Boltzmann Distribution with two gasses

[edpell]

What is the Boltzmann distribution look like if we have a mix of two gasses with very different molecular weights? Say hydrogen at 2 amu and xenon at 132 amu. Do they have equal average momentum and hence the hydrogen has an average velocity 66 times the velocity of the xenon and an energy .5*m*v^2 that is 66 times higher than the xenon? Or do they have equal average energy? But shouldn't they both have just .5*kT of energy per degree of freedom?

 

[klimatos]

What is the Boltzmann distribution look like if we have a mix of two gasses with very different molecular weights? Say hydrogen at 2 amu and xenon at 132 amu. Do they have equal average momentum and hence the hydrogen has an average velocity 66 times the velocity of the xenon and an energy .5*m*v^2 that is 66 times higher than the xenon? Or do they have equal average energy? But shouldn't they both have just .5*kT of energy per degree of freedom?

Since the two gases are mixed, at equilibrium they will have the same temperature and hence the same mean kinetic energy of translation. They will not have the same momentum. Molecular momentum is a function of the molecular velocity, but energy of translation is a function of the means of the squares of the velocities. The Maxwell distribution curves of their respective kinetic energies of translation will be different as well. The curve for hydrogen will be lower and broader. The mean total KE per molecule will be 3/2 kT for each gas.

In short, the KE means will be the same, the distributions will be different.

 

[Curl]

The Boltzmann distribution describes a statistic (mean speed) in the microcanonical ensemble. As such, particles of different kinds are non-interacting so in a mixture of two gasses, each species can be treated independently.

If you want a combined statistic for the entire gas, just add up the 2 statistics in the end using the appropriate mathematics.

 

[Rap]

What is the Boltzmann distribution look like if we have a mix of two gasses with very different molecular weights? Say hydrogen at 2 amu and xenon at 132 amu. Do they have equal average momentum and hence the hydrogen has an average velocity 66 times the velocity of the xenon and an energy .5*m*v^2 that is 66 times higher than the xenon? Or do they have equal average energy? But shouldn't they both have just .5*kT of energy per degree of freedom?

If you really mean momentum, and not absolute value of momentum, then they will have the same average momentum per particle, which will be zero. Since the absolute value of momentum is not a conserved quantity, the gases will not have the same average absolute value of momentum. Energy is conserved, so they will have the same average energy per particle.

 

[PJC01]

The Boltzmann distribution describes the distribution of energy among particles in a system at a given temperature. In the case of a mixture of two gases with different molecular weights, the distribution will be different for each gas due to their different masses.

The average momentum of each gas will depend on its molecular weight, as stated in the question. This means that the hydrogen gas will have a higher average velocity compared to the xenon gas. However, this does not necessarily mean that the hydrogen gas will have a higher average energy.

The average energy of a gas is determined by its temperature and the number of degrees of freedom it has. In the case of a monatomic gas, such as hydrogen and xenon, the number of degrees of freedom is equal to 3. This means that both gases will have the same average energy of 0.5*kT per degree of freedom, where k is the Boltzmann constant and T is the temperature.

Therefore, while the hydrogen gas may have a higher average velocity and momentum, it does not necessarily have a higher average energy compared to the xenon gas. Both gases will have the same average energy per degree of freedom, but the distribution of this energy will be different due to their different molecular weights.