A given mass of a gas is compressed isothermally untill its pressure is doubled. It is then allowed to expand adiabatically until its original volume is restored and its pressure is then found to be 0.75 of its initial pressure . The ratio of the specific heats of the gas is approximately .

Options

A. `1.20`
B. 1.41
C. 1.67
D. 1.83

Answer

B

Explanation

Isothermal compression : `P_(L) =2P_(1),T- Constant `
`P_(1) V_(1) =P_(2) V_(2)`
`implies P_(1) V_(1) = (2P_(1)) V_(2)`
`therefore (V_(1)) /(V_(2)) = 2` …(1)
Adiabatic expension :` V_(2) = 2V_(1), P_(2) = 0.75 P_(1)`
`P_(1) V_(1)^(gamma) = P_(2) V_(2)^(gamma)`
`implies (2P)V_(1)^(gamma) = (0.75 P) (2 V_(1))^(gamma)`
` implies 2 = 0.75 xx 2^(gamma)`
`implies (8)/(3) = 2^(gamma)`
`implies log ((8)/(3)) = log (2^(gamma))`
`implies log8 - log3 = gamma log 2`
`therefore gamma = (log 8 - log 3 )/(log 2) = 1.41`