As originally formulated by Benjamin Widom in 1963,^[1] the approach can be summarized by the equation:

${\displaystyle \mathbf {B} _{i}={\frac {\rho _{i}}{a_{i}}}=\left\langle \exp \left(-{\frac {\psi _{i}}{k_{B}T}}\right)\right\rangle }$ 

where ${\displaystyle \mathbf {B} _{i}}$  is called the insertion parameter, ${\displaystyle \rho _{i}}$  is the number density of species ${\displaystyle i}$ , ${\displaystyle a_{i}}$  is the activity of species ${\displaystyle i}$ , ${\displaystyle k_{B}}$  is the Boltzmann constant, and ${\displaystyle T}$  is temperature, and ${\displaystyle \psi }$  is the interaction energy of an inserted particle with all other particles in the system. The average is over all possible insertions. This can be understood conceptually as fixing the location of all molecules in the system and then inserting a particle of species ${\displaystyle i}$  at all locations through the system, averaging over a Boltzmann factor in its interaction energy over all of those locations.

Note that in other ensembles like for example in the semi-grand canonical ensemble the Widom insertion method works with modified formulas.^[5]

Chemical potential edit

From the above equation and from the definition of activity, the insertion parameter may be related to the chemical potential by

${\displaystyle \mu _{i}=-k_{B}T\ln \left({\frac {\mathbf {B} _{i}}{\rho _{i}\lambda ^{3}}}\right)=\underbrace {k_{B}T\ln(\rho _{i}\lambda ^{3})} _{\mu _{id}}\underbrace {-k_{B}T\ln \left(\left\langle \exp \left(-{\frac {\psi _{i}}{k_{B}T}}\right)\right\rangle \right)} _{\mu _{ex}}=\mu _{id}+\mu _{ex}}$ 

Equation of state edit

The pressure-temperature-density relation, or equation of state of a mixture is related to the insertion parameter via

${\displaystyle Z={\frac {P}{\rho k_{B}T}}=1-\ln \mathbf {B} +{\frac {1}{\rho }}\int \limits _{0}^{\rho }\ln \mathbf {B} \,d\rho '}$ 

where ${\displaystyle Z}$  is the compressibility factor, ${\displaystyle \rho }$  is the overall number density of the mixture, and ${\displaystyle \ln \mathbf {B} }$  is a mole-fraction weighted average over all mixture components:

${\displaystyle \ln \mathbf {B} =\sum _{i}{x_{i}\ln \mathbf {B} _{i}}}$ 

In the case of a 'hard core' repulsive model in which each molecule or atom consists of a hard core with an infinite repulsive potential, insertions in which two molecules occupy the same space will not contribute to the average. In this case the insertion parameter becomes

${\displaystyle \mathbf {B} _{i}=\mathbf {P} _{ins,i}\left\langle \exp \left(-{\frac {\psi _{i}}{k_{B}T}}\right)\right\rangle }$ 

where ${\displaystyle \mathbf {P} _{ins,i}}$  is the probability that the randomly inserted molecule of species ${\displaystyle i}$  will experience an attractive or zero net interaction; in other words, it is the probability that the inserted molecule does not 'overlap' with any other molecules.

The above is simplified further via the application of the mean field approximation, which essentially ignores fluctuations and treats all quantities by their average value. Within this framework the insertion factor is given as

${\displaystyle \mathbf {B} _{i}=\mathbf {P} _{ins,i}\exp \left(-{\frac {\left\langle \psi _{i}\right\rangle }{k_{B}T}}\right)}$