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