Gas considered as lots of molecules

[WiFO215]

I've read in quite a few textbooks, chemistry and physics, that we can approximate a gas as many molecules which are floating in space and bouncing off each other. I want to know on what basis we make this approximation. What forces do we ignore and why?

Here are some of the questions I asked my prof:
Somehow, you approximate these molecules to be beach balls simply floating around and bouncing, or at least, this was the picture I got. How and why is this justified?

When these things are "bouncing off", what forces of interaction come into play? The teacher replied that Coulomb(static force law) Force is the only force we consider. How can we do this when the charges are in motion. I know the obvious answer is "the other part that comes out of the motion of the charges is negligible". Can someone justify?

Apart from Coulomb attraction, he also said that we don't consider any other forces. Why not?

As far as I understand, we cannot be sitting and calculating 10²³ force equations which give you the motion of these particles. This is why I have to study this subject of statistical mechanics, wherein we use probability to describe our system. But how is this beach balls picture justified?

 

[Andrew Mason]

I've read in quite a few textbooks, chemistry and physics, that we can approximate a gas as many molecules which are floating in space and bouncing off each other. I want to know on what basis we make this approximation.

What makes you think that this is an approximation. It is an accurate description of what is actually occurring. The molecules interact through coulomb repulsion in which energy is conserved - so the collisions are elastic.

Here are some of the questions I asked my prof:
Somehow, you approximate these molecules to be beach balls simply floating around and bouncing, or at least, this was the picture I got. How and why is this justified?

Monatomic gases may be viewed this way. Diatomic and polyatomic molecules have more complicated behaviour so the beach ball analogy is not quite correct.

When these things are "bouncing off", what forces of interaction come into play? The teacher replied that Coulomb(static force law) Force is the only force we consider. How can we do this when the charges are in motion. I know the obvious answer is "the other part that comes out of the motion of the charges is negligible". Can someone justify?

Well, there are only two forces that act outside the nucleus. Gravity and the electro-magnetic force. The nuclear forces are not involved as they act only within a very short range in and near the nucleus.

Apart from Coulomb attraction, he also said that we don't consider any other forces. Why not?

The em force is about 40 some orders of magnitude greater than the gravitational force so there is really no other force than the em force to consider.

As far as I understand, we cannot be sitting and calculating 10²³ force equations which give you the motion of these particles. This is why I have to study this subject of statistical mechanics, wherein we use probability to describe our system. But how is this beach balls picture justified?

Where do you see a conflict between the "beach ball" model and the statistical approach? Both are used in the kinetic theory.
See: http://en.wikipedia.org/wiki/Kinetic_theory

AM

 

[Redbelly98]

The interaction potential between two neutral atoms looks like the following:

http://upload.wikimedia.org/wikipedia/commons/5/5a/12-6-Lennard-Jones-Potential.png

For sufficiently high temperatures, the kinetic energy of the atoms is high enough to ignore the little well in the graph. To a good approximation, the potential is zero when the atoms are far apart (i.e. no appreciable force), and behaves as a rigid wall (the nearly vertical part of the curve) at a certain distance apart. This is akin to rigid spheres that will bounce off each other when their centers are separated by 1 ball diameter, but otherwise do not exert a force on each other.

 

[WiFO215]

What makes you think that this is an approximation. It is an accurate description of what is actually occurring. The molecules interact through coulomb repulsion in which energy is conserved - so the collisions are elastic.

I was thinking of perhaps some magnetic field being created due to the motion of the charges. Will that not happen?

 

[WiFO215]

The interaction potential between two neutral atoms looks like the following:

http://upload.wikimedia.org/wikipedia/commons/5/5a/12-6-Lennard-Jones-Potential.png

For sufficiently high temperatures, the kinetic energy of the atoms is high enough to ignore the little well in the graph. To a good approximation, the potential is zero when the atoms are far apart (i.e. no appreciable force), and behaves as a rigid wall (the nearly vertical part of the curve) at a certain distance apart. This is akin to rigid spheres that will bounce off each other when their centers are separated by 1 ball diameter, but otherwise do not exert a force on each other.

