Closed form expression for the partition function Z using the Canonical Ensemble

[haitrungdo82]

Homework Statement

I'm looking for a closed form expression for the partition function Z using the Canonical Ensemble

Homework Equations

epsilon_j - epsilon_j-1 = delta e
Z = Sum notation(j=0...N) e^(-beta*j*delta e)
beta = 1/(k_B*T)
t = (k_B*T)/delta e
N is the number of excited states

The Attempt at a Solution

I am given Z = Sum notation(j=0...N) e^(-beta*j*delta e). But the question asks me
to graph (in MathCAD) Z vs. t, where t = (k_B*T)/delta e. So, if I substitute t = (k_B*T)/delta e into the original Z, then Z becomes Z = Sum notation (j=0...N) e^-(j/t). Then I sketched Z vs. t for N = 25. It is fine!

However, then I have to sketch Z vs. t for N appoaches infinitive. This is a problem. My Professor told me that, for this case, I have to use the closed form of Z. The only closed form expression that I know is Z = e^[1-(lambda_1/k_B)]. But..."lambda_1" doesn't make sense to me. Specifically, I'm looking for a closed form expression that relates the parameters that I am given, or a definition of "lambda_1" that includes the given parameters. When I know it, I will be able to make a graph.

Could anyone help me, please?

 

[Dick]

The sum of e^(-j/t) over j is a geometric series. It's r^j where r=e^(-1/t). Look up 'geometric series'. You shouldn't have any problem finding a closed form expression for the sum over j from 0 to infinity.

 

[haitrungdo82]

The sum of e^(-j/t) over j is a geometric series. It's r^j where r=e^(-1/t). Look up 'geometric series'. You shouldn't have any problem finding a closed form expression for the sum over j from 0 to infinity.

Hi Dick! I was able to sketch the graph thanks to your advice. Good job!