Consider a system of ( N ) distinguishable non-interacting...

Consider a system of ( N ) distinguishable non-interacting particles, each of which has either of two energies, 0 and ( epsilon ), where ( epsilon>0 ). (For example, ( N ) localized spins which point either up or down in a magnetic field.) (a) Suppose the system is in equilibrium at temperature ( T ). What is the probability ( p_{0}(T) ) that a particular particle is in the lower energy level? What is the probability ( p_{1}(T) ) that the particle is in the upper energy level? (b) Calculate the total energy ( U ) and the heat capacity ( C ) as functions of temperature. (c) Sketch ( U ) and ( C ) as functions of temperature.

Transcript

00:01 For a, you want to find the probabilities p0 of t and p1 of t.

00:07 So for a system in thermal equilibrium at temperature t, the probability that a system is in a particular state follows the boltzmann distribution.

00:19 So the probability pi that a particle is in state i with energy epsilon i is given by pi equals e to negative epsilon i over kb times t over z, where kb is boltzmann's constant, and z is the partition function for a single particle, which sums over all possible states of the particle.

00:51 So for our two -state system, we have epsilon 0 equals 0, epsilon 1 equals epsilon, and the partition function z is therefore e to negative epsilon 0 over kb t plus e to negative epsilon 1 over kb t, which is equal to 1 plus e to negative epsilon over kb t.

01:21 So the probabilities are for p0 of t, this is the particle being in the lower energy level, so energy equals 0.

01:32 This is equal to e to negative 0 over kb times t over z which is equal to 1 over 1 plus e to negative epsilon k over kb times t and p1 of t is equal to e to negative epsilon epsilon over kb t over z or equals to e to negative epsilon over kb times t over 1 plus e to negative epsilon over kb times t.

02:22 Okay, now we're gonna do part.

02:25 We're gonna calculate the total energy u and the heat capacity c.

02:29 So the total energy u of the system is the sum of the expected energies of all particles.

02:35 Since the particles are non -interacting and identical, the total energy is simply n times the expected energy of a single particle...

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