# Statistical Thermodynamics and Rate Theories/Equations for reference

## Equation Sheet 1

### Translational States

The translational energy of a particle in a 3 dimensional box is given by the equation:

${\displaystyle E_{n_{x},n_{y},n_{z}}={h^{2} \over 8m}\left({{{n_{x}}^{2}} \over {a^{2}}}+{{{n_{y}}^{2}} \over {b^{2}}}+{{{n_{z}}^{2}} \over {c^{2}}}\right)}$

Where h is Planck's constant ${\displaystyle (6.626068\times 10^{-34}Js)}$, m is the mass of the particle in kg, n is the translational quantum number for the denoted direction of translation (x, y, z) and a, b, and c are the length of the box in the x, y, and z directions respectively. The translational quantum number n may possess any positive integer value.

### Rotational States

The moment of inertia for a rigid rotor is given by the equation:

${\displaystyle I=\sum _{i}{m_{i}}{r_{i}^{2}}}$

where m is the mass of each atom in the molecule and r is the distance in meters from that atom to the molecule's center of mass. For a diatomic molecule this formula may be simplified to:

${\displaystyle I=\mu {r_{e}^{2}}}$

where ${\displaystyle r_{e}}$ is the internuclear distance and μ is the reduced mass for the two atoms:

${\displaystyle \mu ={{m_{1}}{m_{2}} \over {m_{1}+m_{2}}}}$

The energy of a rigid rotor occupying a rotational quantum state J (J = 0,1,2,...) is given by the equation:

${\displaystyle E_{J}={{\hbar }^{2} \over 2\mu {r_{e}^{2}}}J(J+1)}$

where ${\displaystyle \hbar ={h \over 2\pi }}$. The degeneracy of each rotational state is given by ${\displaystyle g_{J}=2J+1}$.

The frequency of radiation corresponding to the energy of rotation at a given rotational quantum state is given by:

${\displaystyle {\tilde {\nu }}=2{\tilde {B}}(J+1)}$

where ${\displaystyle {\tilde {B}}}$ is the rotational constant, and can be related to the moment of inertia by the equation:

${\displaystyle {\tilde {B}}={h \over {8{\pi }^{2}cI}}}$

where c is the speed of light. To yield ${\displaystyle {\tilde {\nu }}}$ and ${\displaystyle {\tilde {B}}}$ values in units of wavenumbers (${\displaystyle cm^{-1}}$), c should be expressed in units of cm/s, i.e. ${\displaystyle c=2.99792458\times 10^{10}cm/s}$.

### Vibrational States

The energy of a simple harmonic oscillator is given by the equation:

${\displaystyle E_{n}=h\nu (n+{1 \over 2})}$

where n = 0,1,2,... is the vibrational quantum number and ν is the fundamental frequency of vibration, given by:

${\displaystyle \nu ={1 \over 2\pi }{\sqrt {k \over \mu }}}$

where k is the bond force constant.

### Electronic States

An electron in an atom may be described by four quantum numbers: the principle quantum number, ${\displaystyle n=1,2,...}$; the angular momentum quantum number, ${\displaystyle l=0,1,...,(n-1)}$; the magnetic quantum number, ${\displaystyle m_{l}=-l,-l+1,...,l-1,l}$ and the spin quantum number, ${\displaystyle m_{s}=\pm {1 \over 2}}$. For a hydrogen-like atom (possessing only a single electron), the energy of said electron is given by the equation:

${\displaystyle E_{n}=-({{{m_{e}}e^{4}} \over {32{\pi }^{2}{\epsilon }_{0}^{2}{\hbar }^{2}}}){1 \over n^{2}}}$

where ${\displaystyle m_{e}}$ is the mass of an electron, e is the charge of an electron, and ${\displaystyle {\epsilon }_{0}}$ is the permittivity of free space.

For a system with multiple electrons, the total spin for the system is given the sum:

${\displaystyle S=\sum _{i}m_{s,i}}$

The electronic degeneracy of the system may then be determined by ${\displaystyle g_{el}=2S+1}$.

