Every single-particle state is of a fixed energy, formula_40. As the sub-ensemble associated with a single-particle state varies by the number of particles only, it is clear that the total energy of the sub-ensemble is also directly proportional to the number of particles in the single-particle state; where formula_41 is the number of particles, the total energy of the sub-ensemble will then be formula_42. Beginning with the standard expression for a grand partition function and replacing formula_43 with formula_42, the grand partition function takes the form