Understanding Boltzman Equation: Probability of Energy in an Ideal Gas

[shirin]

Hi
1)Boltzman 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)Boltzman 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.

 

[pmacias]

1) Yes, the Boltzman equation tells us the probability of finding an atom in an ideal gas with a specific energy E_i at a given temperature T. It does not tell us the probability of the entire gas having that energy, but rather the probability of finding one specific atom with that energy in the gas. This equation is important for understanding the behavior of ideal gases and how their energy is distributed among the particles.

2) The ratio of the probability of finding atoms with a specific energy is the same as the ratio of their number when the number of atoms goes to infinity because at that point, the gas is considered to be in a state of thermodynamic equilibrium. In this state, the energy of the particles is distributed according to the Boltzman distribution, which means that the probability of finding an atom with a specific energy is directly proportional to the number of atoms with that energy. As the number of atoms approaches infinity, the ratio of the probabilities becomes equal to the ratio of the numbers. This is known as the law of large numbers and is a fundamental principle in statistical mechanics.