# Statistical Thermodynamics and Rate Theories/Molecular partition functions

The partition function of a system, Q, provides the tools to calculate the probability of a system occupying state i . The general form of a partition function is a sum over the states of the system,

${\displaystyle Q=\sum _{i}\exp \left({\frac {-E_{i}}{k_{B}T}}\right)}$

This requires that the energies levels of the entire system must be known and the calculations have to be calculated from sums over states. This limits the types of the systems that we can derive properties for.

For ideal gases, we assume that the energy states of molecules are independent of those in other molecules. The molecular partition function, q, is defined as the sum over the states of an individual molecule.

${\displaystyle q=\sum _{i}\exp \left({\frac {-\epsilon _{i}}{k_{B}T}}\right)}$

For an ideal gas of a indistinguishable particles, the partition function of the system (Q) can be expressed in terms of the molecular partition function (q) and the number of particles in the system (N).

 Partition function of an ideal gas of indistinguishable particles ${\displaystyle Q={\frac {q^{N}}{N!}}}$

## Molecular Partition Functions

The energy levels of a molecule can be approximated as the sum of energies in the various degrees of freedom of the molecule,

${\displaystyle \epsilon =\epsilon _{trans}+\epsilon _{rot}+\epsilon _{vib}+\epsilon _{elec}}$

Correspondingly, we can divide molecular partition function (q),

${\displaystyle q=\sum _{i}\exp \left({\frac {-(\epsilon _{trans}+\epsilon _{rot}+\epsilon _{vib}+\epsilon _{elec}}{k_{B}T}}\right)}$
${\displaystyle q=\sum _{i}\exp \left({\frac {-\epsilon _{trans,i}}{k_{B}T}}\right)\sum _{i}\exp \left({\frac {-\epsilon _{rot,i}}{k_{B}T}}\right)\sum _{i}\exp \left({\frac {-\epsilon _{vib,i}}{k_{B}T}}\right)\sum _{i}\exp \left({\frac {-\epsilon _{elec,i}}{k_{B}T}}\right)}$
${\displaystyle q=\sum _{i}\exp \left({\frac {-\epsilon _{trans,i}}{k_{B}T}}\right)\times \sum _{i}\exp \left({\frac {-\epsilon _{rot,i}}{k_{B}T}}\right)\times \sum _{i}\exp \left({\frac {-\epsilon _{vib,i}}{k_{B}T}}\right)\times \sum _{i}\exp \left({\frac {-\epsilon _{elec,i}}{k_{B}T}}\right)}$
${\displaystyle q=q_{trans}\times q_{rot}\times q_{vib}\times q_{elec}}$

## Translational Partition Function

The translational partition function, qtrans, is the sum of all possible translational energy states. For a molecule in three dimensional space the energy term in the general partition function equation is replaced with the particle in a 3D box equation. All molecules have three translational degrees of freedom, one for each axis the molecule can move along in three dimensional space.

 Particle in a 3D Box Equation ${\displaystyle E_{n_{x},n_{y},n_{z}}={\frac {h^{2}}{8mL^{2}}}(n_{x}^{2}+n_{y}^{2}+n_{z}^{2})}$

${\displaystyle q_{trans}=\sum _{j=1}^{\infty }\ {g_{i}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)}}$

${\displaystyle q_{trans}=\left[\sum _{j=1}^{\infty }\ {\exp \left({{\frac {-h^{2}}{k_{B}T8mL^{2}}}n^{2}}\right)}\right]^{3}}$

This is the open form of the translational partition function, ignoring degeneracy.

Making the assumption that the energy levels are continuous, the partition function can be represented in a closed form (derivation).

${\displaystyle q_{trans}=\left({\frac {2\pi mk_{B}T}{h^{2}}}\right)^{3/2}V}$

The assumption that energy levels are continuous is valid because the space between energy levels is extremely small, resulting in minimal error. This form is convenient as it does not include a sum to infinity and can therefore be solved with relative ease. The translational partition function can be simplified further by defining a DeBroglie wavelength, ${\displaystyle \Lambda }$, of a molecule at a given temperature.

