# 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 the macroscopic properties. Classical thermodynamics describes macroscopic properties of a system composed of atoms or molecules such as pressure, enthalpy and internal energy. On the other hand, quantum mechanics utilizes quantized values of microscopic properties of a system, such as rotational and vibrational movement, to calculate the energy of the system. 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 volume, energy and total number of particles are held constant. The total number of microcanonical ensembles (𝒜), can be averaged to obtain a mean result, known as the 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 → ∞. This law states that the average value of a mechanical variable 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, which relates the energy of said microstates to internal energy of a system as calculated by classical thermodynamics.

## Gibbs's Postulate

Gibbs's Postulate relates the internal energy (U) of a system, determined by thermodynamics, to the average ensemble energy (E), determined by statistical mechanics.

${\displaystyle U=\langle E\rangle }$

The average energy over the 𝒜 copies in the ensemble is given by the following equation.

${\displaystyle \langle E\rangle ={\frac {1}{A}}\sum _{i}^{A}E^{i}}$
${\displaystyle E^{i}=hv(n_{1}^{i}+n_{2}^{i}+n_{3}^{i}+...)}$

Through the Gibbs Postulate, the average ensemble energy can be used to define the thermodynamic values of Helmholtz energy (${\displaystyle A}$ ), entropy (${\displaystyle S}$ ), pressure (${\displaystyle p}$ ), and thermodynamic potential (${\displaystyle \mu }$ ) through the following relationships.

${\displaystyle {A=U-TS}}$
${\displaystyle {S=-\left({\frac {\partial A}{\partial T}}\right)_{N,V,}}}$
${\displaystyle {p=-\left({\frac {\partial A}{\partial V}}\right)_{N,T,}}}$
${\displaystyle {\mu =\left({\frac {\partial A}{\partial N}}\right)_{T,V,}}}$

## 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 ${\displaystyle n}$ , with ${\displaystyle n=0,1,2,3...}$ , there is an equal probability of the first system occupying the states ${\displaystyle n_{1}=1,n_{2}=0,n_{3}=2}$ , as there is the second system occupying the states ${\displaystyle n_{1}=6,n_{2}=4,n_{3}=9}$ . Across the distribution for each system there is an indiscriminate occupation of each quantum state which is just as likely as any other.