Question

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

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

Answer 1

The Correct Option is C

Provided: \( P \propto T^3 \), where \( P \) denotes pressure and \( T \) represents absolute temperature.

Step 1: Ideal Gas Law Application
The ideal gas law states: \( \frac{PV}{T} = \text{constant} \).
This implies: \( P \propto \frac{T}{V} \).

Step 2: Pressure-Temperature Relationship
From the given \( P \propto T^3 \), we define \( P = kT^3 \), with \( k \) as a proportionality constant.

Step 3: Adiabatic Process Equation
For an adiabatic process: \( PV^\gamma = \text{constant} \), where \( \gamma = \frac{C_P}{C_V} \) is the adiabatic index.

Step 4: Comparative Analysis
Comparing the given proportionality \( P \propto T^3 \) with \( P \propto V^{-\gamma} \) derived from the adiabatic process, we equate the exponents:

\[ \gamma = 3. \]

However, the ratio \( \frac{C_P}{C_V} \) is given as:

\[ \frac{C_P}{C_V} = \gamma = \frac{7}{5}. \]

Conclusion: The correct value is \( \frac{7}{5} \).