**Vibrational Contribution to Heat Capacity of Ideal Gases****i) Deriving the Expression for Vibrational Contribution:**Starting with the expression for the average vibrational energy including zero-point energy, derive the following expression for the contribution of vibrational motion to the constant volume heat capacity of an ideal gas:\[\overline{C}_{V,\text{vib}} = R \left( \frac{\Theta_{\text{vib}}}{T} \right)^2 \frac{e^{-\Theta_{\text{vib}}/T}}{(1 - e^{-\Theta_{\text{vib}}/T})^2}\]**ii) Calculating for Nitrogen:**Determine the vibrational contribution to the heat capacity of nitrogen (\(\tilde{\nu} = 2100 \, \text{cm}^{-1}\)) at 375K. Compare this to the value from the high-temperature equipartition value of \(\frac{1}{2}R\), and explain why your calculated value is similar or different.

To determine the expression for the contribution of vibrational motion to the heat capacity of an…

i) Starting with the expression for the average vibrational energy including zero-point energy, derive the expression below for the contribution of vibrational motion to the constant volume heat capacity of an ideal gas. Cv,vib = R -R evib T 2 e-vib/T (1 – e-Ovib/T)²

i) Starting with the expression for the average vibrational energy including zero-point energy, derive the expression below for the contribution of vibrational motion to the constant volume heat capacity of an ideal gas. Cv,vib = R -R evib T 2 e-vib/T (1 – e-Ovib/T)²

Chemistry

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ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

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Question

**Vibrational Contribution to Heat Capacity of Ideal Gases**

**i) Deriving the Expression for Vibrational Contribution:**

Starting with the expression for the average vibrational energy including zero-point energy, derive the following expression for the contribution of vibrational motion to the constant volume heat capacity of an ideal gas:

\[
\overline{C}_{V,\text{vib}} = R \left( \frac{\Theta_{\text{vib}}}{T} \right)^2 \frac{e^{-\Theta_{\text{vib}}/T}}{(1 - e^{-\Theta_{\text{vib}}/T})^2}
\]

**ii) Calculating for Nitrogen:**

Determine the vibrational contribution to the heat capacity of nitrogen (\(\tilde{\nu} = 2100 \, \text{cm}^{-1}\)) at 375K. Compare this to the value from the high-temperature equipartition value of \(\frac{1}{2}R\), and explain why your calculated value is similar or different.

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