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 potential**,*µ.*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:

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

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