### Thermodynamic Relations

There exist a number of equations which allow for the relation of thermodynamic variables, such that it is possible to determine values for many of these variables mathematically starting with just a few:

${\displaystyle H=U+pV}$

where H is enthalpy, U is internal energy, p is pressure and V is volume;

${\displaystyle G=H-TS}$

where G is Gibbs energy, T is temperature and S is entropy;

${\displaystyle A=U-TS}$

where A is Helmholtz energy;

${\displaystyle \Delta U=q+w}$

where q is heat and w is work;

${\displaystyle dS={dq_{rev} \over T}}$

where ${\displaystyle dq_{rev}}$ is the heat associated with a reversible process;

${\displaystyle pV=nRT}$

where n is the number of moles of a gas and R is the ideal gas constant (${\displaystyle 8.314JK^{-1}mol^{-1}}$).

The heat capacity for a gas at constant volume may be estimated by the differential:

${\displaystyle C_{v}=\left({\partial U \over \partial T}\right)_{V,n}}$

while pressure may be estimated by the similar calculation:

${\displaystyle p=-({\partial U \over \partial V})_{n}}$

${\displaystyle C_{v}}$ allows for the determination of q:

${\displaystyle q=C_{v}\Delta T}$

${\displaystyle C_{V}v}$ may also be related to the heat capacity at constant pressure:

${\displaystyle C_{p}=C_{V}+nR}$

which in turn allows for the determination of enthalpy:

${\displaystyle \Delta H=C_{p}\Delta T}$

Overall heat capacity in each case may be related to molar heat capacity by relating the number of moles of gas:

${\displaystyle C_{V}=nC_{V}^{m}}$

${\displaystyle C_{p}=nC_{p}^{m}}$

Finally, the internal energy contribution from translational, rotational and vibrational energies of a gas may be determined by the equation:

${\displaystyle U={1 \over 2}n_{trans}nRT+{1 \over 2}n_{rot}nRT+n_{vib}nRT}$

where ${\displaystyle n_{trans},n_{rot},n_{vib}}$ are the translational, rotational and vibrational degrees of freedom for the molecule, respectively.

## Equation Sheet 2

Formula to calculate the internal energy is given by the formula

${\displaystyle U=\langle E\rangle =\sum _{j}{E_{j}}{\exp(-E_{j}/{k_{B}}T)}/Q}$

Where U is the internal energy of the system ${\displaystyle {E_{j}}}$ is the energy of the system, ${\displaystyle k_{B}}$ is the Boltzmann constant (1.3807 x 10^-23 joule s per kelvin (J/K)) and T is the temperature in Kelvin. Q is the partition function of the system

### Canonical Ensemble

Internal Energy of Canonical Ensemble

${\displaystyle U=\langle E\rangle =k_{B}T^{2}\left({\partial \ln Q \over \partial T}\right)_{N,V}}$

Where U is the internal energy of the system,${\displaystyle k_{B}}$ is the Boltzmann constant (1.3807 x 10^-23 joule s per kelvin (J/K)) and T is the temperature in Kelvin. Q is the partition function of the system

Entropy of the Canonicial Ensemble

${\displaystyle S={\langle E\rangle \over T}+k_{B}\ln Q}$

Where S is the entropy of the system <E> is the energy of the system, ${\displaystyle k_{B}}$ is the Boltzmann constant (1.3807 x 10^-23 joule s per kelvin (J/K)) and T is the temperature in Kelvin. Q is the partition function of the system

Helmholtz Free Energy of the Canonical Ensemble

${\displaystyle A=-k_{B}T\ln Q}$

A is the Helmhotlz free energy, ${\displaystyle k_{B}}$ is the Boltzmann constant (1.3807 x 10^-23 joule s per kelvin (J/K)), T is the temperature in Kelvin, and Q is the partition function of the system.

### Partition Functions

Function to calculate the partition function Q can be calculated by

${\displaystyle Q={q^{N} \over N!}}$

In which Q is the partition function, q is the molecular partition functions, and N is the number of molecules.

Calculation of the molecular partition function

${\displaystyle q=q_{trans}q_{rot}q_{vib}q_{elec}}$

In which q is the molecular partition function, ${\displaystyle q_{trans}}$is the partition functions of the translational degree of freedom, ${\displaystyle q_{rot}}$ is the partition functions of the rotational degree of freedom, ${\displaystyle q_{elec}}$ is the partition functions of the electrical degree of freedom, ${\displaystyle q_{vib}}$ is the partition functions of the vibrational degree of freedom.