This is the potential due to what field? Electromagnetic? Or is there some separate field that I don't know of? I ask this, as it seems like a very funny graph. I've seen this before in textbooks but never understood what field would produce such a type of potential. Does this take into account that there will be magnetic forces on the two interacting molecs?

What happens to this "wall" diagram if the two molecs happen to interact and form some new product? Is there a different diagram for that case?

 

[jtbell]

This is the potential due to what field? Electromagnetic? Or is there some separate field that I don't know of?

Wikipedia has some information on the origin and significance of the shape of the graph:

http://en.wikipedia.org/wiki/Lennard-Jones_potential

It's important to note that this is an empirical formula, i.e. a "guesswork" made up to fit experimental results. It can be justified only in a qualitative way by appealing to the fundamental underlying interactions.

 

[Redbelly98]

This is the potential due to what field? Electromagnetic? Or is there some separate field that I don't know of? I ask this, as it seems like a very funny graph. I've seen this before in textbooks but never understood what field would produce such a type of potential. Does this take into account that there will be magnetic forces on the two interacting molecs?

It is electromagnetic, as far as I know magnetic forces play an insignificant role. Electrostatic forces and, as the wiki article jtbell linked to mentioned, Pauli exclusion of the electrons in the different atoms, are the dominant mechanisms here.

What happens to this "wall" diagram if the two molecs happen to interact and form some new product? Is there a different diagram for that case?

It's the same diagram, but for this to happen the temperature (i.e. kinetic energy of the atoms) must be low enough so that the system becomes trapped in the small potential well.

 

[HallsofIvy]

But I suspect the only "force" you are to consider here is the force the wall imparts on a molecule bouncing off it. Imagine a molecule of mass m and velocity v (perpendicular to the wall) bouncing off the wall. Since the collision is "perfectly elastic", the molecule now has velocity -v and so momentum -v. The total change in momentum, from mv to -mv is -2mv.

Now, if that molecule is in a box and the distance from that wall to the opposite one is L, the molecule must go a distance 2L to hit the other wall, bounce back and hit the first wall again. A speed v, that takes 2L/v seconds and so the molecule hits the wall v/2L times every second. (-2mv change in momentum per hit)(v/2L hits per second)= -mv²/L "change in momentum per second" and that is force. The wall exerts a force of -mv²/L on the molecule and the molecule exerts a force of mv²/L on the wall.

We can carry this further. With N molecules in random motion, we can treat it as if N/3 were moving along each of the three axes so all of the molecules will be exerting a total force of mNv²/3L on the wall. Since pressure is "force divided by area", if that wall has height and width H and W, its area is HW, and the pressure on that wall is P= mNv²/3LHW= mNv²/3V where "V" here is the Volume of the box. You might recognize that in the form PV= mNv²/3 as the "ideal gas law" PV= NRT where P is pressure, V is volume, N is the number of molecules, R is the "gas constant" and t is the temperature (in Kelvins). That also shows that the temperature is proportional to v²- that is to the internal kinetic energy of the gas.

By the way, in 1905, Albert Einstein wrote four massively important papers. In addition to papers on the special theory of relativity, the general theory of relativity, and the photoelectric effect, his paper analyzing Brownian Motion of small objects as being due to their being struck by molecules was considered by many to be the first clear evidence of the "molecular theory".

And he did that while working as a clerk in a patent office!

 

[Vanadium 50]

The interatomic forces are complicated and practically impossible to derive from first principles. However, the biggest correction to the ideal gas law is not that Redbelly's curve doesn't look exactly like a right angle: vertical at r=1 and horizontal at 0 for r>0. It's that the gas molecules have non-zero volume, and for cold and/or dense gasses, this is important:

$$PV = nRT$$

becomes

$$P \left(V - nb\right) = nRT$$

where b is the volume occupied by the molecules in a mole of gas. For air at STP, it's about a 0.2% effect.

