Search any question & find its solution
Question:
Answered & Verified by Expert
A gas obeying the equation of state $\mathrm{PV}=\mathrm{RT}$ undergoes a hypothetical reversible process described by the equation, $\mathrm{PV}^{5 / 3} \exp \left(-\frac{\mathrm{PV}}{\mathrm{E}_{0}}\right)=\mathrm{c}_{1}$ where $c_{1}$ and $E_{0}$ are dimensioned constants. Then, for this process, the thermal compressibility at high temperature
Options:
Solution:
2731 Upvotes
Verified Answer
The correct answer is:
approaches a constant value.
$\begin{array}{l}
\mathrm{PV}^{5 / 3}=\mathrm{c}_{1} \mathrm{e}^{\frac{\mathrm{PV}}{\mathrm{E}_{0}}} \\
\ln \mathrm{P}+\frac{5}{3} \quad \ln \mathrm{V}=\ln \mathrm{c}_{1}+\frac{\mathrm{PV}}{\mathrm{E}_{0}} \\
\frac{\mathrm{dP}}{\mathrm{P}}+\frac{5}{3} \frac{\mathrm{dV}}{\mathrm{V}}=0+\frac{\mathrm{PdV}+\mathrm{VdP}}{\mathrm{E}_{0}} \\
\mathrm{dP}\left[\frac{1}{\mathrm{P}}-\frac{\mathrm{V}}{\mathrm{E}_{0}}\right]=\mathrm{d} \mathrm{V}\left[\mathrm{P}-\frac{5}{3 \mathrm{~V}}\right] \\
\mathrm{c}=-\frac{1}{\mathrm{~V}} \frac{\mathrm{d} \mathrm{V}}{\mathrm{dP}}=\frac{\left[\frac{1}{\mathrm{E}_{0}}-\frac{1}{\mathrm{PV}}\right]}{\left[\mathrm{P}-\frac{5}{3 \mathrm{~V}}\right]}
\end{array}$
At very high temperature $c=\frac{1}{E_{0}}$
\mathrm{PV}^{5 / 3}=\mathrm{c}_{1} \mathrm{e}^{\frac{\mathrm{PV}}{\mathrm{E}_{0}}} \\
\ln \mathrm{P}+\frac{5}{3} \quad \ln \mathrm{V}=\ln \mathrm{c}_{1}+\frac{\mathrm{PV}}{\mathrm{E}_{0}} \\
\frac{\mathrm{dP}}{\mathrm{P}}+\frac{5}{3} \frac{\mathrm{dV}}{\mathrm{V}}=0+\frac{\mathrm{PdV}+\mathrm{VdP}}{\mathrm{E}_{0}} \\
\mathrm{dP}\left[\frac{1}{\mathrm{P}}-\frac{\mathrm{V}}{\mathrm{E}_{0}}\right]=\mathrm{d} \mathrm{V}\left[\mathrm{P}-\frac{5}{3 \mathrm{~V}}\right] \\
\mathrm{c}=-\frac{1}{\mathrm{~V}} \frac{\mathrm{d} \mathrm{V}}{\mathrm{dP}}=\frac{\left[\frac{1}{\mathrm{E}_{0}}-\frac{1}{\mathrm{PV}}\right]}{\left[\mathrm{P}-\frac{5}{3 \mathrm{~V}}\right]}
\end{array}$
At very high temperature $c=\frac{1}{E_{0}}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.