Question about the collisions of the molecules in an ideal gas

[tbn032]

(The equation of ideal gas is PV=NRT.if P=1atm,N=1mole,T=0°K,R=gas constant then volume = zero. Hence, the volume of an individual molecule of ideal gas is zero)
An individual molecule of ideal gas is assumed to have zero volume. The molecules of ideal gas are assumed to be dimensionless points. Then how does the dimensionless points collide with each other in accordance with kinetic theory of gases.

I assume that the individual molecule of the gas should have non-zero volume such that it is able to collides with other molecules or the wall of the container. If the molecule has zero volume(i.e. a dimensionless point), then how can it collide with other molecules (how can points collide with each other )?

 

[BvU]

how can points collide with each other ?

Is there any reason they should ?

$\$

 

[tbn032]

Is there any reason they should ?

But the molecules of ideal does collide with each other. The molecules of ideal gas are dimensionless points, how can these points (molecules of ideal gas) collide.

 

[PeroK]

(The equation of ideal gas is PV=NRT.if P=1atm,N=1mole,T=0°K,R=gas constant then volume = zero. Hence, the volume of an individual molecule of ideal gas is zero)
An individual molecule of ideal gas is assumed to have zero volume. The molecules of ideal gas are assumed to be dimensionless points. Then how does the dimensionless points collide with each other in accordance with kinetic theory of gases.

I assume that the individual molecule of the gas should have non-zero volume such that it is able to collides with other molecules or the wall of the container. If the molecule has zero volume(i.e. a dimensionless point), then how can it collide with other molecules (how can points collide with each other )?

A collision between charged particles is modeled using the electric field created by each particle.

In any case, collisions between point particles can be modeled using only the energy-momentum conservation laws. You only have to assume they collide. Which is a fair assumption.

 

[tbn032]

A collision between charged particles is modeled using the electric field created by each particle

But each particle of ideal gas is electrically neutral.

 

[PeroK]

But each particle of ideal gas is electrically neutral.

Yes, but generally the electrons are outside the nucleus, so those fields interact once the particles are close enough.

 

[hutchphd]

In any case, collisions between point particles can be modeled using only the energy-momentum conservation laws. You only have to assume they collide. Which is a fair assumption.

Do they need to interact directly with each other at all? To be confined they must interact with the walls of the container (presumably at temperature).

 

[tbn032]

Do they need to interact directly with each other at all? To be confined they must interact with the walls of the container (presumably at temperature).

I think they need to collide directly with each other to achieve maxwell-boltzmann distribution curve.

 

[BvU]

I think they need to collide directly with each other to achieve maxwell-boltzmann distribution curve.

You started with the ideal gas law.

The equation of ideal gas is PV=NRT

Now you are shifting to the MB distribution -- for which some way of re-distributing kinetic energy is indeed required, I suppose.

 

[hutchphd]

Now you are shifting to the MB distribution -- for which some way of re-distributing kinetic energy is indeed required, I suppose.

The same issue arises for Stefan-Boltzmann (black body spectrum). There are manifestly no direct interactions but the walls do it.

 

[Lord Jestocost]

(The equation of ideal gas is PV=NRT.if P=1atm,N=1mole,T=0°K,R=gas constant then volume = zero. Hence, the volume of an individual molecule of ideal gas is zero)
An individual molecule of ideal gas is assumed to have zero volume. The molecules of ideal gas are assumed to be dimensionless points. Then how does the dimensionless points collide with each other in accordance with kinetic theory of gases.

I assume that the individual molecule of the gas should have non-zero volume such that it is able to collides with other molecules or the wall of the container. If the molecule has zero volume(i.e. a dimensionless point), then how can it collide with other molecules (how can points collide with each other )?

The ideal gas law is merely an approximation, never exactly true for a real gas in a real world. Nevertheless, some properties of a low-density gas can be summarized in the ideal gas law.

 

[DrJohn]

(The equation of ideal gas is PV=NRT.if P=1atm,N=1mole,T=0°K,R=gas constant then volume = zero. Hence, the volume of an individual molecule of ideal gas is zero)
An individual molecule of ideal gas is assumed to have zero volume. The molecules of ideal gas are assumed to be dimensionless points. Then how does the dimensionless points collide with each other in accordance with kinetic theory of gases.

I assume that the individual molecule of the gas should have non-zero volume such that it is able to collides with other molecules or the wall of the container. If the molecule has zero volume(i.e. a dimensionless point), then how can it collide with other molecules (how can points collide with each other )?

You will be dealing with a liquid or a solid at the temperature you quoted - absolute zero.
The Ideal Gas Law applies to gases. Clue's in the name.

