Question

Match the List-I with List-II.

Choose the correct answer from the options given below:

• A. A-III, B-IV, C-I, D-II
• B. A-III, B-II, C-IV, D-I
• C. A-II, B-IV, C-I, D-III
• D. A-IV, B-II, C-III, D-I

Hint:

For ideal gases, the ratio \( \frac{C_P}{C_V} \) is determined by the number of degrees of freedom of the gas molecules. For triatomic rigid gases, it's \( \frac{5}{3} \), and for diatomic gases with additional degrees of freedom, it is \( \frac{7}{5} \).

Answer 1

The Correct Option is A

1. Triatomic rigid gas (A): For a rigid triatomic gas, the ratio \( \frac{C_P}{C_V} \) is \( \frac{5}{3} \) because there are no additional degrees of freedom for rotation or vibration. Thus, A matches with I. 
2. Diatomic non-rigid gas (B): For a non-rigid diatomic gas, the ratio \( \frac{C_P}{C_V} \) is \( \frac{7}{5} \), so B matches with II. 
3. Monoatomic gas (C): For a monoatomic gas, the ratio \( \frac{C_P}{C_V} \) is \( \frac{4}{3} \), corresponding to C matching with III. 
4. Diatomic rigid gas (D): For a rigid diatomic gas, the ratio \( \frac{C_P}{C_V} \) is \( \frac{9}{7} \), matching with D matching with IV. 
Final Answer $\text{A-III, B-IV, C-I, D-II}$.

Answer 2

Given the formula for the adiabatic index \(\gamma\): \[ \gamma = 1 + \frac{2}{f} \] Where \( f \) is the degrees of freedom: - For a triatomic rigid gas, \( f = 6 \): \[ \gamma = 1 + \frac{2}{6} = \frac{4}{3} \] - For a diatomic non-rigid gas, \( f = 7 \): \[ \gamma = 1 + \frac{2}{7} = \frac{9}{7} \] - For a diatomic rigid gas, \( f = 5 \): \[ \gamma = 1 + \frac{2}{5} = \frac{7}{5} \] - For a monoatomic rigid gas, \( f = 3 \): \[ \gamma = 1 + \frac{2}{3} = \frac{5}{3} \] \[ \boxed{\gamma = \frac{5}{3}} \]