The Boltzmann distribution of uniformly spaced energy levels

In summary, the question asks for the fraction of particles in the ground state of a system with uniformly spaced energy levels at 3.2x10^-10 J apart, at a temperature of 300K. The Boltzmann distribution formula is used to calculate this, with an expression for the energy state of each level. A mistake in the original function led to a negative answer, but after correction, the answer remains the same.
  • #1
corr0105
7
0
The question states:
A system has energy levels uniformly spaced at 3.2x10^-10 J apart. Thepopulations of the energy levels are given by the Boltzmann distribution. What fraction of particles is in the ground state at T=300K.

I know that the Boltzmann distribution is:
[tex]_{}p*[/tex]j=probability that a particle is in state j
=exp(-[tex]_{}E[/tex]j/KT) / [tex]\sum[/tex]exp(-[tex]_{}E[/tex]j/KT)

I created my own expression for the energy state Ej:
[tex]_{}E[/tex]j=3.2x10^-20(j-1)
My thought process behind this was that to get the energy state of each level you have to multiply by the space between each level times 1 minus the level. In other words, E1=0*3.2x10^-20, E2=1(3.2x10^-10)...etc.

I set this pj* equal to: # particles in ground state (g) / # states (t)
Assuming that the particles in the ground state would have energy E=0 I plugged all of the given values into the equation: T=300, Ej=0, K=boltzmann's constant. Because the problem did not give the number of states my answer is in terms of t. I got an answer of:
-0.25875/(1+t)
Obviously this is wrong because you cannot have a negative number of particles.


ANY IDEA WHAT I'VE DONE WRONG?
thanks in advance for any help
 
Last edited:
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  • #2
Your Boltzmann distribution function has a couple of minus signs missing.
 
  • #3
I'm sorry, you are correct. The post has been edited. It was a typo, but the answer I got still stands.
 

Related to The Boltzmann distribution of uniformly spaced energy levels

1. What is the Boltzmann distribution?

The Boltzmann distribution, also known as the Maxwell-Boltzmann distribution, is a probability distribution that describes the distribution of particles over different energy states in a system at a given temperature. It is based on the assumption that particles in a system are in thermal equilibrium with their surroundings.

2. How is the Boltzmann distribution derived?

The Boltzmann distribution is derived from the principles of statistical mechanics, specifically the Boltzmann factor, which relates the energy of a particle to the temperature of the system. It takes into account the total number of particles, the energy levels available, and the temperature of the system.

3. What is the significance of uniformly spaced energy levels in the Boltzmann distribution?

The Boltzmann distribution assumes that the energy levels in a system are uniformly spaced, meaning that the difference in energy between each level is the same. This simplifies the equations used to calculate the distribution and allows for easier analysis of the system.

4. How does temperature affect the Boltzmann distribution?

The Boltzmann distribution is directly affected by temperature. As the temperature of a system increases, the distribution shifts towards higher energy states, meaning that more particles will occupy higher energy levels. Similarly, as the temperature decreases, the distribution shifts towards lower energy states.

5. In what types of systems is the Boltzmann distribution applicable?

The Boltzmann distribution is applicable to a wide range of systems, including gases, liquids, and solids. It is commonly used in thermodynamics and statistical mechanics to describe the behavior of particles in these systems. It is also applicable to chemical reactions and quantum mechanical systems.

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