Statistical Thermodynamics and Rate Theories/Equations for reference

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 $$ (6.626068 \times 10^{-34} J s) $$, 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 mi is the mass an atom in the molecule and ri 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 re 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, $$ m_s = \plusmn {1 \over 2} $$.

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 ($$ 8.314 J K^{-1} mol^{-1} $$).

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

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:

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

This simplifies the translational molecular partition function to:

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

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

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, Θr, which has units of Kelvin:

$$ \Theta_r = {h^2 \over {8 {\pi}^2 k_B \mu r_{e}^2}} = {{\hbar}^2 \over {2 k_B \mu r_{e}^2}} $$ Using the characteristic temperature, the rotational molecular partition function is simplified to:

$$ q_{rot} = {T \over {\sigma \Theta_r}} $$

Molecular Vibrational Partition Function
$$ q_{vib} = {1 \over {1 - \exp\left({{-hv} \over {k_bT}}\right)}} $$

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 Θυ 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:

$$ q = {\left({2 \pi m k_B T \over h^2} \right)^{3 \over 2}V \times \left({2 k_B T \mu r_{e}^2 \over \hbar^2}\right) \times \left({1 \over 1 - \exp \left({-hv \over k_BT}\right)}\right) \times g_1 } $$

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 qA, qB , qC , and qD 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;

$$ K_p (t) = {{\rho_C}^{\nu_C} {\rho_D}^{\nu_D} \over {\rho_A}^{\nu_A} {\rho_B}^{\nu_B}} = (k_BT)^{\nu_C+\nu_D-\nu_A-\nu_B}K_c(T) $$ The chemical potential can be determined by

$$ \mu_i= -k_BT^2 \ln({q_i(V,T) \over N_i}) $$

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

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