Statistical Thermodynamics and Rate Theories/Postulates of Statistical Thermodynamics

Postulates of Statistical Thermodynamics

Statistical thermodynamics is a branch of science which utilizes statistics in order to relate the microscopic properties of a system to macroscopic properties. Classical thermodynamics describes macroscopic properties of a system composed of atoms or molecules such as pressure, enthalpy and internal energy quite accurately, while quantum mechanics utilizes quantized values of microscopic properties of a system, such as rotational and vibrational movement to calculate energy of the system, among other properties. Using two different methodologies, both thermodynamics and quantum mechanics can be utilized to evaluate similar properties of a system without any clear connection between the two fields. Statistical thermodynamics provides this connection by averaging the microscopic values across a large set of microcanonical ensembles at an instant in time to arrive at macroscopic values. These microcanonical ensembles are physical systems in which the total number of particles, total volume and total energy are held constant. By having a large set of these identical microcanonical ensembles, the total number of microcanonical ensembles given by 𝒜, the quantum states of the atoms in each separate system can be averaged to obtain a mean result, replicating that of thermodynamic value, otherwise called a mechanical value.

First Postulate of Statistical Thermodynamics

The First Postulate of Statistical Thermodynamics states that the time average of a mechanical variable M in the thermodynamic system of interest is equal to the ensemble average of M as the limit of 𝒜 → ∞. In context, this law states that the average value of a mechanical variable, such as pressure, taken from the ensembles of microstates matches the mechanical value predicted by classical thermodynamics so long as the number of microstates 𝒜 is an exceedingly high number. The first postulate of statistical thermodynamics can be extended to arrive at the Gibbs postulate, a postulate which relates the energy of said microstates to internal energy of a system as calculated by classical thermodynamics.

Gibbs's Postulate

Gibbs's Postulate is one which relates the internal energy, U, of a system as determined by thermodynamics to the average ensemble energy, E, as determined by statistical mechanics.

$U=\langle E\rangle$ Through this relation, the average ensemble energy can be used to define the thermodynamic values of Hemholtz energy, $A$ , entropy, $S$ , pressure, $p$ , and thermodynamic potential, $\mu$ through the following relationships.

${A=U-TS}$ ${S=-\left({\frac {\partial A}{\partial T}}\right)_{N,V,}}$ ${p=-\left({\frac {\partial A}{\partial V}}\right)_{N,T,}}$ ${\mu =\left({\frac {\partial A}{\partial N}}\right)_{T,V,}}$ Second Postulate of Statistical Thermodynamics

The Second Postulate of Statistical Thermodynamics states that for an ensemble representative of an isolated system, the systems of the ensemble are distributed uniformly. All states consistent with the specified microcanonical system will occur with equal probability. This is otherwise known as the principle of equal 'a priori' probabilities. For example, in two microcanonical systems with three particles capable of occupying a quantum level $n$ , with $n=0,1,2,3...$ , there is an equal probability of the first system occupying the states $n_{1}=1,n_{2}=0,n_{3}=2$ , as there is the second system occupying the states $n_{1}=6,n_{2}=4,n_{3}=9$ . Across the distribution for each system there is an indiscriminate occupation of each quantum state which is just as likely as any other.