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:

$E_{n_x,n_y,n_z}=\frac{h^2}{8m}(\frac{{n_x}^2}{a^2}+\frac{{n_y}^2}{b^2}+\frac{{n_z}^2}{c^2})$

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:

$\mu =\frac{m_1{m}_2}{m_1+m_2}$

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

$\mu =\frac{m_1{m}_1}{m_1+m_1}=\frac{m_1^2}{2m_1}=\frac{m_1}{2}$

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

$E_J=\frac{{\hslash }^2}{2\mu r_e^2}J(J+1)$

where $\hslash =\frac{h}{2\pi }$ 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:

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

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:

$E_n=h\nu (n+\frac{1}{2})$

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

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

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, $m_s=\pm \frac{1}{2}$.

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

$E_n=-\left(\frac{m_e{e}^4}{32{\pi }^2{ϵ}_0^2{\hslash }^2}\right)\frac{1}{n^2}$

where $m_e$ is the mass of an electron, e is the charge of an electron, and ${ϵ}_0$ 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;

$\mathrm{\Delta }U=q+w$

where q is heat and w is work;

$dS=\frac{d{q}_{rev}}{T}$

where $d{q}_{rev}$ 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 ($8.314J{K}^{-1}mo{l}^{-1}$).

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

$C_v={\left(\frac{\partial U}{\partial T}\right)}_{V,n}$

while pressure may be estimated by the similar calculation:

$p=-(\frac{\partial U}{\partial V}{)}_n$

$C_v$ allows for the determination of q:

$q=C_v\mathrm{\Delta }T$

$C_v$ 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:

$\mathrm{\Delta }H=C_p\mathrm{\Delta }T$

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

$C_v=n{C}_v^m$

$C_p=n{C}_p^m$

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

$U=\frac{1}{2}{n}_{trans}nRT+\frac{1}{2}{n}_{rot}nRT+n_{vib}nRT$

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:

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

And for non-linear molecules:

$U=\frac{3}{2}nRT+\frac{3}{2}nRT+(3n_{atom}-6)nRT$ Template:BookCat

Equation Sheet 2

Formula to calculate the internal energy is given by the formula

$U=⟨E⟩=\frac{\sum _j{E}_j\exp (-E_j/k_{B}T)}{Q}$

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:

$U=⟨E⟩=k_B{T}^2{\left(\frac{\partial \ln Q}{\partial T}\right)}_{N,V}$

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:

$S=\frac{⟨E⟩}{T}+k_B\ln Q$

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:

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

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

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

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:

$\mathrm{\Lambda }={\left(\frac{2\pi m{k}_{B}T}{h^2}\right)}^{-1/2}$

This simplifies the translational molecular partition function to:

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

Molecular Rotational Partition Function

$q_{rot}=\frac{8{\pi }^2{k}_{B}T\mu r_e^2}{\sigma h^2}=\frac{2k_{B}T\mu r_e^2}{\sigma {\hslash }^2}$

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, $\hslash$ is defined as $\frac{h}{2\pi }$, 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:

${\mathrm{\Theta }}_r=\frac{h^2}{8{\pi }^2{k}_B\mu r_e^2}=\frac{{\hslash }^2}{2k_B\mu r_e^2}$

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

$q_{rot}=\frac{T}{\sigma {\mathrm{\Theta }}_r}$

Molecular Vibrational Partition Function

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

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:

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

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:

${\mathrm{\Theta }}_{\nu }=\frac{h\nu }{k_B}$

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

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

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:

$q_{elec}=g_1\exp \left(\frac{D_0}{k_{B}T}\right)$

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:

$q={\left(\frac{2\pi m{k}_{B}T}{h^2}\right)}^{\frac{3}{2}}V\times \left(\frac{2k_{B}T\mu r_e^2}{{\hslash }^2}\right)\times \left(\frac{1}{1-\exp \left(\frac{-hv}{k_{B}T}\right)}\right)\times g_1$

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.

$q=\frac{V}{{\mathrm{\Lambda }}^3}\times \frac{T}{\sigma {\mathrm{\Theta }}_r}\times \frac{1}{1-\exp \left(\frac{-{\mathrm{\Theta }}_{\nu }}{T}\right)}\times g_1$

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:

$K_c(T)=\frac{((q_C/V{)}^{{\nu }_C}(q_D/V{)}^{{\nu }_D})}{((q_A/V){]}^{{\nu }_A}(q_B/V{)}^{{\nu }_B})}=\frac{{{\rho }_C}^{{\nu }_C}{{\rho }_D}^{{\nu }_D}}{{{\rho }_A}^{{\nu }_A}{{\rho }_B}^{{\nu }_B}}$

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;

$K_p(t)=\frac{{{\rho }_C}^{{\nu }_C}{{\rho }_D}^{{\nu }_D}}{{{\rho }_A}^{{\nu }_A}{{\rho }_B}^{{\nu }_B}}=(k_{B}T{)}^{{\nu }_C+{\nu }_D-{\nu }_A-{\nu }_B}{K}_c(T)$

The chemical potential can be determined by

${\mu }_i=-k_B{T}^2\ln (\frac{q_i(V,T)}{N_i})$

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

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