The next level of approximation is to consider that gas molecules weakly attract each other (Redbelly's curve). If I parameterize this attraction by a constant a, with units Nm⁴, it serves to "suck the gas in", reducing its pressure, so I get:

$$\left(P + \frac{n^2a}{V^2} \right) \left(V - nb\right) = nRT$$

This is called the van der Waals gas equation. Like I said, it works better than the ideal gas equation for low temperatures or high pressures.

 

[WiFO215]

But I suspect the only "force" you are to consider here is the force the wall imparts on a molecule bouncing off it. Imagine a molecule of mass m and velocity v (perpendicular to the wall) bouncing off the wall. Since the collision is "perfectly elastic", the molecule now has velocity -v and so momentum -v. The total change in momentum, from mv to -mv is -2mv.

Now, if that molecule is in a box and the distance from that wall to the opposite one is L, the molecule must go a distance 2L to hit the other wall, bounce back and hit the first wall again. A speed v, that takes 2L/v seconds and so the molecule hits the wall v/2L times every second. (-2mv change in momentum per hit)(v/2L hits per second)= -mv²/L "change in momentum per second" and that is force. The wall exerts a force of -mv²/L on the molecule and the molecule exerts a force of mv²/L on the wall.

We can carry this further. With N molecules in random motion, we can treat it as if N/3 were moving along each of the three axes so all of the molecules will be exerting a total force of mNv²/3L on the wall. Since pressure is "force divided by area", if that wall has height and width H and W, its area is HW, and the pressure on that wall is P= mNv²/3LHW= mNv²/3V where "V" here is the Volume of the box. You might recognize that in the form PV= mNv²/3 as the "ideal gas law" PV= NRT where P is pressure, V is volume, N is the number of molecules, R is the "gas constant" and t is the temperature (in Kelvins). That also shows that the temperature is proportional to v²- that is to the internal kinetic energy of the gas.

Why do you want to go through so much trouble Halls? Since Redbelly's curve gave us the potential, why not just take it's gradient, and we will land up with the force. Wouldn't that do? Seems like you are computing something in a harder way to me and are making more approximations (all are moving with V when they hit the wall, elastic collision, exactly 1/3 going in each direction (shouldn't this be 1/6 considering that they are moving in positive and negative directions??) etc.)

 

[WiFO215]

The interatomic forces are complicated and practically impossible to derive from first principles. However, the biggest correction to the ideal gas law is not that Redbelly's curve doesn't look exactly like a right angle: vertical at r=1 and horizontal at 0 for r>0. It's that the gas molecules have non-zero volume, and for cold and/or dense gasses, this is important:

$$PV = nRT$$

becomes

$$P \left(V - nb\right) = nRT$$

where b is the volume occupied by the molecules in a mole of gas. For air at STP, it's about a 0.2% effect.

Huh? I don't understand the logic you used to derive the volume correction. 'b' is the volume, okayyyy. Soooooo...?

The next level of approximation is to consider that gas molecules weakly attract each other (Redbelly's curve). If I parameterize this attraction by a constant a, with units Nm⁴, it serves to "suck the gas in", reducing its pressure, so I get:

$$\left(P + \frac{n^2a}{V^2} \right) \left(V - nb\right) = nRT$$

This is called the van der Waals gas equation. Like I said, it works better than the ideal gas equation for low temperatures or high pressures.

Weakly attract? Weakly attract with what force? Didn't someone just say EM force is all we need to consider? If not, doesn't the Lennard Jones curve take care of that bit?

 

[WiFO215]

The form of the repulsion term has no theoretical justification

Oh boy. So after reading that article, I take it that we don't build the potential diagram using any of the known forces, but try to fit the experiment as best as possible.

Following which, I have a question.

