0}e\times p\left(\frac{pv}{E_0}\right)}{\frac{5}{3}{v}^{\frac{2}{3}}-\frac{C_1}{E_0}e\times p\left(\frac{pv}{E_0}\right)}\right\}$

At high Tempreature $E_0\to$ high.

$\propto |HighT=\frac{1}{p}\left\{\frac{v^{\frac{2}{3}}}{\frac{5}{3}{v}^{\frac{2}{3}}}\right\}$ [Independent of T]

$\left[E_o\right]=RT$

It approcahes a constant value