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 .......

Explanation

For isothermal expansion,
`PxxV=p_(i)2V` implies `P_(1)=(P)/(2)`
For adiabatic expansion,
`PV^(y)=P_(a)xx(2V)^(y) implies P_(a)=(P)/(2^(y))=(P)/(2^(1.67))`
`:. (P_(a))/(P_(i))=(P)/(2^(1.67))xx(2)/(P)=(1)/(2^(0.67))`