Question

Show that Helmholtz free energy F and F' of a set n localized particles, each of which can exist in levels of energy 0, ε, 2ε, and 3ε, having the degeneracy 1, 3, 3, and 1 respectively, is given by:

F = -3NkBT ln (1 + exp(-ε/kBT))

Answer 1

The Helmholtz free energy F and F' of a set of n localized particles can be calculated using the formula -3NkBT ln (1 + exp(-ε/kBT)).

Step-by-step explanation:

The Helmholtz free energy, denoted as F, of a set of n localized particles with different energy levels and degeneracy can be calculated using the formula:

F = -3NkBT ln (1 + exp(-ε/kBT))

where N is the number of particles, kB is the Boltzmann constant, T is the temperature, ε is the energy, and exp is the exponential function.

In the given question, the energy levels are 0, ε, 2ε, and 3ε, and the corresponding degeneracies are 1, 3, 3, and 1 respectively. Plugging in these values into the formula will yield the Helmholtz free energy F and F'.