# Statistical weight

In statistical mechanics, the statistical weight is the relative probability of a particular feature of a state. It refers to the Configuration entropy of a system [1]. The configuration is the instantaneous arrangement of all the particles of the system throughout the distinct energy levels of the system. The macrostate of the system will reflect those configurations of the particles which are more likely; we assume that the probability of a particle being at each level is equal[2]. It is denoted W which can also be referred to as multiplicity and is the tenet of entropy.

W = ​N!no! n1! n2! where N is the total number of particles and n the number of particles in any one of the energy levels available. ! is the factorial expression whereby x! = x(x-1)(x-2)... ..1 and 0! =1 and is the mathematical version of whether a tiny number of particles have a vast amount of energy or a ginormous number of particles have a low energy value. The weight relates the microstate to the number of ways particles can be arranged to match the macrostate (T,V,P); it expresses the nanoscopic in the scale of our reality.

If the energy associated with the feature is ΔE, the statistical weight is given by the Boltzmann factor e−ΔE/kT, where k is the Boltzmann constant and T is the temperature in kelvins.

The statistical weight is a convenient shorthand that is often used in transfer matrix solutions of problems in statistical mechanics.

In statistical mechanics, we always seek the number of microstates corresponding to a given macrostate ( N, V, U). it is called the thermodynamic probability or the Statistical Weight of the macrostate and is denoted by W( N, V, U). This variable is related to the thermodynamic variable entropy S . You will observe that the relation between the entropy S and the thermodynamic probability W forms the basis of entire statistical analysis.

Derivation of Relation between entropy and thermodynamic probability .

We now search for a microscopic explanation for S in terms of . We start from the TdS equation

T dS = dU + P dV �dN .....(1)

So,

S =� f (E; V;N) .....(2)

Also

� W = g(E; V;N) ......(3)

Note : In adiabatic changes, for example, volume increase with no heat added (dQ = 0). dS = 0 i.e. S is constant.Also dnj = 0 so W is also constant .

During irreversible changes, the total S in the Universe increases. The total W of the Universe also increases because the equilibrium macrostate must be the maximum W.

## References

1. ^ Atkins, PW; Julio de Paula (2002). Physical Chemistry (7th ed.). OUP.
2. ^ Gasser, R.P.H.; W.G. Richards (1974). Oxford Chemistry series: Entropy and Energy Levels (1st ed.). OUP.
• Atkins, PW; Julio de Paula (2002). Physical Chemistry (7th ed.). OUP.
• Gasser, R.P.H.; W.G. Richards (1974). Oxford Chemistry series: Entropy and Energy Levels (1st ed.). OUP.