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 $2^{\gamma -1}$


When the compression is isothermal for gas in $A$,
$p_2{V}_2=p_1{V}_1$
$p_2=\frac{p_1{V}_1}{V_2}=p_1\frac{V_1}{V_1/2}=2p_1$

For gas in $B$, when compression is adiabatic,
$p_2^{\prime }{V}_2^{\gamma }=p_1{V}_1^{\gamma }$
$p_2^{\prime }=p_1{\left(\frac{V_1}{V_2}\right)}^{\gamma }=2^{\gamma }{P}_1$
$\frac{p_2^{\prime }}{p_2}=\frac{2^{\gamma }{p}_1}{2p_1}=2^{\gamma -1}$