These parameters can be fitted to reproduce experimental data or accurate quantum chemistry calculations.

How do they do this? Is there some sort of potential measuring experiment for this set up? If there are really 10²³ molecs, what sort of experiment would they use to isolate 2 molecs and study them?

 

[Vanadium 50]

Huh? I don't understand the logic you used to derive the volume correction. 'b' is the volume, okayyyy. Soooooo...?

b is the volume per mole, and n is the number of moles.

Weakly attract? Weakly attract with what force? Didn't someone just say EM force is all we need to consider? Doesn't the Lennard Jones curve take care of that bit?

These forces are ultimately electromagnetic, but the actual calculation is (as I said) very difficult to calculate. For example, two neutral spherical atoms, say neon, will attract via London dispersion forces. Furthermore, the Lennard-Jones potential might be more accurately described as a family of potentials. Each gas has a different curve.

 

[WiFO215]

b is the volume per mole, and n is the number of moles.

But sir, when we "made" the equation along the lines of how HallsofIvy did, we took V to be volume of the box and the volume of the molec itself was ignored. From where do we bring it into the derivation? Is there a similar derivation to get to what you are saying?

 

[WiFO215]

Anyone?

 

[jtbell]

The Wikipedia article about the van der Waals equation has some discussion of its derivations.

http://en.wikipedia.org/wiki/Van_der_Waals_equation

 

[WiFO215]

Thanks a lot jtbell! Now if someone could answer post #12, I'm done (for now).

 

[Redbelly98]

Is there some sort of potential measuring experiment for this set up? If there are really 10²³ molecs, what sort of experiment would they use to isolate 2 molecs and study them?

Good question. I don't know definitely, but can hazard a guess that the measured vibrational spectrum on a collection of dimers (2-atom molecules) would yield details of the potential curve's shape, along with the theoretically justified 1/r⁶ behavior at long range.

I.e., one would measure the transition energies between quantized states within the potential well. This is just an educated guess on my part, maybe somebody else knows more definitively.

 

[Vanadium 50]

I don't know how it's done, but if I wanted to study the left hand side of that potential, I wouldn't be working with gasses - I'd want to be working with liquids. The idea would be to get as much of the material as I could to the left hand side of the plot.

 

[Redbelly98]

I don't know how it's done, but if I wanted to study the left hand side of that potential, I wouldn't be working with gasses - I'd want to be working with liquids. The idea would be to get as much of the material as I could to the left hand side of the plot.

Are you sure that would help? The potential is for the interaction between two and only two atoms. Wouldn't having many atoms close together, as in a liquid, hopelessly distort the measured potential?

Anirudh, I did some googling, found this image:

http://images.absoluteastronomy.com/images/encyclopediaimages/m/mo/morse-potential.png

The spacing of the energy levels becomes closer as one goes higher in energy, this is what I was talking about two posts ago.

 

[WiFO215]

Are you sure that would help? The potential is for the interaction between two and only two atoms. Wouldn't having many atoms close together, as in a liquid, hopelessly distort the measured potential?

Anirudh, I did some googling, found this image:

http://images.absoluteastronomy.com/images/encyclopediaimages/m/mo/morse-potential.png

The spacing of the energy levels becomes closer as one goes higher in energy, this is what I was talking about two posts ago.

I don't understand either of your posts sir.

 

[Redbelly98]

Whoops.

Let's start with, are you familiar with the concept of energy levels, for example in the hydrogen atom?

 

[WiFO215]

I possesses a very rudimentary knowledge - there are only certain orbits around which an electron is permitted to revolve and these have certain Kinetic and Potential (E-static) energies associated with them.

 

[Redbelly98]

I possesses a very rudimentary knowledge - there are only certain orbits around which an electron is permitted to revolve and these have certain Kinetic and Potential (E-static) energies associated with them.

That's pretty close to the mark. More accurately there are only certain orbitals, and each has certain amount of Total Energy. (Where total energy is the sum of the kinetic and potential energies.)

