Understanding Boltzman Equation: Probability of Energy in an Ideal Gas

[shirin]

Hi
1)Boltzmann equation states that P(E_i)=g_i * exp(-E_i/kT) / Sigma_j(g_j * exp(-E_j/kT)).
Does it tell us that the probability that the energy of an atom which we have selected from an ideal gas with tempreture K be E_i? Whether it is the probability the energy of whole gas be E_i?
2) why the ratio of the probability of finding atoms with a specific energy is the same as the ration of their number, when the number of atoms goes to infinity?

 

[Ken G]

Hi
1)Boltzmann equation states that P(E_i)=g_i * exp(-E_i/kT) / Sigma_j(g_j * exp(-E_j/kT)).
Does it tell us that the probability that the energy of an atom which we have selected from an ideal gas with tempreture K be E_i? Whether it is the probability the energy of whole gas be E_i?

The first one-- it's the probability that a given particle that is randomly chosen will have energy E_i.

2) why the ratio of the probability of finding atoms with a specific energy is the same as the ration of their number, when the number of atoms goes to infinity?

Because if you are choosing a particle at random, the probability it will have a certain attribute equals the fraction of the particles that have that attribute.

 

[shirin]

about question 2:
I don't unserstand why it happens when the number of particles goes to infinity. I mean isn't it true for a limited number of particles?

 

[Ken G]

Yes, it is not necessary to have a large number of particles. All that does is "fill in" the distribution, so the probabilities also correspond to the actual distribution you get.