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

Answer: (A)

2$\gamma$–1

Answer 2

The gas in container A is compressed isothermally, $\therefore \mathrm { P } _ { 1 } \mathrm {~V} _ { 1 } = \mathrm { P } _ { 2 } \mathrm {~V} _ { 2 }$

or $\mathrm { P } _ { 2 } = \frac { \mathrm { P } _ { 1 } \mathrm {~V} _ { 1 } } { \mathrm {~V} _ { 2 } } = \mathrm { P } _ { 1 } \frac { \mathrm { V } _ { 1 } } { \mathrm {~V} _ { 1 } / 2 } = 2 \mathrm { P } _ { 1 } \quad \left( \because \mathrm { V } _ { 2 } = \mathrm { V } _ { 1 } / 2 \right)$

again the gas in container B is compressed adiabatically,

$\therefore \mathrm { P } _ { 1 } \mathrm {~V} _ { 1 } ^ { \gamma } \mathrm { P } _ { 2 } ^ { \prime } \left( \mathrm { V } _ { 2 } ^ { \prime } \right) ^ { \gamma }$

Hence $\frac { \mathrm { P } _ { 2 } ^ { \prime } } { \mathrm { P } _ { 2 } } = \frac { 2 ^ { \gamma } \mathrm { P } _ { 1 } } { 2 \mathrm { P } _ { 1 } } = 2 ^ { \gamma - 1 }$