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 is the mass of each atom in the molecule and r 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 two atoms:

$\mu ={{m_{1}}{m_{2}} \over {m_{1}+m_{2}}}$ 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 $\hbar ={h \over 2\pi }$ . 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 yield ${\tilde {\nu }}$ and ${\tilde {B}}$ values in units of wavenumbers ($cm^{-1}$ ), c should be expressed in units of cm/s, i.e. $c=2.99792458\times 10^{10}cm/s$ .

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, $m_{l}=-l,-l+1,...,l-1,l$ and the spin quantum number, $m_{s}=\pm {1 \over 2}$ . For a hydrogen-like atom (possessing only a single electron), the energy of said electron is given by the equation:

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

$q=C_{v}\Delta T$ $C_{V}v$ 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.

Equation Sheet 2

Formula to calculate the internal energy is given by the formula

$U=\langle E\rangle =\sum _{j}{E_{j}}{\exp(-E_{j}/{k_{B}}T)}/Q$ 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 joule s per kelvin (J/K)) and T is the temperature in Kelvin. Q is the partition function of the system

Canonical Ensemble

Internal Energy of Canonical Ensemble

$U=\langle E\rangle =k_{B}T^{2}\left({\partial \ln Q \over \partial T}\right)_{N,V}$ Where U is the internal energy of the system,$k_{B}$ is the Boltzmann constant (1.3807 x 10^-23 joule s per kelvin (J/K)) and T is the temperature in Kelvin. Q is the partition function of the system

Entropy of the Canonicial Ensemble

$S={\langle E\rangle \over T}+k_{B}\ln Q$ Where S is the entropy of the system <E> is the energy of the system, $k_{B}$ is the Boltzmann constant (1.3807 x 10^-23 joule s per kelvin (J/K)) and T is the temperature in Kelvin. Q is the partition function of the system

Helmholtz Free Energy of the Canonical Ensemble

$A=-k_{B}T\ln Q$ A is the Helmhotlz free energy, $k_{B}$ is the Boltzmann constant (1.3807 x 10^-23 joule s per kelvin (J/K)), T is the temperature in Kelvin, and Q is the partition function of the system.

Partition Functions

Function to calculate the partition function Q can be calculated by

$Q={q^{N} \over N!}$ In which Q is the partition function, q is the molecular partition functions, and N is the number of molecules.

Calculation of the molecular partition function

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

Transitional partition functions

$q_{trans}(V,T)=(({2\pi mk_{B}T \over h^{2}})^{3}/2V)={V \over \Lambda ^{3}}$ q_trans is the partition functions of the translational degree of freedom, $k_{B}$ is the Boltzmann constant (1.3807 x 10^-23 joule s per kelvin (J/K)), m is the mass of the molecule, T is the temperature in Kelvin, V is the volume of the system, and \lambda is the deBrogie wavelength which can be given by the equation;

$\Lambda =(({2\pi mk_{B}T \over h^{2}})^{-1/2})$ Molecular Rotational Partition Function

$q_{rot}(T)={2k_{B}T\mu r_{e}^{2} \over \sigma h^{2}}={T \over \sigma \theta _{r}}$ Where $q_{rot}$ is the partition functions of the rotational degree of freedom, T is the temperature in Kelvin, ${k_{B}}$ is the Boltzmann constant (1.3807 x 10^-23 joule s per kelvin (J/K)), $r_{e}$ is the bond length of the molecule, μ is the reduced mass, σ is the symmetry factor (σ=2 for homonuclear molecules and σ=1 for heteronuclear), $\theta _{r}$ represents the determined by the function

$\theta _{r}={hv \over k_{B}}$ In which h is planks constant (6.62607004 × 10^-34 m^2 kg/s), $\theta _{r}$ , v is the frequency of the rotation of the molecule and {k_B} is the Boltzmann constant (1.3807 x 10^-23 Joule s per kelvin (J/K))

Molecular Vibrational Partition Function

$q_{vib}(T)={1 \over 1-\exp(-\theta _{v}/T)}$ in which $q_{vib}$ is the partition function of the vibrations of the molecule, T is the temperature in Kelvin, and $\theta _{v}$ is given by the function;

$\theta _{v}={hv \over k_{b}}$ Molecular Electronic Partition function

$q_{elec}=g_{1}\exp({D_{0} \over k_{B}T})$ in which $q_{elec}$ is the electronic partition function of the molecule, $D_{0}$ is the ,

Also it can be determined by using an approximation of the ground state, making $q_{elec}=g_{1}$ Knowing these four equations, the entire molecular partition function can be written as

$q=g_{1}\times {1 \over 1-\exp(-\theta _{v}/T)}\times \left({2\pi mk_{B}T \over h^{2}}\right)^{3}/2V\times {2k_{B}T\mu r_{e}^{2} \over \sigma h^{2}}$ which is equivalent to the above function $q=q_{trans}q_{rot}q_{vib}q_{elec}$ Chemical Equilibrium

Determination of equilibrium constant $K_{c}$ is found using the 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 $q_{c},q_{D},q_{B},andq_{A}$ are partition functions of each species of 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

pressure (p) can then be determined by

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