Question

Match the following (f is the number of degrees of freedom):

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

Hint:

When matching gases with heat capacity ratios, remember that monoatomic gases have \( \gamma = \frac{5}{3} \), rigid diatomic gases have \( \frac{7}{5} \), and polyatomic gases have lower \( \gamma \) values due to additional vibrational modes.

Answer 1

The Correct Option is D

Step 1: Understanding Degrees of Freedom and Heat Capacity Ratios The ratio of specific heats \( \frac{C_p}{C_v} \) for different gases is derived from the degrees of freedom \( f \) using: \[ \gamma = \frac{C_p}{C_v} = \frac{f+2}{f} \] where \( f \) is the number of degrees of freedom of a molecule. Step 2: Identifying Correct Values - Monoatomic gases (e.g., noble gases) have \( f = 3 \), so: \[ \gamma = \frac{3+2}{3} = \frac{5}{3} \quad \Rightarrow \quad \text{(II)} \] - Diatomic (rigid) gases (e.g., \( O_2, N_2 \) at low temperatures) have \( f = 5 \), so: \[ \gamma = \frac{5+2}{5} = \frac{7}{5} \quad \Rightarrow \quad \text{(III)} \] - Diatomic (non-rigid) gases (considering vibrational motion) have \( f = 6 \), so: \[ \gamma = \frac{6+2}{6} = \frac{4+f}{3+f} \quad \Rightarrow \quad \text{(IV)} \] - Polyatomic gases (e.g., \( CO_2, H_2O \)) have more vibrational modes, and for typical cases: \[ \gamma = \frac{9}{7} \quad \Rightarrow \quad \text{(I)} \] Step 3: Correct Matching \[ A \rightarrow II, \quad B \rightarrow III, \quad C \rightarrow IV, \quad D \rightarrow I \] Thus, the correct answer is \( \mathbf{A-II, B-III, C-IV, D-I} \).