An ideal gas with pressure P, volume V and temperature T is expanded isothermally to a volume 2V and a final pressure `P_i,` If the same gas is expanded adiabatically to a volume 2V, the final pressure `P_a.` The ratio of the specific heats of the gas is 1.67. The ratio `(P_a)/(P_1)` is .......

Options

A. `2^(-1//3)`
B. `2^(1//3)`
C. `2^(2//3)`
D. `2^(-2//3)`

Answer

D

Explanation

For isothermal process:
`PV = P_(i). 2V`
`P =2P_(i)`………(i)
For adiabatic process:
`PV^(gamma) = P_(a)(2V)^(gamma)` (`therefore` for monatomic gas `gamma = 5//3`)
or, `2P_(i)V^(5/3) = P_(a)(2V)^(5/3)` [From (i)]
`rArr P_(a)/P_(i) =2/2^(5/3) rArr P_(a)/P_(i) = 2^(-2/3)`