 

[hutchphd]

Yes. My apologies for missing the paragraph about T=0. For a classical ideal gas it goes to zero.
In the real world of course a host of interesting things happen at low Temperature. Perhaps most interesting is the blatantly quantum behavior of He⁴ and He³ at T=a few K even though they do not strongly interact

 

[Borek]

Wiki (not an ideal source, but the easiest to consult) lists following properties of the ideal gas:

• The molecules of the gas are indistinguishable, small, hard spheres
• All collisions are elastic and all motion is frictionless (no energy loss in motion or collision)
• Newton's laws apply
• The average distance between molecules is much larger than the size of the molecules
• The molecules are constantly moving in random directions with a distribution of speeds
• There are no attractive or repulsive forces between the molecules apart from those that determine their point-like collisions
• The only forces between the gas molecules and the surroundings are those that determine the point-like collisions of the molecules with the walls
• In the simplest case, there are no long-range forces between the molecules of the gas and the surroundings.

Nowhere it states molecules are points. Yes, it can be deduced from these assumed properties, but this is quite typical for simplified models, that they fail somewhere. As long as they produce otherwise meaningful results we don't care too much, we just remember they have to fail at some point.

Which reminds me how the equation for the steady state current for a microelectrode was calculated (that was back in eighties, when I was a wannabe electrochemist), The solution assumed concentration of the substance reacting at the electrode (and producing the current) was not changing. That apparently means amount of substance doesn't change, so the current must be zero - yet the current calculated from this "wrong" assumption nicely agreed with the experiment.

 

[Vanadium 50]

You can in fact include this effect: $P(V-nb)=nRT$, where b is the volume occupied by the gas molecules themselves. (The astute reader will recognize this from the van der Waals gas equation)

The ideal gas law is an approximation where $nb \ll V$, which is pretty good at STP. You have 10²³ elements each a cubic Angstrom or so occupying a few liters.

 

[TeethWhitener]

Where in the ideal gas model does it say that point particles can’t collide with each other/the container walls?

 

[hutchphd]

Where in the ideal gas model does it say that point particles can’t collide with each other/the container walls?

As I understand it the OP question is the assumption of point particles: how can they possibly interact (very small scattering cross-section) ?

 

[PeroK]

As I understand it the OP question is the assumption of point particles: how can they possibly interact (very small scattering cross-section) ?

Or, a zero cross section!

 

[TeethWhitener]

I’m more curious about where specifically in the vast body of knowledge called “kinetic theory” the OP is getting hung up on the idea of point particles not interacting with one another (that is, is OP’s question about a specific step in a derivation where an assumption is made or is it just a general question).

Incidentally, it certainly makes sense that cross sections and mean free paths and whatnot are associated with molecular volumes, but I don’t immediately see that there’s anything inherent in the math that says a non-zero cross section (interactions) necessarily implies a non-zero van der Waals b term (volume dependence on number of particles; i.e., non-point particles). Willing to be proven wrong though; I haven’t explored the issue too deeply.

 

[hutchphd]

I think the (reasonable) confusion was about how the energy gets distributed if they are truly point particles (and therefore cannot interact).

but I don’t immediately see that there’s anything inherent in the math that says a non-zero cross section (interactions) necessarily implies a non-zero van der Waals b

As I recall, for finite hard spheres this "excluded volume" will show up in the microcanonical counting of available states for N particles in an box. The number of states is proportional to the available volume.

 

[Vanadium 50]

Option A is to assume non-point particles and take the limit as the size goes to zero,
Option B is to assume action-at-a-distance.

It's an approximation, for Pete's sake, one that historically came about empirically. Expecting a fully self-consistent theory to somehow pop out of it is unrealistic.

 

[hutchphd]

I would add option C which is that the molecules will interact with the pressure vessel walls (thay gotta) and thereby with each other (like a radiant cavity). But your point is absolutely true.

 

[256bits]

You started with the ideal gas law.Now you are shifting to the MB distribution -- for which some way of re-distributing kinetic energy is indeed required, I suppose.

Two different different theories as far as I understand.

In the simplest derivation of the kinetic theory of gas atoms/ molecules the velocity of the particles do not interact with each other, but move between parallel in the x, y z directions. A velocity distribution can be assumed, and one takes the root mean square of the velocities to even out the contributions from slower and faster molecules.
The step MB distribution takes it further with momentum exchange between molecules to explain what the velocity distribution should be.

 

[Mayhem]

The point of the ideal gas law is that everything is simplified to the point that pressure can be seen as these molecules colliding with the inside of the container, resulting in temperature change. The ideal gas law is probably not used in an industrial setting. I did the calculations, and compared to the Van der Waals model of gases, only helium can be treated as an ideal gas to a good approximation.

 

[hutchphd]

I did the calculations, and compared to the Van der Waals model of gases, only helium can be treated as an ideal gas to a good approximation.

You are answering a question that has not been asked. "A good approximation" is a meaningless phrase. This is not a moral question.
All calculations are approximations and the issue is whether your approximation is sufficient to answer the question at hand.
But you are correct that Noble Gases are "better approximated" by the Ideal Gas equation than other gases for well understood reasons.