Question

Consider two containers $A$ and $B$ containing identical gases at the same pressure, volume and temperature. The gas in container $A$ is compressed to half of its original volume isothermally while the gas in container $B$ is compressed to half of its original value adiabatically. The ratio of final pressure of gas in $B$ to that of gas in $A$ is

• A. $2^{\gamma -1}$
• B. ${\left(\frac{1}{2}\right)}^{\gamma -1}$
• C. ${\left(\frac{1}{1-\gamma }\right)}^2$
• D. $-{\left(\frac{1}{\gamma -1}\right)}^2$

Answer 1

The correct option is A $2^{\gamma -1}$

Consider the $P-V$ diagram shown for the container $A$ and container $B$.

Both the process involves compression of the gas

(i) Isothermal compression (Gas $A$) (during $1\to 2$)

$P_1{V}_1=P_2{V}_2$

$⟹P_0(2V_0{)}^{\gamma }=P_2(V_0{)}^{\gamma }$

$⟹P_0\left(2V_0\right)=P_2\left(V_0\right)$

(II) Adiabatic compression (Gas $B$) (during $1\to 2)$

$P_1{V}_1^{\gamma }=P_2{V}_2^{\gamma }$

$⟹P_0(2V_0{)}^{\gamma }=P_2(V_0{)}^{\gamma }$

$⟹P_2=(\frac{2V_0}{V_0}{)}^{\gamma }$ $P_0=(2{)}^{\gamma }{P}_0$

Hence $\frac{(P_2{)}_B}{(P_2{)}_A}=$ Ratio of final pressure $=\frac{(2{)}^{\gamma }{P}_0}{2P_0}=2^{\gamma -1}$

where, $\gamma$ is the ratio of specific heat capacities for the gas.