Having discrete, allowed energies is also a property of the two-atom potential we have been discussing, as long as the energy is low enough that the atoms are bound together. That is what is depicted by the energy curve here:

The curved line is the potential energy, while the horizontal lines are the allowed energies for the two atoms. Note, a horizontal line gives the Total Energy (kinetic + potential).

In other words, at any time the total energy must be one of those values given by one of the horizontal lines, and can't be anything lying in between them.

Are you with this so far?

 

[WiFO215]

Having discrete, allowed energies is also a property of the two-atom potential we have been discussing, as long as the energy is low enough that the atoms are bound together.

Okay. Bound together means the total energy of the system is below the horizontal "zero potential" line.

The curved line is the potential energy, while the horizontal lines are the allowed energies for the two atoms. Note, a horizontal line gives the Total Energy (kinetic + potential).

In other words, at any time the total energy must be one of those values given by one of the horizontal lines, and can't be anything lying in between them.

Given that the total energy is such and such, I suppose I could figure out how much potential energy it has got from analyzing the curve and thereby calculate the kinetic energy. I suppose these two quantities will characterize the system as all I can imagine in the case of two atoms is only linear motion.

What determines the spacing between two horizontal lines? I suppose that depends on the two atoms/ molecs in question? That should be important, as you say that is what they measure (post #18). Does the spacing between the lines tell you the shape of the curve also?

 

[Redbelly98]

Just a reminder:

Good question. I don't know definitely, but can hazard a guess...

That being said . . .

Okay. Bound together means the total energy of the system is below the horizontal "zero potential" line.

Yes.

Given that the total energy is such and such, I suppose I could figure out how much potential energy it has got from analyzing the curve and thereby calculate the kinetic energy. I suppose these two quantities will characterize the system as all I can imagine in the case of two atoms is only linear motion.

We're getting a little off track here. Are you aware of Heisenberg's Uncertainty Principle ? Suffice to say, we can never measure both the atoms' separation and velocities to sufficient accuracy. So forget any thoughts you might have about subtracting the kinetic (depends on velocity, that we can't know accurately) energy from the total energy to find the potential energy at a specific location (which we also cannot know with good precision.)

What determines the spacing between two horizontal lines? I suppose that depends on the two atoms/ molecs in question? That should be important, as you say that is what they measure (post #18).

The spacing is determined by the solution of the Schrodinger Equation for the potential curve. If you're not too familiar with the Schrodinger Equation, I'll just say that it is a differential equation whose solution gives the allowed (total) energies, and leave it at that for now. So it would tell us the values of the horizontal lines in the graph, and more specifically the spacings between those lines. It is the spacings, or energy differences, that get measured in the lab.

Does the spacing between the lines tell you the shape of the curve also?

That is the tricky part. Since we don't know the shape of the curve to begin with, we can't solve the Schrodinger Equation (SE) for the curve to calculate the energies. Instead, the energy spacings would be measured as a starting point, and we must do the reverse of solving the SE, to find out the shape of the potential energy curve.

But I think there is more than one possible curve that could produce a given set of energies. (Can anybody else confirm/refute this?). If so, we would need some other information. I am supposing that, since the 1/r⁶ long-range part of the curve is justified on theoretical (and perhaps experimental as well?) grounds, one might determine what can be added to a 1/r⁶ potential to fit the measured energy spacings. But here we are getting outside of what I, personally, know about how these curves are determined.

 

[WiFO215]

The spacing is determined by the solution of the Schrodinger Equation for the potential curve. If you're not too familiar with the Schrodinger Equation, I'll just say that it is a differential equation whose solution gives the allowed (total) energies, and leave it at that for now. So it would tell us the values of the horizontal lines in the graph, and more specifically the spacings between those lines. It is the spacings, or energy differences, that get measured in the lab.

