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

Given that:

\[ P \propto T^3, \]

where \( P \) is the pressure and \( T \) is the absolute temperature.

Step 1: Using the Ideal Gas Law
From the ideal gas law, we have:

\[ \frac{PV}{T} = nR = \text{constant}. \]

Therefore:

\[ P \propto \frac{T}{V}. \]

Step 2: Relating Pressure and Temperature
Given that:

\[ P \propto T^3, \]

we can write:

\[ P = kT^3, \]

where \( k \) is a proportionality constant.

Step 3: Applying the Adiabatic Process Equation
For an adiabatic process, the relation is given by:

\[ PV^\gamma = \text{constant}, \]

where \( \gamma = \frac{C_P}{C_V} \) is the adiabatic index.

Step 4: Comparing the Relations
From the given proportionality:

\[ P \propto T^3 \quad \text{and} \quad P \propto V^{-\gamma}. \]

Equating the exponents:

\[ \gamma = 3. \]

Thus, the ratio of \( \frac{C_P}{C_V} \) is:

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

Therefore, the correct answer is \( \frac{7}{5} \).