In: Other
Describe the concept of a ‘canonical ensemble’ and why it is useful in statistical thermodynamics. Prove that for an ensemble of distinguisable independent molecules the canonical partition function Q=q^N
If a closed system of specified volume, composition, and temperature & consider that it is replicated N times . These identical closed systems are regarded as being in thermal contact with one another so that they can exchange energy. The total energy of all the systems is E & because they are in thermal equilibrium with one another, they all have the same temperature, T. This imaginary collection of replications of the actual system with a common temperature is called the canonical ensemble.
We know the canonical partition function (Q):
Q =∑ e(- aEi)
where Ei is the energy of some members & let there are N members of the canonical ensemble which has Ei as it's energy:
The total energy of a collection of N independent molecules is the sum of the energies of the molecules. Therefore, we can write the total energy of a state i of the system as
E = Ei (1) + Ei (2) + ··· + ei (N)
In this expression, Ei (1) is the energy of molecule 1 when the system is in the state i, Ei (2) the energy of molecule 2 when the system is in the same state i, and so on. The canonical partition function is then
Q =∑ e[- aEi (1) -aEi (2)...... -aEI(N)]
since each Ei are same,
so, Q = [∑ e(- aEi) ]N
Hence Q = qN