Hmm. My expertise level is mechanics at the level of Kleppner and electromagnetism at the level of Griffiths. So I don't know very much about SE.

That is the tricky part. Since we don't know the shape of the curve to begin with, we can't solve the Schrodinger Equation (SE) for the curve to calculate the energies. Instead, the energy spacings would be measured as a starting point, and we must do the reverse of solving the SE, to find out the shape of the potential energy curve.

Here we come back to my original question of how and what they measure. You said something about spectrometers which I didn't understand.

But I think there is more than one possible curve that could produce a given set of energies. (Can anybody else confirm/refute this?). If so, we would need some other information. I am supposing that, since the 1/r⁶ long-range part of the curve is justified on theoretical (and perhaps experimental as well?) grounds, one might determine what can be added to a 1/r⁶ potential to fit the measured energy spacings.

Okay. This is what I understood : The energy spacings are given by solutions to some PDE. Can't we use the Existence-Uniqueness theorem to pin down the curve who's giving rise to the spacings? If so, shouldn't there be one and only one curve?

 

[Redbelly98]

Here we come back to my original question of how and what they measure. You said something about spectrometers which I didn't understand.

Okay. The system can jump between different energy levels, and a photon will be either emitted or absorbed whenever that happens. By measuring the wavelengths of the emitted photons, we can tell what energy spacings are present, since the energy spacings are given by

ΔE = hc/λ​

Okay. This is what I understood : The energy spacings are given by solutions to some PDE. Can't we use the Existence-Uniqueness theorem to pin down the curve who's giving rise to the spacings? If so, shouldn't there be one and only one curve?

Perhaps. PDE's are not really my area of expertise.I just went back to the wiki article that jtbell linked to earlier. It says:

... the r⁻⁶ term describes attraction at long ranges. The [r⁻¹²] term has no theoretical justification ... [but is] convenient due to the ease and efficiency of computing r¹² as the square of r⁶

Thus I suspect you just need to adjust the coefficients on the r⁻⁶ and r⁻¹² terms -- a total of two parameters -- and obtain a "best fit" to the known energy spacings, which come from the measured wavelengths of emitted photons. Again, I suspect.

 

[WiFO215]

Again, I suspect.

Sir, may I comment you have a very fine attribute in that you admit when you do not know something. I have had very few people who have taught me who have this quality. Thank you.

Getting back to topic, how do I [we?] learn further from here?

 

[Redbelly98]

...Thank you.

You're welcome.

Getting back to topic, how do I [we?] learn further from here?

I would recommend:
1. Keep asking questions here, and
2. Take a course in quantum mechanics, if you are still in school.

 

[Redbelly98]

... how do I [we?] learn further from here?

Another, shorter-term suggestion for you:

Besides Physics Forums, you could look online at wikipedia or hyperphysics. A few topics to look for are:

The Schrodinger Equation (the time-dependent and also time-independent versions)
wavefunctions
eigenvalues, eigenvectors, and eigenfunctions​

These are some of the fundamental building blocks to learning quantum mechanics.

As for me, if I had the time I'd probably look for journal articles dealing with the possible measurements we have been discussing here. But my current, personal interests are to learn more about electric motors, power generation, and thermodynamics.

Good luck to you.

 

[WiFO215]

Another, shorter-term suggestion for you:

Besides Physics Forums, you could look online at wikipedia or hyperphysics. A few topics to look for are:

The Schrodinger Equation (the time-dependent and also time-independent versions)
wavefunctions
eigenvalues, eigenvectors, and eigenfunctions​

These are some of the fundamental building blocks to learning quantum mechanics.

I know what eigenvalues/ functions are. My Linear Algebra is not too bad and getting better. I don't know much about the other two though. I'll look into those. I am currently to learn about classical statistical mechanics and quantum mechanics and that is where all these questions popped up from. Texts always seem to leave many holes in my understanding.

Good luck to you.

Thanks for all your help! Does anyone else have comments on this?