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 an isothermal process,
PV = constant
`P_(1)V_(1) = P_(2)V_(2)`
`PV = P_(1)(2V)`
`rArr (P)/(P_(1)) = 2" "...(i)`
While for an adiabatic process `PV^(gamma)` = constant
`PV^(gamma) = P_(2)(2V)^(gamma)`
`rArr (P)/(P_(2))= 2^(gamma)" "...(ii)`
Dividing equation (i) by equation (ii), we get,
`((P_(2))/(P_(1)))=(2)/(2^(gamma))=(1)/(2^(gamma-1))=(1)/(2^(0.67))`