# Average Velocity of gas molecules calculated with a Boltzmann distribution

• Viona
In summary, the average velocity of gas molecules calculated by Boltzmann distribution is equal to the average speed of a single molecule over time, provided that the number of molecules sampled is high enough. However, not all molecules have the same average velocity as it depends on the parameters involved in the calculation, such as the molecular mass. Additionally, the average velocity will vary for different molecular species in a gas volume at a given temperature. There is also a connection between the average kinetic energy and the average velocity, with the latter scaling with the inverse square root of the molecular mass. Finally, the ergodic hypothesis states that the instantaneous average speed over all molecules is equal to the time-average for a single molecule, although this has not been proven.
Viona
Hello

What is the meaning of the average velocity of gas molecules calculated by Boltzmann distribution (in kinetic theory of gases)?
Does all molecules have the same average velocity?

Viona said:
What is the meaning of the average velocity of gas molecules calculated by Boltzmann distribution (in kinetic theory of gases)?
I guess you mean speed, because the average velocity is normally 0.

The meaning is the same as for any average. Were you to sample a bunch of molecules, the average speed of these molecules would be equal to the average speed calculated by the Boltzmann distribution, provided that the number of sampled molecules is high enough (for smaller numbers, you will get fluctuations around the average value).

Viona said:
Does all molecules have the same average velocity?
It depends. Check the parameters that enter into the calculation of the average speed. Provided that these parameters are equal, then the average speed will be the same.

Viona said:
Does all molecules have the same average velocity?
No.

I don't understand your question. Using an average calculation in any circumstance is needed only when the values are not all equal.

Dale
The average thermal speed is useful for a number of things, but I don't really understand what the question is here. As for your question about all molecules, are you asking if different molecular species have the same average velocity? If so, then generally no. In a gas volume at a given temperature, the average kinetic energy of all molecules is the same (assuming it has reached equilibrium), so the average velocity of any gas species will scale with the inverse square root of the molecular mass.

anorlunda said:
No.

I don't understand your question. Using an average calculation in any circumstance is needed only when the values are not all equal.

cjl said:
The average thermal speed is useful for a number of things, but I don't really understand what the question is here. As for your question about all molecules, are you asking if different molecular species have the same average velocity? If so, then generally no. In a gas volume at a given temperature, the average kinetic energy of all molecules is the same (assuming it has reached equilibrium), so the average velocity of any gas species will scale with the inverse square root of the molecular mass.
I think I am confused: is the average velocity calculated using Boltzmann distribution is similar to, for example, measuring the velocity for one molecule many times and take the average?

Viona said:
I think I am confused: is the average velocity calculated using Boltzmann distribution is similar to, for example, measuring the velocity for one molecule many times and take the average?

First answer @cjl 's question. Are you asking about a gas with one molecule specie. or multiple species ? (such as air with O2, and N2)

If the average kinetic energy is E, then E=mv2, and the corresponding velocity is ##v=\sqrt{E/m}## . But m is different for each specie.

If you are just asking about the definition of average for multibody cases, try to answer the question yourself and post your conclusion.

I believe the instantaneous average speed over all the molecules is equal to the time-average for a single molecule, but I'm not sure I could prove it.

## 1. What is the Boltzmann distribution?

The Boltzmann distribution is a statistical distribution that describes the distribution of energy among particles in a system. It is commonly used in physics to describe the behavior of gas molecules.

## 2. How is the average velocity of gas molecules calculated using the Boltzmann distribution?

The average velocity of gas molecules can be calculated by taking the square root of the average of the squared velocities of all the gas molecules in the system, as described by the Boltzmann distribution.

## 3. Why is the Boltzmann distribution important in understanding gas behavior?

The Boltzmann distribution allows scientists to predict the behavior of gas molecules in a system, such as their average velocity, which is important in understanding and studying various physical phenomena, such as gas pressure and temperature.

## 4. What factors affect the average velocity of gas molecules calculated with the Boltzmann distribution?

The average velocity of gas molecules is affected by temperature, mass of the gas molecules, and the size of the container they are in. These factors can all influence the distribution of energy among the gas molecules, as described by the Boltzmann distribution.

## 5. Is the Boltzmann distribution applicable to all types of gases?

Yes, the Boltzmann distribution is applicable to all types of gases, as long as they are in thermal equilibrium and follow the laws of classical mechanics. It is a fundamental concept in statistical mechanics and is widely used in the study of gases.

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