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 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:

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

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

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

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

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

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

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

where ${\displaystyle \hbar ={h \over 2\pi }}$  and 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 avoid confusing in obtaining values for the frequency of radiation, ${\displaystyle {\tilde {\nu }}}$ , and the rotational constant, ${\displaystyle {\tilde {B}}}$ , c is often expressed in units of cm/s (${\displaystyle c=2.99792458\times 10^{10}cm/s}$ )

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, ${\displaystyle m_{l}=-l,-l+1,...,l-1,l}$ ; the spin quantum number, ${\displaystyle m_{s}=\pm {1 \over 2}}$ .

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

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

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:

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

${\displaystyle C_{v}}$  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.

For linear molecules the internal energy simplifies to:

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

And for non-linear molecules:

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

Equation Sheet 2

Formula to calculate the internal energy is given by the formula

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

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

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

Where ${\displaystyle k_{B}}$  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:

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

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

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

Where ${\displaystyle k_{B}}$  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:

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

Where q is the molecular partition functions.

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 molecular partition function of the translational degree of freedom, ${\displaystyle q_{rot}}$  is the molecular partition function of the rotational degree of freedom, ${\displaystyle q_{vib}}$  is the molecular partition function of the vibrational degree of freedom, and ${\displaystyle q_{elec}}$  is the molecular partition function of the electronic degree of freedom.

Molecular Translational Partition Function

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

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

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

This simplifies the translational molecular partition function to:

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

Molecular Rotational Partition Function

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

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

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

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

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

Molecular Vibrational Partition Function

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

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

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

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:

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

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

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

Molecular Electronic Partition Function

${\displaystyle q_{elec}=g_{1}}$

Where ${\displaystyle q_{elec}}$  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:

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

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

Simplified Molecular Partition Function

All molecular partition functions combined are defined as:

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

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.

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

Which is equivalent to:

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

Chemical Equilibrium

Determination of equilibrium constant ${\displaystyle K_{c}}$  is found by the following 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 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;

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

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

The pressure, p, can then be determined by

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