Question

Two identical cylinders each contain the same amount of the same ideal gas with the same initial temperature, Tlow. The gas in the first cylinder undergoes an isovolumetric pressure increase and reaches a final temperature of Thigh. The gas in the second cylinder undergoes an isobaric expansion reaching the same final temperature, Thigh. What is the ratio of the change of entropy of the gas in the first cylinder to the change of entropy of the gas in the second cylinder? (∆S1/∆S2 = ?)

Answer 1

1st cylinder :

change at constant volume

we know that

at constant volume

dS = nCv ln (T2 / T1)

so

in this case

dS1 = nCv ln (Thigh / T1)

2nd cylinder :

change at constant pressure

we know that

at constant pressure

dS = nCp ln (Thigh / T1)

so

in this case

dS2 = nCp ln ( Thigh / T1)

now

dS1 / dS2 = nCv ln (Thigh / T1) / nCp ln ( Thigh / T1)

so

dS1 / dS2 = Cv / Cp

so

the ratio os Cv / Cp of the gas

where

Cv = molar specific heat at constant volume

Cp = molar specific heat at constant pressure