# Chemistry and Maths #1: Statistical Thermodynamics

Statistical mechanics is a branch of physics that uses probability theory to study the behaviour of a mechanical system whose state is uncertain. A common use of statistical mechanics is in the study of thermodynamic behaviour of large systems. Statistical thermodynamics “provides a connection between the macroscopic properties of materials in thermodynamic equilibrium, and the microscopic behaviours and motions occurring inside the material“.

There are three main ensembles – isolated systems with a finite volume – of statistical mechanics:

• Microcanonical Ensemble – describes an isolated system. This ensemble contains each possible state that’s consistent with that energy and composition with equal probability.
• Canonical Ensemble – describes a system in contact with a heat bath. This ensemble contains states of varying energy, but with identical composition.
• Grand Canonical Ensemble – describes a system in contact with a heat and particle bath. This ensemble contains stated of varying energy and varying numbers of particles.

### Microcanonical Ensemble

Fixed variables:

• Total number of particles in the system, N.
• System’s volume, V.
• Total energy in the system, E.

Every microstate that has energy E has the same probability:

where W is the number of microstates.

Entropy can be defined for this ensemble using the Boltzmann entropy formula:

### Canonical Ensemble

Fixed Variables:

• Number of particles in the system, N.
• Absolute temperature, T.
• System’s volume, V.

In this ensemble, each microstate is assigned a probability, P, using the following formula:

where k is Boltzmann’s constant.

The number, F, defined as the Helmholtz free energy, is a constant for the ensemble as is calculated by:

### Grand Canonical

Fixed Variables:

• Chemical potentialµ. This is a form of potential energy that can be absorbed or released during a chemical reaction.
• Absolute temperature, T.
• System’s Volume, V.

The probability, P, given to each distinct microstate is given by:

where Ω is the ‘grand potential’.

The grand potential is a constant for this ensemble and can be calculated using the following equation:

Sources 1 | 2

I have another post on chemistry coming on Friday! Hope you enjoy. M x