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Statistical Thermodynamics: Calculating Thermodynamic Functions DH, DG, and DS

Copyright (c) RDMCHEM LLC 2019

Overview

In this activity, you will apply statistical thermodynamics to calculate the enthalpy, free energy, and entropy of reaction for the combustion of ethene to form ethylene oxide at 298 K:

 Rxn (1)

To simplify the calculations, you will treat each species as an ideal gas. The thermal entropy (S) (J/mol K) and the internal energy (E) (J/mol) are given by:

 (1)

and

 (2)

where q is the molecular partition function  , where the subscript t, e, r, and v refer to the translational, electronic, rotational, and vibrational contributions, respectively.  To calculate q, we further assume each species is in the ground electronic state (with degeneracy 2 spin + 1) with rovibrational energies given by harmonic oscillator and rigid rotor approximations. These assumptions lead to the following expressions for q, S, and E:

Translational:

 (3) (4) (5)

where P = 1 atm = 100.325 kPa.

Electronic:

 (6) (7)

The thermal contribution to internal energy is 0 since the ${q}_{e}$ does not depend on temperature.

Rotational:

Linear Molecule:

 (8) (9) (10)

where ${\mathrm{σ}}_{r}$ is the rotational symmetry number and ${\mathrm{Θ}}_{r}$ is the rotational temperature.  Each molecule we will consider in this activity is characterized by

Nonlinear Polyatomic Molecule:

 (11) (12) (13)

where  are the rotational temperatures corresponding to the principal moments of inertia.

Vibrational:

 (14) (15) (16)

where s = 3N - 5 normal modes for a linear molecule and s = 3N - 6 for a nonlinear polyatomic.

Once the translational, electronic, rotational, and vibrational thermal contributions to the total entropy and internal energy have been calculated, total entropy (S), internal energy (E), enthalpy (H), and free-energy (G) can be calculated as follows:

 (17) (18) (19) (20)

Note we have corrected for the vibrational zero-point-energy for the internal energy.  We will calculate S, H, and G for each of the reactants and product in Rxn (1) and calculate ΔH, ΔG, and ΔS as

 (21)

etc.



Initialize

We begin with initializing the QuantumChemistry package, along with the ScientificConstants and LinearAlgebra packages.

 > $\mathrm{restart}:\mathrm{with}\left(\mathrm{QuantumChemistry}\right):\mathrm{with}\left(\mathrm{ScientificConstants}\right):\mathrm{with}\left(\mathrm{LinearAlgebra}\right):\mathrm{Digits}≔15:$

Specify Molecule

Here we specify the coordinates of the molecule for which we want to calculate thermodynamics as well as some related calculate parameters, such as symmetry number, charge, AObasis, etc.  Note that the geometries provided in .xyz data files are at the HartreeFock / STO-3G level.   If you wish to use a different method/basis combination, begin with the provided .xyz coordinates for each molecule and uncomment the geometry optimization line below.

 > $#\mathrm{molec≔ReadXYZ}\left("ethylene.xyz"\right);$
 > $\mathrm{molec}≔\mathrm{ReadXYZ}\left("O2.xyz"\right);$
 ${\mathrm{molec}}{≔}\left[\left[{"O"}{,}{0.}{,}{0.}{,}{0.}\right]{,}\left[{"O"}{,}{0.}{,}{0.}{,}{1.22169673}\right]\right]$ (3.1)
 > $#\mathrm{molec≔ReadXYZ}\left("ethyleneoxide.xyz"\right);$
 >
 >
 > $\mathrm{molec_charge}≔0:$
 > $\mathrm{freqmethod}≔'\mathrm{HartreeFock}':$
 > $\mathrm{freqbasis}≔"sto-3g":$
 > $\mathrm{energymethod}≔'\mathrm{HartreeFock}':$
 > $\mathrm{energybasis}≔"sto-3g":$
 >
 > $\mathrm{temp}≔298.15:$
 >
 > 

Calculate Thermodynamics

If  not using Hartree-Fock and sto-3g, uncomment the following line:

 >

Calculate electronic energy in J/mol using AOmethod, AObasis, spin, and charge defined above:

 >
 ${\mathrm{molec_energy}}{≔}{-147.55157168}$
 ${\mathrm{molec_energy}}{≔}{-3.87396598}{}{{10}}^{{8}}$ (4.1)

To calculate the vibrational energies, perform a vibrational analysis:

 >
 ${\mathrm{emodes}}{,}{\mathrm{vmodes}}{≔}\left[\begin{array}{c}{2057.72134813}\end{array}\right]{,}\left[\begin{array}{c}{5.41846411}{}{{10}}^{{-8}}\\ {5.41846015}{}{{10}}^{{-8}}\\ {-0.70710693}\\ {1.26430759}{}{{10}}^{{-7}}\\ {-5.41846392}{}{{10}}^{{-8}}\\ {0.70710663}\end{array}\right]$
 $\left[\begin{array}{c}1842.27792298\end{array}\right]$ (4.2)

Calculate principal moments of inertia:

 >
 ${\mathrm{I_A}}{≔}{0.}$
 ${\mathrm{I_B}}{≔}{1.98266764}{}{{10}}^{{-46}}$
 ${\mathrm{I_C}}{≔}{1.98266764}{}{{10}}^{{-46}}$ (4.3)

Now we are ready to calculate partition functions!

