Partition function and heat capacity

[quietrain]

Homework Statement

QN1. A system consists of N weakly interacting subsystems. Each subsystem possesses only two energy levels E1and E2, each of them non-degenerate. Obtain an exact expression for the heat capacity of the system.

QN2. A system possesses three energy levels E1 =E , E2 = 2E and E3 = 3E, with degeneracies g (E1 ) = g(E3) = 1, g(E2) = 2. Find the heat capacity of the system.

The Attempt at a Solution

when solving the partition function Z for these questions , why is it that for the first question, i can let E1 to be = 0 and hence E2 = E

but for the 2nd question, i can't let E1 = 0 even though the ans says i can? is the ans a mistake?

since:

qn1: Z = e^-BE₁+e^-BE₂ = 1+e^-BE

so my question is, why for question 2, Z must be like the following
Z= e^-BE+2e^-2BE+e^-3BE

and why i cannot let E₁=0 which gives
Z= 1+2e^-BE+e^-2BE

thanks!

 

[mark726]

Hello,

Thank you for your question. In order to obtain an exact expression for the heat capacity, we need to consider the number of possible energy states for each subsystem. In the first question, each subsystem has only two energy levels, hence we can let E1 = 0 and E2 = E. This means that the two energy levels have degeneracies of 1 and 1, respectively. Therefore, the partition function can be written as Z = 1 + e^(-βE).

In the second question, we have three energy levels with degeneracies of 1, 2, and 1, respectively. This means that there are two possible states for the second energy level, and hence we cannot simply let E1 = 0. Instead, we need to consider all possible combinations of energy levels, which leads to the partition function Z = 1 + 2e^(-βE) + e^(-β2E) = 1 + 2e^(-βE) + e^(-β3E).

I hope this helps to clarify the difference between the two questions. If you have any further questions, please don't hesitate to ask.