An ideal gas with pressure P, volume V and temperature T is expanded isothermally to a volume 2V and final pressure `P_(I)`. The same gas is expanded adiabatically to a volume 2V and final pressure in this case is `P_(A)`. In terms of the ratio of the two specific heats for the gas `gamma`, the ratio `P_(I)//P_(A)` is

Options

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

Answer

A

Explanation

For an isothermal process
PV = constant `:. PV = P_(i) 2V `
`P_(I) = (P)/(2) `
For an adiabatic process
`PV^(gamma) ` = constant `:. PV^(gamma) = P_(A) (2V )^(gamma)`
or `P_(A) = (P)/(2^(gamma))`
Divide (i) by (ii) we get
`(P_(I))/(P_(A)) = (2^(gamma))/(2) = 2^(gamma-1)`