Chemistry
10th Edition
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
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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![### Information for Linear HCN Molecule at 300K
#### Physical Properties
- **Moment of Inertia**: \(18.82 \times 10^{-47} \, \text{kg} \, \text{m}^2\)
| Vibration Type | Wavenumber (cm\(^{-1}\)) |
|-------------------------|--------------------------|
| HC-N (stretch) | 2097 |
| H-C-N (bend) (degeneracy of 2) | 713 |
| C-H (stretch) | 3311 |
#### Problems
i) **Calculate:**
a) Vibrational temperatures
b) Rotational temperatures
c) The volume per state for translational motion, \(\Lambda^3\).
ii) **Comparison:**
- Compare the values for the partition functions for each degree of freedom (D.O.F) of motion included in translation, rotation, and vibration (neglect electronic energy for now).
iii) **Given Equation:**
\[
\frac{\partial f}{\partial T} = \frac{\partial f}{\partial \beta} \ast \frac{\partial \beta}{\partial T}
\]
- Show that:
\[
\frac{\partial f}{\partial \beta} = -k_B T^2 \frac{\partial f}{\partial T}
\]
iv) **Calculations:**
a) Calculate the average energy of HCN at 300K, neglecting electronic energy. Use the result from (ii) to do this. Calculate your molecular energy and then convert to kJ/mol.
b) Compare this value to the expected value for an ideal gas from the equipartition principle at high temperature, which would apply to all D.O.F. Explain why you do or do not see a difference from the ideal, high temperature case.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faba5746d-e0e8-46ca-a0bb-ec14000055d1%2Fb4f26324-53f8-46a3-9110-6e8b87d78dbb%2Fimo55xo_processed.png&w=3840&q=75)
Transcribed Image Text:### Information for Linear HCN Molecule at 300K
#### Physical Properties
- **Moment of Inertia**: \(18.82 \times 10^{-47} \, \text{kg} \, \text{m}^2\)
| Vibration Type | Wavenumber (cm\(^{-1}\)) |
|-------------------------|--------------------------|
| HC-N (stretch) | 2097 |
| H-C-N (bend) (degeneracy of 2) | 713 |
| C-H (stretch) | 3311 |
#### Problems
i) **Calculate:**
a) Vibrational temperatures
b) Rotational temperatures
c) The volume per state for translational motion, \(\Lambda^3\).
ii) **Comparison:**
- Compare the values for the partition functions for each degree of freedom (D.O.F) of motion included in translation, rotation, and vibration (neglect electronic energy for now).
iii) **Given Equation:**
\[
\frac{\partial f}{\partial T} = \frac{\partial f}{\partial \beta} \ast \frac{\partial \beta}{\partial T}
\]
- Show that:
\[
\frac{\partial f}{\partial \beta} = -k_B T^2 \frac{\partial f}{\partial T}
\]
iv) **Calculations:**
a) Calculate the average energy of HCN at 300K, neglecting electronic energy. Use the result from (ii) to do this. Calculate your molecular energy and then convert to kJ/mol.
b) Compare this value to the expected value for an ideal gas from the equipartition principle at high temperature, which would apply to all D.O.F. Explain why you do or do not see a difference from the ideal, high temperature case.
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