Question

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

• A. \(\frac{5}{3}\)
• B. \(\frac{3}{2}\)
• C. \(\frac{7}{5}\)
• D. \(\frac{9}{7}\)

Answer 1

The Correct Option is B

Given:
\(P \propto T^3 \implies P T^{-3} = \text{constant}.\)

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

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

Simplify:  
\(P^{1-\gamma} T^\gamma = \text{constant}.\)

Substitute \( P \propto T^3 \):
\(P^{1-\gamma} T^\gamma = T^3 \implies P^{1-\gamma} T^{\gamma - 3} = \text{constant}.\)

Reorganize to find the relationship between \( \gamma \) and the exponents:  
\(P^{1-\gamma} T^{\gamma - 3} = \text{constant}.\)

Equating powers of \(T\):
\(\frac{\gamma}{1 - \gamma} = -3.\)

Solve for \( \gamma \):  
\(\gamma = -3 + 3\gamma.\)

Simplify:  
\(3 = 2\gamma \implies \gamma = \frac{3}{2}.\)

The Correct answer is: \(\frac{3}{2}\)