An ideal gas with pressure P, volume V and temperature T is expanded isothermally to a volume 2V and a final pressure `P_(1)`. The same gas is expanded adiabatically to a volume 2V, the final final pressure 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. `2gamma`

Answer

A

Explanation

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