Question

Consider two containers A and B containing 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 volume 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. $\frac{1}{\gamma -1}$

Answer 1

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

Given data:

$P_{A1}=P_{B1};V_{B1}=V_{A1};T_{A1}=T_{B2}$

Let the final pressure of gas in A be $P_{A2}$ and gas in B be $P_{B2}$.

For isothermal compression of gas in A,

$P_{A1}{V}_{A1}=P_{A2}{V}_{A2}$

$P_{A2}=\frac{P_{A1}{V}_{A1}}{V_{A2}}=\frac{P_{A1}{V}_{A1}}{V_{A1}/2}=2P_{A1}$

For adiabatic compression of gas in B,

$P_{B2}{V}_{B1}^{\gamma }=P_{B2}{V}_{B2}^{\gamma }$

$P_{B2}=P_{B1}{\left(\frac{V_{B1}}{V_{B2}}\right)}^{\gamma }=P_{B1}{2}^{\gamma }$

So, $\frac{P_{B2}}{P_{A2}}=\frac{P_{B1}{2}^{\gamma }}{2P_{A1}}=2^{\gamma -1}$