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

Options

A. `2^(gamma-1)`
B. `(1/2)^(gamma-1)`
C. `((1)/(1-gamma))^2`
D. `((1)/(gamma-1))^2`

Answer

A

Explanation

Consider the p-V diagram shown for the container A (isothermal) and for the container B (adiabatic).

Both the process involving compression of the gas.
For isothermal process (gas A) (during `1 rarr 2`)
`p_1 V_1=p_2 V_2`
`implies p_0(2V_0)=p_2(V_0)`
`implies p_2=2p_0`
For adiabatic process. (gas B) (during `1 rarr 2`)
`p_1V_1^(gamma)=p_2V_2^(gamma)`
`implies p_0(2V_0)^(gamma)=p_2(V_0)^(gamma)`
`implies p_2=((2V_0)/(V_0))^(gamma) p_0 = (2)^(gamma) p_0`
Hence, `((p_2)_B)/((p_2)_A)`= Ratio of pressure = `((2)^(gamma)p_0)/(2p_0)=2^(gamma-1)` where, `gamma` is ratio of specific heat capacities for the gas.