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

To solve this problem, we need to understand the relationship between pressure \(P\), temperature \(T\), and the specific heat ratio \(\gamma = \frac{C_p}{C_v}\) during an adiabatic process. Given that the pressure of the gas is proportional to the cube of its absolute temperature, we can express this relationship as:

\(P \propto T^3\) 

For an adiabatic process, the relation between pressure and temperature is given by:

\(PT^{-\frac{\gamma}{\gamma - 1}} = \text{constant}\)

Since \(P \propto T^3\), we can equate the exponents to get:

\(T^3 \cdot T^{-\frac{\gamma}{\gamma - 1}} = \text{constant}\)

\(3 - \frac{\gamma}{\gamma - 1} = 0\)

Solving the above equation:

\(3 = \frac{\gamma}{\gamma - 1}\)

\(3(\gamma - 1) = \gamma\)

\(3\gamma - 3 = \gamma\)

\(2\gamma = 3\)

\(\gamma = \frac{3}{2}\)

This gives us the specific heat ratio \(\frac{C_p}{C_v} = \gamma = \frac{3}{2}\).

Therefore, the correct answer is:

\(\frac{3}{2}\)

Explanation of Options:

• The option \(\frac{5}{3}\) corresponds to diatomic gases with additional energy degrees. Our derivation does not support this value for the given condition.
• The option \(\frac{3}{2}\) was derived based on the given condition and is correct.
• The option \(\frac{7}{5}\) is typically related to diatomic gases as well, which don't match the condition stated.
• The option \(\frac{9}{7}\) does not fit the temperature-proportional relationship provided.

Answer 2

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: \(\frac{3}{2}\)