# 6: Computation of Thermodynamic Quantities

- Page ID
- 62029

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## Computation of Thermodynamic Quantities Based on Molecular Structure Calculations

## Introduction:

The thermochemistry of common substances is largely known through carefully measured experiment. Thermochemical data for substances are available in data tables in both written and digital formats (e.g. Chemical and Engineering Data Reference Tables, CRC Handbook). Knowledge of the standard values of state functions such as the heat of formation, heat capacity, molar entropy, and free energy are useful in determining appropriate reaction conditions. This laboratory assignment is designed to provide a basic introduction to modern methods of computational thermochemistry, to thermochemical data, and to a comparison of the accuracy of computed values to experimental data. Note that this laboratory can be done in shifts, since there are only three machines with Spartan loaded in the PChem lab.

## Objectives:

- Learn to use computational chemistry software to optimize geometries and to model molecular behavior.
- Perform computations of thermodynamic functions using equipartition theory and compare to literature values.
- Evaluate the performance of various theoretical models used in computational chemistry with regards to thermochemistry

## Procedure:

(Equipment Required: Computer with Spartan)

For this lab, you will compute IR vibrational frequencies and thermodynamic state function values for different molecules using several theoretical techniques. You will then compare the results to literature values for ideal and real gases as well as compare the performance of different theoretical models.

### Running Spartan for Thermochemical Calculations.

- Make a new file.
- Build molecule by selecting atoms and clicking on screen.
- Go to Setup Calculations, Select “Single Point Energy”, depending on your case: molecular mechanics, semi-empirical, and Hartree-Fock (HF/3-21G), check IR and thermodynamics. You need not change anything else.
- Go to submit and “submit”.
- Use Monitor to see if model is executing. (Should be very fast so might not be necessary).
- Go to display and record thermodynamic values. Record the “contributions” to enthalpy, entropy, Cv … Note that Total Enthalpy = Total vibration + ideal gas + translation + rotation.
- Go to display spectra and write down the vibrational frequencies.

### Computational Thermochemistry Simulations

- Build and minimize molecular models for CH
_{2}O, HCN, CO_{2}and CH_{4}using molecular mechanics, semi-empirical, and Hartree-Fock (HF/3-21G*). - Tabulate the thermochemical data for each simulation.
- Tabulate the vibrational (IR) data for each simulation.
- Perform a literature search to identify the thermodynamic constants and the vibrational frequencies of each molecule. Evaluate the % error, bias, and average absolute error for the thermodynamic constants and separately for the vibrational bands from literature values.
- Compare and discuss the differences between the calculated thermodynamic values for the enthalpy, entropy and molar heat capacity and their corresponding literature values for each molecule. [You should construct a simple table of values]
- Discuss the trends that you observe. For example, what was the trend in the %error for each constant as a function of computational method? What about the other statistical measures?
- Compare and discuss the differences between the calculated thermodynamic values for the vibrational frequencies for each molecule. [You should construct a simple table of values here as well] Discuss the trends that you observe. For example, what was the trend in errors for each constant as a function of computational method? How did the errors vary by molecular size? How did the errors vary as a function of vibrational mode?
- Compute the vibrational contributions to the molar heat capacities of hydrogen cyanide and methane using a) the computed values of the vibrational frequencies and b) the literature values of the vibrational frequencies. To what do you attribute the differences in these quantities? Represent your data in tabulated form. Explain the trends and discuss the results. HINT: Use the vibrational temperatures for each gas to compute the molar heat capacities and note that the total vibrational heat capacity is equal to the sum of the heat capacities for the individual vibrational degrees of freedom.

The vibrational temperature, \(\Theta _{vib}\), is defined by the equation:

\[\Theta _{vib}=\frac{hc\nu}{k}\]

where h is Planck's constant, c is the speed of light, v is the frequency and k is Boltzmann's constant.

