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Physics Question on Thermodynamics

During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio of CpCv\frac{C_p}{C_v} for the gas is :

A

53\frac{5}{3}

B

32\frac{3}{2}

C

75\frac{7}{5}

D

97\frac{9}{7}

Answer

32\frac{3}{2}

Explanation

Solution

Given:
P∝T3  ⟹  PT−3=constant.P \propto T^3 \implies P T^{-3} = \text{constant}.

From the adiabatic relation:
PVγ=constant.P V^\gamma = \text{constant}.

Using the ideal gas law:
P(nRTP)γ=constant.P \left(\frac{nRT}{P}\right)^\gamma = \text{constant}.

Simplify:
P1−γTγ=constant.P^{1-\gamma} T^\gamma = \text{constant}.

Substitute P∝T3P \propto T^3:
P1−γTγ=T3  ⟹  P1−γTγ−3=constant.P^{1-\gamma} T^\gamma = T^3 \implies P^{1-\gamma} T^{\gamma - 3} = \text{constant}.

Reorganize to find the relationship between γ\gamma and the exponents:
P1−γTγ−3=constant.P^{1-\gamma} T^{\gamma - 3} = \text{constant}.

Equating powers of TT:
γ1−γ=−3.\frac{\gamma}{1 - \gamma} = -3.

Solve for γ\gamma:
γ=−3+3γ.\gamma = -3 + 3\gamma.

Simplify:
3=2γ  ⟹  γ=32.3 = 2\gamma \implies \gamma = \frac{3}{2}.

The Correct answer is: 32\frac{3}{2}