1. A molecule in thermal equilibrium with a bath at temperature, T, has two accessible energy states, with energies $U_1$ and $U_2$, multiplicities $W_1$ and $W_2$ (e.g. each energy state consists of $W_1$ and $W_2$ microstates, respectively), and the corresponding probabilities of each energy state are $p_1$ and $p_2$. Note that these probabilities correspond to the whole energy state, not individual microstates.
a. If $p_1$ represents state probability for state 1, and we define $P_{micro1}$ to be the probability of a specific microstate from energy state 1, what is the ratio $p_1/P_{micro1}$?
b. What is $P_{micro2}/P_{micro1}$ where $P_{micro2}$ is the probability of seeing a specific microstate from energy state 2?
c. If $U_1 = U_2$ and $W_1 = 10^{-2} \cdot W_2$, then what is $p_2/p_1$ at equilibrium?
d. If $W_1 = W_2$ and $p_2/p_1 = e^{-2}$, then what is $U_2 - U_1$ at equilibrium?
e. If the system undergoes a temperature change from $T_1$ to $T_2$, describe what happens to the energies, multiplicities, and probabilities of the two states at equilibrium.
f. If the system is at equilibrium and has N independent molecules, write an expression for the system entropy, S, in terms of the state energies, multiplicities, state probabilities, and microstate probabilities. Note, you may not need all of these parameters to correctly express the system entropy.
g. What is the average energy of the molecule at temperature T.
Determine if the following statements are true or false for this system of molecules
h. If the system consists of N molecules total, the average number of molecules in energy state 1 is $Np_1$ at equilibrium.
i. The number of molecules in energy state 1 plus the number of molecules in energy state 2 is exactly N.
j. The probability that there are exactly zero molecules in energy state 1 is $(p_2)^N$