Question

When an ideal gas at pressure $P$, temperature $T$ and volume $V$ is isothermally compressed to $V/n$, its pressure becomes $P_1$. If the same amount of same gas is compressed adiabatically to $V/n$, its pressure becomes $P_2$. The ratio $P_1/P_2$ is :

• A. $n$
• B. $n^{\gamma }$
• C. $\frac{1}{n^{\gamma -1}}$
• D. $1$

Answer 1

The correct option is C $\frac{1}{n^{\gamma -1}}$

When the gas is compressed isothermally,
$PV=P_1{V}_1$
$P_1=\frac{PV}{V_1}=\frac{PV}{V/n}=nP$
When the gas is compressed adiabatically,
$P{V}^{\gamma }=P_2{V}_2^{\gamma }$
$P_2=P{\left(\frac{V}{V_2}\right)}^{\gamma }=P{\left(\frac{V}{V/n}\right)}^{\gamma }=P{n}^{\gamma }$
Now, $\frac{P_1}{P_2}=\frac{nP}{n^{\gamma }P}=n^{1-\gamma }=\frac{1}{n^{\gamma -1}}$