Question

During an adiabatic process of a monatomic gas, the relation between absolute temperature (\( T \)) and pressure (\( P \)) is \( P \propto T^X \). The value of \( X \) is

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

Hint:

For an adiabatic process, use \( P V^\gamma = \text{constant} \). Substitute \( V = \frac{n R T}{P} \) from the ideal gas law to relate \( P \) and \( T \). For a monatomic gas, \( \gamma = \frac{5}{3} \).

Answer 1

The Correct Option is D

For an adiabatic process, the relation between pressure (\( P \)) and temperature (\( T \)) can be derived using the adiabatic condition and the ideal gas law. The adiabatic condition for an ideal gas is: \[ P V^\gamma = \text{constant} \] For a monatomic gas, the adiabatic index \( \gamma = \frac{C_P}{C_V} = \frac{5}{3} \) (since \( C_V = \frac{3}{2}R \), \( C_P = C_V + R = \frac{5}{2}R \)). Using the ideal gas law \( P V = n R T \), we can express the volume as: \[ V = \frac{n R T}{P} \] Substitute \( V \) into the adiabatic condition: \[ P \left( \frac{n R T}{P} \right)^\gamma = \text{constant} \] \[ P \cdot \frac{(n R T)^\gamma}{P^\gamma} = \text{constant} \quad \Rightarrow \quad P^{1-\gamma} T^\gamma = \text{constant} \] \[ P^{1-\gamma} = \text{constant} \cdot T^{-\gamma} \quad \Rightarrow \quad P = \text{constant} \cdot T^{\frac{\gamma}{\gamma-1}} \] Given \( P \propto T^X \), the exponent \( X = \frac{\gamma}{\gamma-1} \). Substitute \( \gamma = \frac{5}{3} \): \[ \gamma - 1 = \frac{5}{3} - 1 = \frac{2}{3}, \quad X = \frac{\frac{5}{3}}{\frac{2}{3}} = \frac{5}{3} \times \frac{3}{2} = \frac{5}{2} \] So, the value of \( X \) is \( \frac{5}{2} \).

Answer 2

Step 1: Understand the given process
- The process is adiabatic for a monatomic gas.
- Relation between pressure \(P\) and temperature \(T\) is given as \(P \propto T^X\).

Step 2: Recall relevant relations for adiabatic process
For an adiabatic process,
\[ P V^\gamma = \text{constant} \] and \[ T V^{\gamma - 1} = \text{constant} \] where \(\gamma = \frac{C_p}{C_v}\) is the adiabatic index.

Step 3: Express \(V\) in terms of \(T\)
From the temperature relation, \[ T V^{\gamma - 1} = \text{constant} \implies V^{\gamma - 1} \propto \frac{1}{T} \implies V \propto T^{-\frac{1}{\gamma - 1}} \]

Step 4: Express \(P\) in terms of \(T\)
Using \(P V^\gamma = \text{constant}\), \[ P \propto V^{-\gamma} \propto \left(T^{-\frac{1}{\gamma - 1}}\right)^{-\gamma} = T^{\frac{\gamma}{\gamma - 1}} \]

Step 5: Calculate \(\gamma\) for monatomic gas
For a monatomic gas, \[ \gamma = \frac{C_p}{C_v} = \frac{5}{3} \]

Step 6: Substitute \(\gamma\) to find \(X\)
\[ X = \frac{\gamma}{\gamma - 1} = \frac{\frac{5}{3}}{\frac{5}{3} - 1} = \frac{\frac{5}{3}}{\frac{2}{3}} = \frac{5}{3} \times \frac{3}{2} = \frac{5}{2} \]

Final Answer: \[ P \propto T^{\frac{5}{2}} \] So, the value of \(X\) is \(\frac{5}{2}\).