Question

More than one distinct quantum state of a system can have a particular value of energy. Here we remember to count all the states in figuring out the probability that a small system has a particular energy value.

The molecule can be in one of six states. It exchanges energy with a thermal reservoir of other molecules at a temperature T. State 1 has E_A, states 2 and 3 each have energy E_B = 3E_A, and states 4,5, and 6 each have energy E_C=5E_A. Now to simplify the calculations, let's set the temperature so that k_BT=E_A.

What is the probability that the molecule has energy E_B?
P(E_B) =

Answer 1

average KE of the molecule = 3/2 k_B T , T is the tempearature

we have 1 state with E_A = k_BT - degeneracy g₁ = 1

2 - states with E_B = 3k_BT - degeneracy g₂=2

3 stats with E_c = 5k_BT - degeneracy g₃ =3

The total number of particles occupying any state with energy E_i is

n_i = g_i A exp(-E_i /k_B T)

n_a /n_b = g₁ /g₂ exp( EB -EA)/k_BT = e² /2

n_a /n_c = g₁ /g3 exp( E_C -EA)/k_BT = e⁴ /5

Probabaility finding the molecule in EB = n_b /(n_a + n_b + n_c )

= 2 e^-2 /(1+ 2 e^-2 + 5e^-4 )

= 0.199