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. 0.43
B. 0.53
C. 0.63
D. 0.73

Answer

C

Explanation

When it expands isothermally,
`PV=P_(i)(2V)" "thereforeP_(i)=(PV)/(2V)=P/2`
When it expands adiabatically, `PV^(r)` = constant
`PV^(1.67)=(P_(a))(2V)^(1.67)`
`thereforeP_(a)=(V/(2V))^(1.67)P=(1/2)^(1.67)P=P/2^(1.67)`
`thereforeP_(a)/P_(i)=P/2^(1.67)xx1/(P(1/2))=2/2^(1.67)`
`=2/3.18=0.628=0.63`