${\displaystyle q_{trans}={\frac {V}{\Lambda ^{3}}}}$

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

## Rotational Partition Function

The rotational partition function, ${\displaystyle q_{rot}}$, is the sum of all possible rotational energy levels. This sum is found by substituting the equation for the energy levels of a linear rigid rotor:

${\displaystyle E_{j}={\frac {\hbar ^{2}}{2\mu r^{2}}}J(J+1),J=0,1,2,...}$

Into the partition function, to produce an open form of the rotational partition function:

${\displaystyle q_{rot}=\sum _{j=1}^{\infty }\ g_{i}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)}$

By solving this sum using a definite integral from zero to infinity, a closed form of this function can be found, making numerical evaluation much easier:

${\displaystyle q_{rot}={\frac {2k_{B}T\mu r^{2}}{{\text{ħ}}^{2}}}}$

The full derivation of the closed form of the rotational partition function of a linear rotor is given here.

This function applies only to heteronuclear diatomic molecules. However, this equation can be altered by adding a variable to alter the equation based on the nature of the diatomic molecule being studied:

${\displaystyle q_{rot}={\frac {2k_{B}T\mu r^{2}}{{\text{σ}}{\text{ħ}}^{2}}}}$

where ${\displaystyle \sigma }$ is 1 for heteronuclear diatomics, and 2 for homonuclear diatomics. When studying a diatomic species using this equation, the main variable being altered is the temperature, T. By combining the constants into a single constant, the characteristic temperature, ${\displaystyle \Theta }$ where,

${\displaystyle \Theta ={\frac {{\text{ħ}}^{2}}{2k_{B}\mu r^{2}}}}$

determining the rotational partition function can be calculated much easier using,

${\displaystyle q_{rot}={\frac {T}{\sigma \Theta }}}$

## Vibrational Partition Function

A Partition Function (Q) is the denominator of the probability equation. It corresponds to the number of accessible states in a given molecule. Q represents the partition function for the entire system, which is broken down and calculated from each individual partition function of each molecule in the system. These individual partition functions are denoted by q. All molecules have four different types of partition functions: translational, rotational, vibrational, and electronic. Looking only at the vibrational aspect of the system, there is a specific unique equation used to calculate its partition function:

The vibrational partition function of a linear molecule is,

${\displaystyle q_{vib}={\frac {1}{1-\exp \left({\frac {-hv}{k_{B}T}}\right)}}}$

In general, the molecule partition function can be written as an infinite sum. This is called the open form of the equation:

${\displaystyle q=\sum _{j}g_{i}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)}$

It is much easier and more convenient to write this as a closed sum. This turns the equation into an approximative algebraic expression, with the following parameters in variables:

• degeneracy ${\displaystyle g_{j}=1}$
• quantum number ${\displaystyle n=1,2,3...}$
• energy levels ${\displaystyle \varepsilon _{n}=hv(n+({\frac {1}{2}}))}$

The energy levels, ${\displaystyle \Delta E_{J}}$ are defined relative to the ground state of the system (i.e., the zero point energy is subtracted from each level),

${\displaystyle \Delta E_{J}=\varepsilon _{j}-\varepsilon _{0}}$
${\displaystyle =hv(j+{\frac {1}{2}})-hv(0+{\frac {1}{2}})=hvj+{\frac {1}{2}}hvj-{\frac {1}{2}}hv=hvj}$

By exploring some substitutions and derivations, the equation listed above for a linear molecule is achieved. The substitutions made include:

• ${\displaystyle g_{j}=1}$
• ${\displaystyle E_{j}=hvj}$
• ${\displaystyle j=0,1,2...}$

At the same time, it is important to note that v represents the vibrational frequency of the molecule. It can be calculated on its own prior by the following relation:

${\displaystyle v={\frac {1}{2\pi }}\left({\frac {k}{\mu }}\right)^{\frac {1}{2}}}$

where k represents the spring constant of the molecule and μ represents the reduced mass of the same molecule.

## Electronic Partition Function

The electronic partition function (qel) of a system describes the electronic states of the system at thermodynamic equilibrium. This can be written as a sum over states,

${\displaystyle q_{el}=\sum g_{j}\exp(-\beta \epsilon _{j})}$

however due to high energy levels being present under most circumstances, the electronic partition function can be reduced to:

${\displaystyle q_{el}=g_{0}}$

Thus, the electronic partition function can usually be approximated as the ground state degeneracy of the atom or molecule.

## Molecular Partition Function

The molecular partition function, q, is the total number of states accessible to the atom or molecule. It is the product of the vibrational, rotational and translational partition functions,

${\displaystyle q=q_{trans}\times q_{rot}\times q_{vib}\times q_{el}}$