How to Calculate the Total Number of Particles in Different Energy States?

AI Thread Summary
To calculate the expected number of particles in the second energy state at temperature T, one must consider the three energy levels: 0, E, and 4E. The discussion highlights the need to apply statistical mechanics principles, particularly the Boltzmann distribution, to determine the probability of particles occupying each energy state. The initial approach involves combinatorial methods for distributing particles among the states, but it lacks the incorporation of temperature effects. The importance of revisiting relevant textbook material, specifically chapter 6, is emphasized to properly address the problem. Understanding the relationship between energy states and temperature is crucial for accurate calculations.
NewtonApple
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Homework Statement



A system of N particles has three possible energy levels namely; 0, E and 4E. How many particles does one expect in the second state at temperature T?

Homework Equations



It's a sample problem for our finals. Our Text book is Statistical Mechanics by Roger Bowley and Mariana Sanchez.

The Attempt at a Solution



Three Energy levels

E_{1}=0, E_{2}=E, E_{3}=4E

Let us first fill the E_{1} state with 3 particle.

N distinguishable ways of selecting the first particle
N-1 different ways to select second particle
N-2 different ways to select third particle

So the total number of ways to place first three particles in state E_{1} is

N(N-1)(N-2)=\frac{N!}{(N-3)!}​

Generally for n_{1} particles placed in E_{1} is,
\frac{N!}{n_{1}!(N-1)!}

for E_{2} state,

\frac{(N-n_{1})!}{n_{2}!(N-n_{1}n_{2})!}​

for E_{3} state,

\frac{(N-n_{1}n_{2})!}{n_{3}!(N-n_{1}n_{2}n_{3})!}​

Total number of particles in all three state will be

P=\frac{N!}{n_{1}!n_{2}!n_{3}!}​

Substituting values

P=\frac{N!}{0!1!4!}​
Am I on right track?
 
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NewtonApple said:
Am I on right track?
I don't think so. You haven't even invoked temperature in any way.

If you had only one particle, what would be the probability of finding in in state 2 when the temperature is T?
 
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ok, I try to re attempt it after going through chapter 6 of the book.
 
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