# Finding total number of particles in a given distribution

1. Dec 3, 2007

### Reshma

1. The problem statement, all variables and given/known data
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?

2. Relevant 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

3. 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?

Last edited: Dec 3, 2007
2. Dec 3, 2007

### Gokul43201

Staff Emeritus
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?)

3. Dec 5, 2007

### 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?

4. Dec 5, 2007

### Gokul43201

Staff Emeritus

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?

5. Dec 7, 2007

### Reshma

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 ?

6. Dec 7, 2007

### 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?

Last edited: Dec 7, 2007
7. Dec 11, 2007

### Gokul43201

Staff Emeritus
Extra hint:

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