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Statistical Mechanics

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Statistical mechanics

Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of Mechanics, which is concerned with the motion of particles or objects when subjected to a force. It provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed in every day life, therefore explaining thermodynamics as a natural result of statistics and mechanics (classical and quantum). In particular, it can be used to calculate the thermodynamic properties of bulk materials from the spectroscopic data of individual molecules.

At the heart of statistical mechanics is the partition function (see Derivation of the partition function):

$Z = \sum_i \exp\left(\frac{-E_i}{kT}\right)$

where k is Boltzmann's constant, T is the temperature and E_i reflects each possible energetic state of the system. This is the version for systems which don't allow an exchange of matter. Otherwise, we would have to introduce chemical potentials, μ_j, j=1,...,n and replace the partition function with

$Z = \sum_i \exp\left(\frac{\sum_{j=1}^n \mu_j N_{ij}-E_i}{kT}\right)$

where N_ij is the number of j^th species particles in the i^th configuration. Sometimes, we also have other variables to add to the partition function, one corresponding to each conserved quantity. Most of them, however, can be safely interpreted as chemical potentials. For the rest of this article, we will ignore this complication and pretend chemical potentials don't matter.

The partition function provides a measure of the total number of energetic states available to the system at a given temperature. Similarly,

$\exp\left(\frac{-E_i}{k T}\right)$

provides a measure of the number of energetic states of a particular energy that are likely to be occupied at a given temperautre.

Dividing the second equation by the first equation gives the probability of finding the system in a particular energetic state, i:

$p_i = \frac{\exp(-E_i / kT)}{Z}$

This probability can be used to find the average value, which corresponds to the macroscopic value, of any property, J, that depends on the energetic state of the system by using the formula:

$\langle J \rangle = \sum_i p_i J_i = \sum_i J_i \frac{\exp(-E_i / kT)}{Z}$

where < J > is the average value of property J. This equation can be applied to the internal energy, U, and pressure, P:

$U = \sum_i E_i \frac{\exp(-E_i / kT)}{Z}$

$P = \sum_i P_i \frac{\exp(-E_i / kT)}{Z}$

Subsequently, these equations can be combined with known thermodynamic relationships between U and P to arrive at an expression for P in terms of only temperature, volume and the partition function. Similar relationships in terms of the partition function can be derived for other thermodynamic properties as shown in the following table.

Helmholtz free energy:	A = - kTlnZ
Internal energy:	$U = kT^2 \left( \frac{d\ln Z}{dT} \right)_{N,V}$
Pressure:	$P = kT \left( \frac{d \ln Z}{dV} \right)_{N,T}$
Entropy:	S = klnZ + U / T
Gibbs free energy:	$G = -kT \ln Z + kTV \left( \frac{d \ln Z}{dV}\right)_{N,T}$
Enthalpy:	H = U + PV
Constant Volume Heat Capacity:	$C_V = \left( \frac{dU}{dT} \right)_{N,V}$
Constant Pressure Heat Capacity:	$C_P = \left( \frac{dH}{dT} \right)_{N,P}$
Chemical potential:	$\mu_i = -kT \left( \frac{d \ln Z}{dN_i} \right)_{T,V,N}$

It is often useful to consider the energy of a given molecule to be distributed among a number of modes. For example, translational energy refers to that portion of energy associated with the motion of the center of mass of the molecule. Configurational energy refers to that portion of energy associated with the various attractive and repulsive forces between molecules in a system. The other modes are all considered to be internal to each molecule. They include rotational, vibrational, electronic and nuclear modes. If we assume that each mode is independent (a very questionable assumption!!!!!) the total energy can be expressed as the sum of each of the components:

E = E_t + E_c + E_n + E_e + E_r + E_v

Where the subscripts t, c, n, e, r, and v correspond to translational, configurational, nuclear, electronic, rotational and vibrational modes, respectively. The relationship in this equation can be substituted into the very first equation to give:

$Z = \sum_i \exp\left(-\frac{E_{ti} + E_{ci} + E_{ni} + E_{ei} + E_{ri} + E_{vi}}{kT}\right)$

$= \sum_i \exp\left(\frac{-E_{ti}}{kT}\right) \exp\left(\frac{-E_{ci}}{kT}\right) \exp\left(\frac{-E_{ni}}{kT}\right) \exp\left(\frac{-E_{ei}}{kT}\right) \exp\left(\frac{-E_{ri}}{kT}\right) \exp\left(\frac{-E_{vi}}{kT}\right)$

= Z_tZ_cZ_nZ_eZ_rZ_v

Thus a partition function can be defined for each mode. Simple expressions have been derived relating each of the various modes to various measurable molecular properties, such as the characteristic rotational or vibrational frequencies.

Expressions for the various molecular partition functions are shown in the following table.

Nuclear	$Z_n = 1 \qquad (T < 10^8 K)$
Electronic	$Z_e = W_0 \exp(kT D_e + W_1 \exp(-\theta_{e1}/T) + \cdots)$
vibrational	$Z_v = \prod_j \frac{\exp(-\theta_{vj} / 2T)}{1 - \exp(-\theta_{vj} / T)}$
rotational (linear)	$Z_r = \frac{T}{\sigma} \theta_r$
rotational (non-linear)	$Z_r = \sqrt{\frac{\pi}{\sigma} \frac{T^3}{\theta_A \theta_B \theta_C)^{1/2}}}$
Translational	$Z_t = \frac{(2 \pi mkT)^{3/2}}{h^3}$
Configurational (ideal gas)	Z_c = V

These equations can be combined with those in the first table to determine the contribution of a particular energy mode to a thermodynamic property. For example the "rotational pressure" could be determined in this manner. The total pressure could be found by summing the pressure contributions from all of the individual modes, ie:

P = P_t + P_c + P_n + P_e + P_r + P_v

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