Let's break down the solution into three parts as requested. --- ## **Step 1: Calculate \( C_{P,m} \) by the Classical Equipartition Theorem (at 298 K and 1 atm)** **For a nonlinear molecule:** Total degrees of freedom \( = 3N \) - **Translational:** 3 - **Rotational:** 3 (nonlinear) - **Vibrational:** \( 3N - 6 \) **Each degree of freedom contributes:** - Translational/Rotational: \( \frac{1}{2}R \) per degree - Vibrational: \( R \) per vibrational mode (since each vibrational mode has both kinetic and potential energy) So, for one mole: \[ C_{V,m} = \left(\text{Trans} + \text{Rot} + \text{Vib}\right) = \left(\frac{3}{2}R + \frac{3}{2}R + (\text{number of vibrational modes}) \times R \right) \] \[ C_{V,m} = 3R + (\text{number of vibrational modes}) \times R \] \[ C_{P,m} = C_{V,m} + R = 4R + (\text{number of vibrational modes}) \times R \] ### **Calculations:** #### **Methane, CH₄:** - \( N = 5 \) - Vibrational modes: \( 3N - 6 = 9 \) \[ C_{P,m} = 4R + 9R = 13R \] \[ C_{P,m} = 13 \times 8.314 = 108.1~\text{J/mol·K} \] #### **Ethylene, C₂H₄:** - \( N = 6 \) - Vibrational modes: \( 3N - 6 = 12 \) \[ C_{P,m} = 4R + 12R = 16R \] \[ C_{P,m} = 16 \times 8.314 = 133.0~\text{J/mol·K} \] **End of Step 1:** **Classical predictions:** - CH₄: **108.1 J/mol·K** - C₂H₄: **133.0 J/mol·K** --- ## **Step 2: Compare with Experimental Values and Find Vibrational Contribution at 298 K** ### **Experimental values at 298 K (from literature):** - **CH₄(g):** \( C_{P,m,exp} \approx 35.7~\text{J/mol·K} \) - **C₂H₄(g):** \( C_{P,m,exp} \approx 43.6~\text{J/mol·K} \) ### **Classical vibrational contribution (from Step 1):** - For CH₄: Out of 13R, 9R is vibrational. - For C₂H₄: Out of 16R, 12R is vibrational. ### **Actual vibrational contribution at 298 K:** - In reality, most vibrational modes are not excited at 298 K, so their contribution is much less. ### **Calculate percent of measured value that arises from vibrational motions:** \[ \text{Vibrational contribution, } C_{P,m,vib} = C_{P,m,exp} - C_{P,m,trans+rot} \] Where \( C_{P,m,trans+rot} = 4R \approx 33.3~\text{J/mol·K} \) (for both molecules, as they are both nonlinear). #### **CH₄:** \[ C_{P,m,vib} = 35.7 - 33.3 = 2.4~\text{J/mol·K} \] \[ \text{Percent from vibration} = \frac{2.4}{35.7} \times 100\% = 6.7\% \] #### **C₂H₄:** \[ C_{P,m,vib} = 43.6 - 33.3 = 10.3~\text{J/mol·K} \] \[ \text{Percent from vibration} = \frac{10.3}{43.6} \times 100\% = 23.6\% \] **End of Step 2:** - **CH₄:** About **6.7%** of experimental Cp comes from vibrations at 298 K - **C₂H₄:** About **23.6%** of experimental Cp comes from vibrations at 298 K --- ## **Step 3: Repeat at 800 K and 1 atm** ### **At high T (800 K), vibrational modes are more thermally populated.** #### **Use the same formula:** \[ C_{P,m} = C_{P,m,trans+rot} + C_{P,m,vib} \] ##### **Assuming most vibrational modes are now excited:** - For CH₄: All 9 modes contribute, so \( C_{P,m,vib} \approx 9R \) - For C₂H₄: All 12 modes contribute, so \( C_{P,m,vib} \approx 12R \) \[ C_{P,m}(CH_4, 800~K) \approx 4R + 9R = 13R = 108.1~\text{J/mol·K} \] \[ C_{P,m}(C_2H_4, 800~K) \approx 4R + 12R = 16R = 133.0~\text{J/mol·K} \] ### **Compare with experimental values at 800 K:** - **CH₄(g):** Experimental \( C_{P,m} \) at 800 K ≈ **63 J/mol·K** - **C₂H₄(g):** Experimental \( C_{P,m} \) at 800 K ≈ **79 J/mol·K** ### **Vibrational contribution at 800 K:** \[ C_{P,m,vib} = C_{P,m,exp} - 33.3~\text{J/mol·K} \] #### **CH₄:** \[ C_{P,m,vib} = 63 - 33.3 = 29.7~\text{J/mol·K} \] \[ \text{Percent from vibration} = \frac{29.7}{63} \times 100\% = 47.1\% \] #### **C₂H₄:** \[ C_{P,m,vib} = 79 - 33.3 = 45.7~\text{J/mol·K} \] \[ \text{Percent from vibration} = \frac{45.7}{79} \times 100\% = 57.9\% \] **End of Step 3:** - **CH₄:** About **47%** of Cp at 800 K is vibrational - **C₂H₄:** About **58%** of Cp at 800 K is vibrational --- ## **Final Summary Table** | Molecule | Classical \( C_{P,m} \) (J/mol·K) | Experimental (298 K) | Vibrational % (298 K) | Experimental (800 K) | Vibrational % (800 K) | |----------|----------------------------------|----------------------|-----------------------|----------------------|-----------------------| | CH₄ | 108.1 | 35.7 | 6.7% | 63 | 47% | | C₂H₄ | 133.0 | 43.6 | 23.6% | 79 | 58% | --- ### **Explanations** - **Step 1:** Used the equipartition theorem to predict \( C_{P,m} \) values, assuming all degrees of freedom (including vibrational) are fully excited. This overestimates real values at moderate temperatures due to quantum effects. - **Step 2:** Compared to experimental values at 298 K, most vibrational modes are not excited, so vibrational contribution is much smaller than classical prediction. - **Step 3:** At higher temperature (800 K), more vibrational modes are populated, so the vibrational contribution is much larger, but still less than the full classical prediction for typical organic molecules. --- ### **Final Answers** - **At 298 K:** - **CH₄:** Classical Cp = 108.1, Experimental = 35.7, Vibration ≈ 6.7% - **C₂H₄:** Classical Cp = 133.0, Experimental = 43.6, Vibration ≈ 23.6% - **At 800 K:** - **CH₄:** Experimental Cp ≈ 63, Vibration ≈ 47% - **C₂H₄:** Experimental Cp ≈ 79, Vibration ≈ 58%

The solution relies on the classical equipartition theorem, which predicts molar heat capacities by assuming each quadratic degree of freedom contributes \( \frac{1}{2} R \) (for translational and rotational modes) or \( R \) (for vibrational modes) to the molar heat capacity. For molecules like CH₄ and C₂H₄, the total degrees of freedom are calculated based on their molecular structure, with vibrational modes determined by \( 3N - 6 \). At 298 K, quantum effects limit vibrational excitation, so the experimentally observed heat capacities are much lower than the classical predictions, indicating minimal vibrational contribution. As temperature increases to 800 K, vibrational modes become thermally accessible, resulting in a larger vibrational contribution to the heat capacity. By comparing the classical predictions with experimental data at both temperatures, we can estimate the percentage of the heat capacity arising from vibrational motions, highlighting how vibrational excitation depends on temperature and quantum effects.