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Chemistry Question on Thermodynamics

During a reversible adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio CPCV\frac{C_P}{C_V} for the gas is

A

32\frac{3}{2}

B

72\frac{7}{2}

C

53\frac{5}{3}

D

97\frac{9}{7}

Answer

32\frac{3}{2}

Explanation

Solution

For a reversible adiabatic process, pTγ1−γ=p T^{\frac{\gamma}{1-\gamma}}= constant ....(i) According to question, during a reversible adiabatic process the pressure of a gas is found to be proportional to the cube of its absolute temperature. p∝T3p \propto T^{3} pT−3=p T^{-3}= constant .... (ii) Equating equation (i) and (ii), we get γ1−γ=−3\frac{\gamma}{1-\gamma} =-3 γ=−3(1−γ)\gamma =-3(1-\gamma) γ=−3+3γ\gamma =-3+3 \gamma −2γ=−3-2 \gamma =-3 γ=32\gamma =\frac{3}{2} As we know, the ratio of molar heat capacities at constant pressure and constant volume is respectively by γ\gamma. So, the ratio of CpCv\frac{C_p}{C_v} for the gas is 32\frac{3}{2}.