Statistical Thermodynamics and Rate Theories/Equations for reference

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Equation Sheet 1[edit]

Translational States[edit]

The translational energy of a particle in a 3 dimensional box is given by the equation:

Where h is Planck's constant , 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[edit]

The moment of inertia for a rigid rotor is given by the equation:

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:

where is the internuclear distance and μ is the reduced mass for the two atoms:

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

where . The degeneracy of each rotational state is given by .

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

where is the rotational constant, and can be related to the moment of inertia by the equation:

where c is the speed of light. To yield and values in units of wavenumbers (), c should be expressed in units of cm/s, i.e. .

Vibrational States[edit]

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

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

where k is the bond force constant.

Electronic States[edit]

An electron in an atom may be described by four quantum numbers: the principle quantum number, ; the angular momentum quantum number, ; the magnetic quantum number, and the spin quantum number, . For a hydrogen-like atom (possessing only a single electron), the energy of said electron is given by the equation:

where is the mass of an electron, e is the charge of an electron, and is the permittivity of free space.

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

The electronic degeneracy of the system may then be determined by .

Thermodynamic Relations[edit]

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:

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

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

where A is Helmholtz energy;

where q is heat and w is work;

where is the heat associated with a reversible process;

where n is the number of moles of a gas and R is the ideal gas constant ().

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

while pressure may be estimated by the similar calculation:

allows for the determination of q:

may also be related to the heat capacity at constant pressure:

which in turn allows for the determination of enthalpy:

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

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

where are the translational, rotational and vibrational degrees of freedom for the molecule, respectively.

Equation Sheet 2[edit]

Formula to calculate the internal energy is given by the formula

Where U is the internal energy of the system is the energy of the system, 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[edit]

Internal Energy of Canonical Ensemble

Where U is the internal energy of the system, 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

Where S is the entropy of the system <E> is the energy of the system, 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 is the Helmhotlz free energy, 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[edit]

Function to calculate the partition function Q can be calculated by

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

In which q is the molecular partition function, is the partition functions of the translational degree of freedom, is the partition functions of the rotational degree of freedom, is the partition functions of the electrical degree of freedom, is the partition functions of the vibrational degree of freedom.

Transitional partition functions

q_trans is the partition functions of the translational degree of freedom, 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;

Molecular Rotational Partition Function

Where is the partition functions of the rotational degree of freedom, T is the temperature in Kelvin, is the Boltzmann constant (1.3807 x 10^-23 joule s per kelvin (J/K)), 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), represents the determined by the function

In which h is planks constant (6.62607004 × 10^-34 m^2 kg/s), , 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

in which is the partition function of the vibrations of the molecule, T is the temperature in Kelvin, and is given by the function;

Molecular Electronic Partition function

in which is the electronic partition function of the molecule, is the ,

Also it can be determined by using an approximation of the ground state, making

Knowing these four equations, the entire molecular partition function can be written as

which is equivalent to the above function

Chemical Equilibrium[edit]

Determination of equilibrium constant is found using the equation

In which 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;

The chemical potential can be determined by

in which is the change in helmholtz energy when a new particle is added to the system

pressure (p) can then be determined by