Gas Distribution: Temperature, MFP & More

In summary: I think that's all you need to know for now!In summary, the normal distribution of displacement follows from the fact that the energy of a non-interacting gas is purely kinetic and therefore quadratic in the velocities.
  • #1
Glenn G
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Diffusion of gases seems to follow normal distribution. I imagine deviations from the mean would depend on temperature, mean free path and speed of gas molecules. Any other?

Cheers,
Glenn
 
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  • #2
Glenn G said:
Diffusion of gases seems to follow normal distribution."
Glenn
What do you mean by this? Diffusion in single component gas due to density gradients? 2 component mixture? Or are you talking about the maxwell-boltzmann distribution?
 
  • #3
Sorry I meant that if you modeled the displacement over time of a body of gas and plot where all they have got to in terms of displacement from their original position that it follow the normal distribution (the mean displacement should be zero I'd imagine because they are moving in random directions) is any of this correct?
 
  • #4
Glenn G said:
Sorry I meant that if you modeled the displacement over time of a body of gas and plot where all they have got to in terms of displacement from their original position that it follow the normal distribution (the mean displacement should be zero I'd imagine because they are moving in random directions) is any of this correct?
Well if the body of gas is in equilibrium, then the gas will be homogeneously distributed i.e density even everywhere. So that's where it is trying to get to and doesn't depend on the initial configuration of the gas as long as you wait long enough for equilibrium to happen at a given temperature for fixed volume say. The temperature and pressure determine the mean energy and the temperature alone determines the distribution about this mean value due to statistical fluctuation. For a non interacting gas this corresponds to kinetic energy and thus determines the average speed and the distribution of speeds about this average, which is a gaussian distribution.
Does that help or am I not getting something?
 
  • #5
I believe he means that in a gas at equilibrium the probability for the net displacement of any single molecule over a fixed time is normally distributed. (I can't say that it is always true, but I suppose the central limit theorem makes it hard for it not to be true)

I believe the mean displacement in time depends only on the mean size of the steps (mean free path) and the time of the steps (mean free time). These vary with pressure (shorter paths) and temperature (faster molecules means shorter time between collisions). I suspect any stickiness (chemical bonding when the collision energy happens to be low) would also increase the time.
 
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  • #6
Cutter Ketch said:
I believe he means that in a gas at equilibrium the probability for the net displacement of any single molecule over a fixed time is normally distributed.
Ah yes OK, I probably should have been able to get what was meant but it was the
Glenn G said:
a body of gas and plot where all they have got to
that threw me off.
Yes for a single particle when the fixed time is less than mean free time or if the walls of the container are at infinity.
The main point I was trying to make earlier hints at why this is:
muscaria said:
For a non interacting gas this corresponds to kinetic energy and thus determines the average speed and the distribution of speeds about this average, which is a gaussian distribution.
I was making reference to the Maxwell-Boltzmann distribution i.e that the speed of a particle is normally distributed.. which then leads to a normal distribution of displacement for fixed time.
Cutter Ketch said:
(I can't say that it is always true, but I suppose the central limit theorem makes it hard for it not to be true)
Here the normal distribution arises from the fact that the energy of a non-interacting gas is purely kinetic and therefore quadratic in the velocities. So the probability of measuring a particle with energy ##\epsilon=mv^2/2## is proportional to
$$exp\left[-\frac{mv_x^2+mv_y^2+mv_z^2}{2k_BT}\right]$$
where ##v=\sqrt{v_x^2+v_y^2+v_z^2}##.

Glenn G said:
I imagine deviations from the mean would depend on temperature, mean free path and speed of gas molecules. Any other?
As with all statistical mechanics observables, the spread from the average occurs due to energy fluctuating between the system and heat bath during thermal equilibrium and it is the temperature ##\textit{alone}## which determines this fluctuation (pressure doesn't play any role for instance). Increasing the energy of a system through work (pressure, E-M fields etc..) shifts energy levels upwards, whereas increasing energy by adding heat and raising the temperature doesn't change the energy levels but shifts the occupation towards higher energy levels: work changes the energy levels and leaves distribution unchanged, heat changes the distribution and leaves the energy levels unchanged.
So the deviation from the mean you are asking about, should be entirely fixed by the temperature of the system - for a given particle mass that is.. You can see this is the case from the probability distribution above. You can find everything you want to know from the Boltzmann distribution, e.g. the mean speed (and therefore mean displacement), the spread from the mean etc..
 
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  • #7
Thanks for that very insightful. The fact that work changes energy levels but leaves distribution unchanged whereas heating does the reverse is unexpected for me. Good stuff.
 

Related to Gas Distribution: Temperature, MFP & More

1. What is gas distribution?

Gas distribution refers to the process of how gases are dispersed or spread out in a particular area or environment. This can be influenced by factors such as temperature, pressure, and molecular properties of the gas.

2. How does temperature affect gas distribution?

Temperature plays a critical role in gas distribution as it affects the kinetic energy of gas molecules. As temperature increases, gas molecules gain more energy and move faster, leading to a wider distribution. Conversely, lower temperatures result in slower-moving molecules and a more concentrated distribution.

3. What is the mean free path (MFP) in gas distribution?

The mean free path (MFP) is the average distance a gas molecule can travel before colliding with another molecule. It is influenced by factors such as gas density, temperature, and pressure. In general, as MFP increases, gas molecules have a higher chance of colliding with each other, resulting in a more uniform distribution.

4. How is gas distribution measured?

Gas distribution can be measured using various techniques such as gas chromatography, mass spectrometry, and spectroscopy. These methods allow scientists to analyze the concentration and distribution of different gas molecules in a sample.

5. Can gas distribution be controlled?

Yes, gas distribution can be controlled through various methods such as temperature and pressure regulation, gas diffusion, and gas barriers. These techniques are commonly used in industries such as gas production and distribution, semiconductor manufacturing, and air conditioning systems.

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