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:

P=1/W,

where W is the number of microstates.

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

S_{\rm {B}}=k\log W=k\log {\Big (}\omega {\frac {dv}{dE}}{\Big )}

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:

P=e^{\frac {F-E}{kT}},

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:

F=-k_{B}T\ln Z

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:

P=e^{\frac {\Omega +\mu N-E}{kT}},

where Ω is the ‘grand potential’.

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

\Omega =-k_{B}T\ln {\mathcal {Z}}

Sources 1 | 2

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

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