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:

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

Where h is Planck's constant $(6.626068\times 10^{-34}Js)$, 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:

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

where m_i is the mass an atom in the molecule and r_i 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:

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

where r_e is the internuclear distance and μ is the reduced mass for the diatomic molecule:

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

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

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

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

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

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

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

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

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

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

where c is the speed of light.

To avoid confusing in obtaining values for the frequency of radiation, ${\tilde {\nu }}$, and the rotational constant, ${\tilde {B}}$, c is often expressed in units of cm/s ($c=2.99792458\times 10^{10}cm/s$)

Vibrational States

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

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

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

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

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, $n=1,2,...$; the angular momentum quantum number, $l=0,1,...,(n-1)$; the magnetic quantum number, $m_{l}=-l,-l+1,...,l-1,l$; the spin quantum number, $m_{s}=\pm {1 \over 2}$.

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

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

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

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

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

The electronic degeneracy of the system may then be determined by

$g_{el}=2S+1$.

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:

$H=U+pV$

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

$G=H-TS$

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

$A=U-TS$

where A is Helmholtz energy;

$\Delta U=q+w$

where q is heat and w is work;

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

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

$pV=nRT$

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

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

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

while pressure may be estimated by the similar calculation:

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

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

$q=C_{v}\Delta T$

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

$C_{p}=C_{v}+nR$

which in turn allows for the determination of enthalpy:

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

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

$C_{v}=nC_{v}^{m}$

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

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

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

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

For linear molecules the internal energy simplifies to:

$U={3 \over 2}nRT+nRT+(3n_{atom}-5)nRT$

And for non-linear molecules:

$U={3 \over 2}nRT+{3 \over 2}nRT+(3n_{atom}-6)nRT$

Equation Sheet 2

Formula to calculate the internal energy is given by the formula

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

Where U is the internal energy of the system ${E_{j}}$ is the energy of the system, $k_{B}$ 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

Internal Energy, U, of Canonical Ensemble:

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

Where $k_{B}$ 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:

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

Where E is the ensemble average energy of the system, $k_{B}$ 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

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

Where $k_{B}$ is the Boltzmann constant, T is the temperature in Kelvin, and Q is the partition function of the system.

Partition Functions

Function to calculate the partition function, Q, in a system of N identical indistinguishable particles can be calculated by:

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

Where q is the molecular partition functions.

Molecular Partition Function

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

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

Molecular Translational Partition Function

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

Where $q_{trans}$ is the molecular partition function of the translational degree of freedom, $k_{B}$ 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:

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

This simplifies the translational molecular partition function to:

$q_{trans}={V \over \Lambda ^{3}}$

Molecular Rotational Partition Function

$q_{rot}={8{\pi }^{2}k_{B}T\mu r_{e}^{2} \over \sigma h^{2}}={2k_{B}T\mu r_{e}^{2} \over \sigma {\hbar }^{2}}$

Where $q_{rot}$ is the molecular partition function of the rotational degree of freedom, T is the temperature in Kelvin, ${k_{B}}$ is the Boltzmann constant, $r_{e}$ is the bond length of the molecule, μ is the reduced mass, h is Planck's constant, $\hbar$ is defined as $h \over {2\pi }$, 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:

$\Theta _{r}={h^{2} \over {8{\pi }^{2}k_{B}\mu r_{e}^{2}}}={{\hbar }^{2} \over {2k_{B}\mu r_{e}^{2}}}$

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

$q_{rot}={T \over {\sigma \Theta _{r}}}$

Molecular Vibrational Partition Function

$q_{vib}={1 \over {1-\exp \left({{-hv} \over {k_{b}T}}\right)}}$

Where $q_{vib}$ is the molecular partition function of the vibrational degree of freedom, T is the temperature in Kelvin, $k_{B}$ is the Boltzmann constant, h is Planck's constant, and υ is the vibrational frequency of the molecule defined as:

$\nu ={1 \over {2\pi }}\left({k \over \mu }\right)^{1/2}$

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:

$\Theta _{\nu }={h\nu \over k_{B}}$

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

$q_{vib}={1 \over {1-\exp \left({-\Theta _{\nu } \over T}\right)}}$

Molecular Electronic Partition Function

$q_{elec}=g_{1}$

Where $q_{elec}$ is the molecular partition function of the electronic state and g₁ is the degeneracy of the ground state.

For large temperatures, the equation turns to:

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

Where D₀ is the bond dissociation energy of the molecule, and $k_{B}$ is the Boltzmann constant.

Simplified Molecular Partition Function

All molecular partition functions combined are defined as:

$q={\left({2\pi mk_{B}T \over h^{2}}\right)^{3 \over 2}V\times \left({2k_{B}T\mu r_{e}^{2} \over \hbar ^{2}}\right)\times \left({1 \over 1-\exp \left({-hv \over k_{B}T}\right)}\right)\times g_{1}}$

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.

$q={V \over \Lambda ^{3}}\times {T \over \sigma \Theta _{r}}\times {1 \over 1-\exp \left({-\Theta _{\nu } \over T}\right)}\times g_{1}$

Which is equivalent to:

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

Chemical Equilibrium

Determination of equilibrium constant $K_{c}$ is found by the following equation:

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

In which q_A, q_B, q_C, and q_D are partition functions of each 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;

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

The chemical potential can be determined by

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

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

The pressure, p, can then be determined by

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