Solving for Population of Two Levels in a Two Level System: Stat Mech HW Help

[babbagee]

Homework Statement

Consider a two level system where the ground state is doubly degenerate and the exited state of energy E is four fold degenerate. Write down the partition function and mathematical expressions for the populations of the two levels.

Homework Equations

z=e^-E/kt
n_i = N(e^-E/kt)/z
n_i = NP_i
where N is total number of molecules, P is probability, E is energy, k is boltzman constant

The Attempt at a Solution

Ok I don't really need any form of solution for this question I just needed clarification. It is asking me for the expresssions for the population of the two levels. I am getting counfused about if it is asking me for the number of molecules in each level or is it asking for the probability.

Thanks in advance

 

[christianjb]

write the answer in terms of n_i/N=Pi, the probability or the fraction of molecules.

 

[Blueshift5]

I would like to clarify that the population of a level refers to the number of molecules occupying that particular energy level. In this case, the question is asking for the mathematical expressions for the populations of the two levels in the given two level system.

To solve for the populations, we can use the equations provided in the homework statement. The partition function, z, can be written as z = e^(-E/kT), where E is the energy of the level, k is the Boltzmann constant, and T is the temperature.

To find the population of the ground state, we can use the expression ni = N(e^(-E/kT))/z, where N is the total number of molecules in the system. In this case, since the ground state is doubly degenerate, we can multiply the expression by 2 to account for both degenerate states.

Similarly, for the excited state, we can use the expression ni = N(e^(-E/kT))/z, but since this state is four-fold degenerate, we can multiply the expression by 4.

Therefore, the mathematical expressions for the populations of the two levels are:
- Population of the ground state = 2N(e^(-E/kT))/z
- Population of the excited state = 4N(e^(-E/kT))/z

I hope this clarifies any confusion and helps you in solving the problem. Good luck!