# Statistical Thermodynamics and Rate Theories/Equations for reference

## Equation Sheet 1

[edit | edit source]### Translational States

[edit | edit source]The translational energy of a particle in a 3 dimensional box is given by the equation:

Where *h* is Planck's constant , *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

[edit | edit source]The moment of inertia for a rigid rotor is given by the equation:

where *m _{i}* is the mass an 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:

_{i}

where *r _{e}* is the internuclear distance and

*μ*is the reduced mass for the diatomic molecule:

In the case of a homonuclear diatomic molecule, the reduced mass, *μ*, can be further simplified to:

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

where and the degeneracy of each rotational state is given by .

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

where is the rotational constant, and can be related to the moment of inertia by the equation:

where *c* is the speed of light.

To avoid confusing in obtaining values for the frequency of radiation, , and the rotational constant, , *c* is often expressed in units of cm/s ()

### Vibrational States

[edit | edit source]The energy of a simple harmonic oscillator is given by the equation:

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

where *k* is the bond force constant.

### Electronic States

[edit | edit source]An electron in an atom may be described by four quantum numbers: the principle quantum number, ; the angular momentum quantum number, ; the magnetic quantum number, ; the spin quantum number, .

For a hydrogen-like atom (possessing only a single electron), the energy of the electron is given by the equation:

where is the mass of an electron, *e* is the charge of an electron, and is the permittivity of free space.

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

The electronic degeneracy of the system may then be determined by

.

### Thermodynamic Relations

[edit | edit source]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:

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

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

where *A* is Helmholtz energy;

where *q* is heat and *w* is work;

where is the heat associated with a reversible process;

where *n* is the number of moles of a gas and *R* is the ideal gas constant ().

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

while pressure may be estimated by the similar calculation:

allows for the determination of *q*:

may also be related to the heat capacity at constant pressure:

which in turn allows for the determination of enthalpy:

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

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

where are the translational, rotational, and vibrational degrees of freedom for the molecule, respectively.

For linear molecules the internal energy simplifies to:

And for non-linear molecules:

## Equation Sheet 2

[edit | edit source]Formula to calculate the internal energy is given by the formula

Where *U* is the internal energy of the system is the energy of the system, is the Boltzmann constant (1.3807 x 10^-23 J K^{-1}), *T* is the temperature in Kelvin, and *Q* is the partition function of the system.

### Canonical Ensemble

[edit | edit source]Internal Energy, *U*, of Canonical Ensemble:

Where is the Boltzmann constant, *T* is the temperature in Kelvin, and *Q* is the partition function of the system.

Entropy, *S*, of the Canonicial Ensemble:

Where *E* is the ensemble average energy of the system, is the Boltzmann constant, *T* is the temperature in Kelvin, and *Q* is the partition function of the system.

Helmholtz Free Energy, *A*, of the Canonical Ensemble

Where is the Boltzmann constant, *T* is the temperature in Kelvin, and *Q* is the partition function of the system.

### Partition Functions

[edit | edit source]Function to calculate the partition function, *Q*, in a system of *N* identical indistinguishable particles can be calculated by:

Where *q* is the molecular partition functions.

#### Molecular Partition Function

[edit | edit source]

In which *q* is the molecular partition function, is the molecular partition function of the translational degree of freedom, is the molecular partition function of the rotational degree of freedom, is the molecular partition function of the vibrational degree of freedom, and is the molecular partition function of the electronic degree of freedom.

#### Molecular Translational Partition Function

[edit | edit source]

Where is the molecular partition function of the translational degree of freedom, is the Boltzmann constant, *m* is the mass of the molecule, *T* is the temperature in Kelvin, *V* is the volume of the system.

To simplify the calculation, the de Broglie wavelength, *Λ*, of the molecule at a given temperature may be used. The de Broglie wavelength is defined as:

This simplifies the translational molecular partition function to:

#### Molecular Rotational Partition Function

[edit | edit source]

Where is the molecular partition function of the rotational degree of freedom, *T* is the temperature in Kelvin, is the Boltzmann constant, is the bond length of the molecule, *μ* is the reduced mass, *h* is Planck's constant, is defined as , and *σ* is the symmetry factor (σ = 2 for homonuclear molecules and σ = 1 for heteronuclear molecules).

The constants in the rotational molecular partition function can be simplified to the characteristic temperature, *Θ _{r}*, which has units of Kelvin:

Using the characteristic temperature, the rotational molecular partition function is simplified to:

#### Molecular Vibrational Partition Function

[edit | edit source]

Where is the molecular partition function of the vibrational degree of freedom, *T* is the temperature in Kelvin, is the Boltzmann constant, *h* is Planck's constant, and *υ* is the vibrational frequency of the molecule defined as:

Where *k* is the spring constant of the molecule and *μ* is the reduced mass of the molecule.

The characteristic temperature *Θ _{υ}* may be used to simplify the constants in the molecular vibrational partition function to the following:

Using the characteristic temperature, the vibrational molecular partition function is simplified to:

#### Molecular Electronic Partition Function

[edit | edit source]

Where is the molecular partition function of the electronic state and *g _{1}* is the degeneracy of the ground state.

For large temperatures, the equation turns to:

Where *D _{0}* is the bond dissociation energy of the molecule, and is the Boltzmann constant.

#### Simplified Molecular Partition Function

[edit | edit source]All molecular partition functions combined are defined as:

Which simplifies further when utilizing the de Broglie wavelength for the translational molecular partition function and the characteristic temperatures for the rotational and vibrational molecular partition functions.

Which is equivalent to:

### Chemical Equilibrium

[edit | edit source]Determination of equilibrium constant is found by the following equation:

In which *q _{A}*,

*q*,

_{B}*q*, and

_{C}*q*are partition functions of each species and with corresponding

_{D}*ν*and

*ρ*values for corresponding stoichiometric coefficients and partial pressures.

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

The chemical potential can be determined by

in which is the change in Helmholtz energy when a new particle is added to the system

The pressure, *p*, can then be determined by