Finding total number of particles in a given distribution

[Reshma]

Homework Statement

N particles are distributed amongst three levels having energies 0, kT, 2kT. If the total equilibrium energy of the system is approximately 425kT, what is the value of N?

Homework Equations

Probability of finding a particle at an energy level is:

$$P_n = Aexp\left({-\epsilon_n \over kT}\right)$$

n = Energy level number
A is the normalization constant

The Attempt at a Solution

I calculated the probabilities of finding the particles at the 3 given energy levels.
P1 = 0.6650
P2 = 0.2450
P3 = 0.0900

I know at equilibrium energy the energy per particle is the same and the particle has the highest probability of being in the equilibrium state. But I can't find a formula which relates the equilibrium energy and the total number of particles. Can anyone help?

 

[Gokul43201]

From the total energy for N particles, you can write down an expression for the mean energy per particle. This is nothing but the expectation value for the energy of a particle. How do you calculate the expectation value if you know the energies of the states and their occupation probabilities? (i.e., how do you find the mean value of a discrete distribution?)

 

[Reshma]

Thanks once again for responding to my questions!
The energy value E can be expressed in terms of the single-particle energies $\varepsilon$:
$$E = \sum_{\varepsilon}n_{\varepsilon}\varepsilon$$
where $n_{\varepsilon}$ is the number of particles in the single particle energy state $\varepsilon$. The values of $n_{\varepsilon}$ must satisfy the condition:
$$\sum_{\varepsilon}n_{\varepsilon} = N$$
E = 425kT
What will be the mean energy then?

 

[Gokul43201]

What will be the mean energy then?

Forget about this problem for a moment.

What is the definition of a mean value of anything? Eg: How will you find the mean weight of a student belonging to a class of N students?

 

[Reshma]

Forget about this problem for a moment.

What is the definition of a mean value of anything? Eg: How will you find the mean weight of a student belonging to a class of N students?

The mean value will be the sum of all the weights of N students divided by the number of students. Here the total energy is 425kT and the number of particles is N. So Mean energy per particle = 425kT/N ?

 

[2Tesla]

That's right, and this can also be written as:

$$425kT/N = \frac{\sum_nN_n\epsilon_n}_{N}$$

where $$N_n$$ is the number of particles in energy level n. Can you find a way to write the right-hand side in terms of only known quantities?

 

[Gokul43201]

Extra hint:

$$\frac{1}{N}\sum _n N_n \epsilon _n = \sum _n \left( \frac{N_n}{N} \right) \epsilon _n = \sum _n (~?~) \epsilon _n$$