The vibrational contribution to the molar heat capacity from a given vibrational degree of freedom of frequency ν is given by:

\[\left ( C_{v} \right )_{vib}=N\times R\times \left ( \frac{ \Theta _{vib}}{T} \right )^{2}\times \frac{\exp \left ( \Theta _{vib}/T \right )}{ \left \{ \exp \left ( \Theta _{vib}/T \right ) -1 \right \}^{2}}\]

where N is the number of moles, R is the ideal gas constant and T is temperature.

The total contribution to the heat capacity at constant volume from all vibrational degrees of freedom is:

\[ C_{vib, Total}=\sum \left ( C_{v} \right )_{vib}\]

where the summation is over all vibrational degrees of freedom, 3N-5 (linear molecules) or 3N-6 (non-linear molecules)

- The speed of sound in any gas can be computed from the expression:

\[v_{sound}= \left \{\left ( \frac{C_{p}}{C_{v}} \right ) \times \frac{RT}{M} \right \}^{1/2}\]

Where C_{p} is the heat capacity at constant pressure, C_{v} is the heat capacity at constant volume and M is the molecular mass of the gas. Use your computed values of molar C_{V} and C_{P} to derive a theoretical speed of sound for each of your gases and compare to each other and to their literature values. Evaluate the %error, bias, and average absolute error for the theoretical and “literature-derived” molar heat capacities. What trends, if any, do you observe both amongst the four gases (within a particular theoretical framework) and between individual gases computed across the framework. For the “literature-derived” sound velocities, use the tabulated values of the molar CP and CV at the standard temperature and pressure values of 298 K and 1 atmosphere.

### To Calculate and Display the Distribution of Electrons

- Setup the molecules as above.
- Make a new file.
- Build molecule by selecting atoms and clicking on screen.
- Go to Setup Calculations, Select “Single Point Energy”, and Hartree-Fock (HF/3-21G), check IR and thermodynamics. You need not change anything else.
- Here you only need to run the Hartree-Fock (HF/3-21G),
- Go into “Setup” menu and choose “Surfaces”
- Click “Add” and choose Surface = density; property = Potential; click “OK”; close window.
- Go to submit and “submit”.
- Use Monitor to see if model is executing. (Should be very fast so might not be necessary).
- When completed go to “Results” menu and choose “Surfaces”; click on the box and see the potential surface around your molecule.
- The red regions of the surface indicate the more electronegative regions of the molecule while the blue regions indicate the lower electronegative regions.

### To Display the Vibrational Modes

- Go to “Results” menu and choose IR. Check the box to display the entire spectrum. Click on the boxes to see the molecule vibrate in each of its modes.

Computational Thermochemistry Exercises

Inorganic Molecules.

- Build H
_{2}and run. - Build HCl and run.
- Build HF and run.
- Compare the electron distribution for these three molecules. What does this say about electronegativity?
- Examine the vibrations for the three molecules and compare. Which molecule has the greatest vibrational frequency? Is there any relationship to the electronegativity?

### Organic Molecules

- Build benzene and run.
- Build chlorobenzene and run.
- Build toluene and run.
- Build C
_{6}H_{5}-O-CH_{3}and run. - Compare the electron distribution for these three molecules. What does this say about electronegativity and the ability of groups to extract or donate charge to the ring?
- Examine the vibrations for the four molecules and compare. Is there any thing interesting that you can say?

### Lab Report

- In the lab write-up, be sure to include responses to the following questions as you discuss and analyze your results.
- What difference does the computational technique make on computed values?
- Does molecular bonding appear to play a role in computational accuracy?
- What differences are there between the computed values and literature values?
- How does one account for the difference among computed values?
- How does one account for the difference between computed and literature values?
- Propose at least two instances where computed data may be useful in an experimental laboratory setting.
- For H
_{2}, HCl and HF compare the electron distribution for these three molecules. What does this say about electronegativity? - For H
_{2}, HCl and HF examine the vibrations for the three molecules and compare. Which molecule has the greatest vibrational frequency?Is there any relationship between the vibrational frequency to the electronegativity difference between the elements? - For the four aromatic molecules compare the electron distributions. What does this say about electronegativity and the ability of the groups to extract or donate charge to the ring?
- For the four aromatic molecules examine the vibrations for the three molecules and compare. Is there any thing interesting that you can say?