Translational partition function:

 >
 ${\mathrm{q_t}}{≔}{7.11466749}{}{{10}}^{{6}}$ (4.4)

Electronic partition function:

 >
 ${\mathrm{q_e}}{≔}{1}$ (4.5)

Rotational temperatures and partition function:

 >
 ${\mathrm{theta_A}}{≔}{Float}{}\left({\mathrm{\infty }}\right)$
 ${\mathrm{theta_B}}{≔}{2.03137108}$
 ${\mathrm{theta_C}}{≔}{2.03137108}$
 ${\mathrm{I_A}}{≔}{0}$
 ${\mathrm{q_r}}{≔}{73.38639462}$
 ${\mathrm{num_modes}}{≔}{1}$ (4.6)

Vibrational temperatures and partition function:

 >
 ${\mathrm{q_v}}{≔}{0.01173726}$ (4.7)

ZPE correction in J/mol:

 >
 ${\mathrm{zpe_correction}}{≔}{11019.26903875}$ (4.8)

Now we can calculate total S, E, H, and G according to the equations provided in the Overview:

 >
 ${\mathrm{S_t}}{≔}{151.96894459}$
 ${\mathrm{E_t}}{≔}{3718.43428406}$
 ${\mathrm{S_e}}{≔}{0.}$
 ${\mathrm{S_v}}{≔}{0.01132692}$
 ${\mathrm{E_v}}{≔}{11027.51974508}$
 ${{\mathrm{S0}}}_{{2}}{≔}{196.01147687}$
 ${{\mathrm{E0}}}_{{2}}{≔}{-3.87368354}{}{{10}}^{{8}}$
 ${{\mathrm{H0}}}_{{2}}{≔}{-3.87365875}{}{{10}}^{{8}}$
 ${{\mathrm{G0}}}_{{2}}{≔}{-3.87424316}{}{{10}}^{{8}}$ (4.9)

Repeat the above calculations for each reactant and product in Rxn (1). Don't forget to change the molec_index!

 > 

Calculate DH, DG, and DS (after completing above for all reactants and products)

 >
 ${\mathrm{ΔS}}{≔}{{\mathrm{S0}}}_{{3}}{-}{{\mathrm{S0}}}_{{1}}{-}{98.00573844}$ (5.1)
 >
 ${\mathrm{ΔG}}{≔}{0.00100000}{}{{\mathrm{G0}}}_{{3}}{-}{0.00100000}{}{{\mathrm{G0}}}_{{1}}{+}{193712.15788334}$ (5.2)
 >
 ${\mathrm{ΔH}}{≔}{0.00100000}{}{{\mathrm{H0}}}_{{3}}{-}{0.00100000}{}{{\mathrm{H0}}}_{{1}}{+}{193682.93747242}$ (5.3)
 > 

NIST lists the following values for enthalpy of formation and free-energy of formation:

  ${\mathrm{ΔH}}_{f}$ (kJ/mol) ${\mathrm{ΔG}}_{f}$ (kJ/mol) (calc.) S (1 bar) (J/molK) ethylene 52.4 61.4 219.3 ethylene oxide -52.6 -51.7 243.0 oxygen 0 0 205.15

Using Hess's Law, calculate ΔH, ΔG, and ΔS for Rxn (1) and compare to your calculated results.

 >
 ${\mathrm{ΔSHess}}{≔}{-78.87500000}$ (5.4)
 > $\mathrm{ΔHHess}≔-52.6-52.4;$
 ${\mathrm{ΔHHess}}{≔}{-105.00000000}$ (5.5)
 > $\mathrm{ΔGHess}≔-51.7-61.4;$
 ${\mathrm{ΔGHess}}{≔}{-113.10000000}$ (5.6)
 > $\mathrm{S0}\left[2\right];$
 ${196.01147687}$ (5.7)
 > 



Appendix

 References 1) D. A. McQuarrie and J. D. Simon, Physical Chemistry: A Molecular Approach (University Science, New York, 1997). 2) JPCA scaling factors 2007 3) https://webbook.nist.gov/cgi/inchi/InChI=1S/C2H4/c1-2/h1-2H2 4) https://webbook.nist.gov/cgi/inchi/InChI=1S/C2H4O/c1-2-3-1/h1-2H2