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 is container B is compressed to half of its original vlue adiabatically. The ratio of final pressure of gas of 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 container B (adiabatic). ltBrgt
Both the process involving compression of the gas.
For isothermal process (gas A) (during 1`to`2)
`p_(1)V_(1)=p_(2)V_(2)`
`rArr" "p_(0)(2V_(0))=p_(2)(V_(0))`
`rArr" "p_(2)=2p_(0)` ltBrgt For adiabatic process, (gas B) (during `1to2`)
`rArr" "p_(0)(2V_(0))^(gamma)=p_(2)(V_(0))^(gamma)`
`rArrp_(2)=((2V_(0))/(V_(0)))^(gamma)p_(0)=(2)^(gamma)p_(0)`
Hence, `((p_(2))_(B))/((p_(2))_(A))` = Ratio of final pressure = `((2)^(gamma)p_(0))/(2p_(0))= 2^(gamma-1)`
where, `gamma` is ratio of specific heat capaacities for the gas.