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### GIBBS CANONICAL ENSEMBLE

(SYSTEM IN CONTACT WITH A HEAT RESERVOIR)

Canonical ensemble describes those systems that are not isolated but are in contact with a heat reservoir.

The system under consideration together with a heat reservoir forms a closed system and then system of interest is taken as sub system of this closed system.

Any sub system of an isolated system in statistical equilibrium can be represented by a canonical ensemble.

The probability density of a canonical ensemble depends both on Energy 'E' and temperature 'T'

It is given by

ρ(E,T) = Ae-E/𝜏

Partition function for canonical ensemble

z=(1/h³ⁿ.n!)∫exp(-E(q,p)/KTdΓ

The statistical energy of a system in a canonical ensemble is given by

𝜎 = logZ  +(E/𝜏);  𝜎 = logZ  +T[∂(log Z)/∂T]

Thermodynamical entropy 'S' of a system in a canonical ensemble is given by

S=K𝜎

S=KlogZ + (E/T); U=KT2[∂(log Z)/∂T]

The entropy at absolute zero in a canonical ensemble can be expressed as

S=K.log g

gₒ is statistical weight of ground state

### What is Chemical Potential?

For a chemical system, molar free energy is known as Chemical Potential.

A chemical substance that is free to move from one place to another place, will move spontaneously from a state of higher chemical potential to a state of lower chemical potential.

In the position of equilibrium, the chemical potential is constant through the entire system.

Let us consider a general heterogeneous system consisting of an independent components in several coexisting phases.

To start with, it is convenient to describe a given phase by its chemical composition, which is specified by the no. of mole 'Ni' of each species i, its volume V and its entropy 'S'.

If we consider internal energy (U)

U=U(S,V,N₁,N₂,.....Nᵢ,.....Nn)
μi=❴∂U/∂Nᵢ❵S,V,Nj ; j= except 'i'

'μi' is chemical potential of component 'i' in given phase.

dU=TdS-PdV+Σμᵢi.dNᵢ for i=1...n

We can also consider chemical potential 'μ' in terms of Helmoltz free energy 'F'.

F = F(T,V,N₁,N₂......Nn)

μ1=❴∂F/∂N1❵T,V,N₂,....

μ2=❴∂F/∂N2❵T,V,N₁,N₃,....

The chemical potentials are thus the rate of change of free energy per mole, at constant volume and temperature.

μ can also be expressed as

μi=❴∂G/∂Nᵢ❵T,P,Nj

A System in external field will be in equilibrium if the temperature and chemical potential of each component of the system is constant through out, i.e.

dT₁=0 and dμᵢ=0