Answer to Question #221520 in Mechanical Engineering for john luvis

Question

Using Lagrange’s method of undetermined multipliers, show that the partition function for a canonical ensemble is

Answer 1

A canonical ensemble is a collection of systems characterized by the same values of N, V, and T. The assignment of a fixed temperature is justified by imagining that the systems of the ensemble are initially brought into thermal equilibrium with each other by immersing them in a heat bath at a temperature T.

The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at a fixed temperature, volume, and a number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at a fixed temperature, volume, and chemical potential.