Transitional partition functions

${\displaystyle q_{trans}(V,T)=(({2\pi mk_{B}T \over h^{2}})^{3}/2V)={V \over \Lambda ^{3}}}$

q_trans is the partition functions of the translational degree of freedom, ${\displaystyle k_{B}}$ is the Boltzmann constant (1.3807 x 10^-23 joule s per kelvin (J/K)), m is the mass of the molecule, T is the temperature in Kelvin, V is the volume of the system, and \lambda is the deBrogie wavelength which can be given by the equation;

${\displaystyle \Lambda =(({2\pi mk_{B}T \over h^{2}})^{-1/2})}$

Molecular Rotational Partition Function

${\displaystyle q_{rot}(T)={2k_{B}T\mu r_{e}^{2} \over \sigma h^{2}}={T \over \sigma \theta _{r}}}$

Where ${\displaystyle q_{rot}}$ is the partition functions of the rotational degree of freedom, T is the temperature in Kelvin, ${\displaystyle {k_{B}}}$is the Boltzmann constant (1.3807 x 10^-23 joule s per kelvin (J/K)), ${\displaystyle r_{e}}$ is the bond length of the molecule, μ is the reduced mass, σ is the symmetry factor (σ=2 for homonuclear molecules and σ=1 for heteronuclear), ${\displaystyle \theta _{r}}$ represents the determined by the function

${\displaystyle \theta _{r}={hv \over k_{B}}}$

In which h is planks constant (6.62607004 × 10^-34 m^2 kg/s), ${\displaystyle \theta _{r}}$, v is the frequency of the rotation of the molecule and {k_B} is the Boltzmann constant (1.3807 x 10^-23 Joule s per kelvin (J/K))

Molecular Vibrational Partition Function

${\displaystyle q_{vib}(T)={1 \over 1-\exp(-\theta _{v}/T)}}$

in which ${\displaystyle q_{vib}}$ is the partition function of the vibrations of the molecule, T is the temperature in Kelvin, and ${\displaystyle \theta _{v}}$ is given by the function;

${\displaystyle \theta _{v}={hv \over k_{b}}}$

Molecular Electronic Partition function

${\displaystyle q_{elec}=g_{1}\exp({D_{0} \over k_{B}T})}$

in which ${\displaystyle q_{elec}}$ is the electronic partition function of the molecule, ${\displaystyle D_{0}}$ is the ,

Also it can be determined by using an approximation of the ground state, making ${\displaystyle q_{elec}=g_{1}}$

Knowing these four equations, the entire molecular partition function can be written as

${\displaystyle q=g_{1}\times {1 \over 1-\exp(-\theta _{v}/T)}\times \left({2\pi mk_{B}T \over h^{2}}\right)^{3}/2V\times {2k_{B}T\mu r_{e}^{2} \over \sigma h^{2}}}$

which is equivalent to the above function ${\displaystyle q=q_{trans}q_{rot}q_{vib}q_{elec}}$

### Chemical Equilibrium

Determination of equilibrium constant ${\displaystyle K_{c}}$ is found using the equation

${\displaystyle K_{c}(T)={((q_{C}/V)^{\nu _{C}}(q_{D}/V)^{\nu _{D}}) \over ((q_{A}/V)]^{\nu _{A}}(q_{B}/V)^{\nu _{B}})}={{\rho _{C}}^{\nu _{C}}{\rho _{D}}^{\nu _{D}} \over {\rho _{A}}^{\nu _{A}}{\rho _{B}}^{\nu _{B}}}}$

In which ${\displaystyle q_{c},q_{D},q_{B},andq_{A}}$ are partition functions of each species of species and with corresponding ν and ρ values for corresponding stoichiometric coefficients and partial pressures.

The equilibrium constant in terms of pressure can be expressed as;

${\displaystyle K_{p}(t)={{\rho _{C}}^{\nu _{C}}{\rho _{D}}^{\nu _{D}} \over {\rho _{A}}^{\nu _{A}}{\rho _{B}}^{\nu _{B}}}=(k_{B}T)^{\nu _{C}+\nu _{D}-\nu _{A}-\nu _{B}}K_{c}(T)}$

The chemical potential can be determined by

${\displaystyle \mu _{i}=-k_{B}T^{2}\ln({q_{i}(V,T) \over N_{i}})}$

in which ${\displaystyle \mu _{i}}$ is the change in helmholtz energy when a new particle is added to the system

pressure (p) can then be determined by

${\displaystyle p=k_{B}T({\partial \ln Q \over \partial V})_{N,T}}$