4. Grand canonical ensemble

What is the main characteristic of the grand canonical ensemble?

N is not fixed for each system.

How do we allow N to vary?

Make the walls permeable to particles.

What conditions are fixed for each system in the grand canonical ensemble?

T, V, and μ.

What do the systems exchange with the baths?

Energy (thermal bath) and particles (particle bath).

What happens after equilibrium is reached?

The entire ensemble is isolated.

What set of energy states corresponds to a given particle number N?

What is a_Nj?

Number of systems containing N atoms and in state j.

What is a distribution in the grand canonical ensemble?

What does the postulate of equal a priori probabilities state here?

All states associated with all possible distributions have equal probability.

What is the total number of systems?

What is the total energy of the ensemble?

What is the total number of particles?

What is the number of ways to configure the ensemble?

What symbol is used for the grand canonical partition function?

used to link this ensemble to the partition functions of other ensembles.

What is the expression linking entropy to Ξ?

What is fugacity (activity)?

What does Ω(N,V,E) represent?

The partition function of the microcanonical ensemble.

What happens if you remove the partitions in the grand canonical ensemble?

You obtain a large isolated system with constant energy.

What are the total macroscopic quantities of the isolated system?

• Total energy: E

• Total volume: A×V

• Total number of molecules/atoms: N

What is the entropy in terms of Ω?

What does Ω represent physically?

The number of states available at a particular energy.

What does the Boltzmann equation tell you about entropy?

Entropy increases when more states are available.