Let's solve each part-by-step using the provided graph and physical principles: --- ## (a At 500 K, how many active degrees of freedom does H₂ have? **Step 1:** From the graph, at 500 K, the value of \( \hat{C}_V \) for H₂ is very close to \( 5/2 \). **Step 2:** For an ideal diatomic gas: - Translational degrees of freedom = 3 - Rotational degrees of freedom = 2 (for H₂, a linear molecule) Each degree of freedom contributes \( \frac{1}{2}R \) to \( C_V \): \[ C_V = \frac{\text{# degrees of freedom}}{2} R \] So for H₂: \[ \frac{5}{2}R \implies 5 \text{ degrees of freedom are active at 500 K} \] **Final Answer:** **5 active degrees of freedom** for H₂ at 500 K. --- ## (b) At 500 K, how many active degrees of freedom does I₂ have (I: iodine)? **Step 1:** From the graph, at 500 K, the value of \( \hat{C}_V \) for I₂ is close to \( 7/2 \). **Step 2:** \[ \frac{7}{2}R \implies 7 \text{ active degrees of freedom} \] **Final Answer:** **7 active degrees of freedom** for I₂ at 500 K. --- ## (c) At 500 K, what kind of active degrees of freedom does I₂ have that H₂ does not have? **Explanation:** - Both H₂ and I₂ have 3 translational and 2 rotational degrees of freedom. - At 500 K, I₂ also has **2 vibrational degrees of freedom** contributing to heat capacity (each vibrational mode counts as 2 degrees due to kinetic and potential energy). - For H₂, vibrational modes are not thermally populated at 500 K due to high vibrational energy spacings. **Final Answer:** **Vibrational degrees of freedom** are active in I₂ at 500 K, but not in H₂. --- ## (d) What is the constant-pressure heat capacity per mole for H₂ and I₂ at 500 K? (In terms of \( R \)) Recall: \[ C_P = C_V + R \] - For H₂: \( C_V = \frac{5}{2}R \) - For I₂: \( C_V = \frac{7}{2}R \) **So:** - \( C_P(\text{H}_2) = \frac{5}{2}R + R = \frac{7}{2}R \) - \( C_P(\text{I}_2) = \frac{7}{2}R + R = \frac{9}{2}R \) **Final Answers:** - **H₂:** \( C_P = \frac{7}{2}R \) - **I₂:** \( C_P = \frac{9}{2}R \) --- ## (e) Speculate on what might explain the increase in \( \hat{C} \) above \( 7/2 \) for the diatomic halogen gases at high temperature. **Explanation:** - The classical maximum for a rigid diatomic molecule is \( 7/2 R \) (3 translational, 2 rotational, and 1 vibrational mode). - An increase above \( 7/2 R \) at high \( T \) suggests **higher vibrational modes** are being excited (overtones, anharmonic effects), or **additional electronic excitations** are becoming thermally accessible. - **Anharmonicity** and **excitation of higher vibrational states** increase the heat capacity above the simple \( 7/2 \) value. **Final Answer:** **The increase in \( \hat{C} \) above \( 7/2 \) at high temperature is due to the excitation of higher vibrational states (overtones, anharmonic effects) and possibly low-lying electronic excitations.** --- ## **Summary Table** | Part | Answer | |------|--------| | (a) | **5 active degrees of freedom (H₂ at 500 K)** | | (b) | **7 active degrees of freedom (I₂ at 500 K)** | | (c) | **Vibrational degrees of freedom (I₂, not H₂)** | | (d) | **H₂: \(\frac{7}{2}R\), I₂: \(\frac{9}{2}R\)** | | (e) | **Excitation of higher vibrational/electronic states at high T** | If you need further explanation on any part, let me know!