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volume of gas at absolute zero





Afaceinthematrix
According to my chemistry book, theoretically, at absolute zero, the volume of an ideal gas will be zero. How can a volume be zero? Can someone please explain that to me?
Xanatos
Afaceinthematrix wrote:
According to my chemistry book, theoretically, at absolute zero, the volume of an ideal gas will be zero. How can a volume be zero? Can someone please explain that to me?


It can't. Absolute zero can never be reached. Besides the fact that at some point before zero the gas would condense into a liquid and then a solid, absolute zero would cause an particles in the atom to have an infinite amount of energy. As objects get closer and closer to zero it becomes possible to know both their location and speed with a high degree of certainty giving particles a high energy as electrons get closer and closer to the nucleus. If the volume were to be zero, all of these particles would have an infinite amount of energy which at the current time is thought to be impossible.
Afaceinthematrix
Well I know that it's impossible. I think the lowest temperatures that have been reached were just a couple of degrees kelvin... but the chemistry book did say that theoretically the volume of an ideal gas at absolute zero would theoretically be zero, and I was just trying to understand how that would even theoretically be true.
TomS
The ideal gas is only ideal, if it follows this mathematical rule (that's the definition of an ideal gas).



Dividing it by the pressure, you get: Volume = Mass * Some Constant * Temperature / Pressure

If the Temperture is zero you get: Volume = 0 / Pressure = 0

So the volume is 0, if the temperture is 0.

The ideal gas exists only theoreticaly.
Logically the volume can't be 0 and therefor no gas is ideal.
Xanatos
TomS wrote:
The ideal gas is only ideal, if it follows this mathematical rule (that's the definition of an ideal gas).



Dividing it by the pressure, you get: Volume = Mass * Some Constant * Temperature / Pressure

If the Temperture is zero you get: Volume = 0 / Pressure = 0

So the volume is 0, if the temperture is 0.

The ideal gas exists only theoreticaly.
Logically the volume can't be 0 and therefor no gas is ideal.


Also by the ideal gas law if you have zero volume you have zero molecules. When volume is 0 n(number of moles) also equals zero. Therefore the only thing that can have zero volume is nothing.
Indi
Both TomS and Xanatos are correct about the fact that the ideal gas law just doesn't apply at absolute zero, but for all the wrong reasons. In fact, they both have it backwards. The ideal gas law doesn't fail at absolute zero because the math leads to nonsense answers, the math leads to nonsense answers because the gas law fails. This is physics, not math - nature trumps calculation every time. Even if the math led to meaningful results, the ideal gas law would still fail at absolute zero.

In order to understand why the ideal gas law doesn't apply at absolute zero, you have to understand what the ideal gas law is. The ideal gas law is derived from statistical mechanics (among other methods, but i'd say that way is preferred nowadays) using a simplified model. The model assumes that the gas is a bunch of point particles darting around in random directions at random speeds in a perfectly sealed container, with no forces on those particles except the impacts on each other and on the walls of the container. Now, this is a pretty good assumption most of the time, because most of the forces acting on gas particles are so tiny compared to the forces due to impacts - gravity barely budges the tiny masses given their speeds, the attractions and repulsions between the electric fields of the particles is negligible if the particles are moving fast enough and are far enough apart, and the kinetic energy (speeds) of most particles is way above the levels where quantum effects dominate - and the size of the particles is extremely small compared to the spaces between them.

Now you have the tools you need to understand what conditions the ideal gas law will fail under. For example: the ideal gas law assumes point particles, which works when the spaces between particles is huge compared to the size of the particle - but when the size of the particles is large, this doesn't apply. Thus, the ideal gas law performs poorly when applied to gases with large molecules. It also doesn't apply when the particles are squeezed closely together - that is, under very high pressure.

Annnnnd... when the kinetic energies of the particles is small enough that other effects - gravity, the electrostatic attractions/repulsions between particles, and quantum effects - begin to become more noticeable... the law breaks down. Temperature is a measure of average kinetic energy. Thus, when the temperature is low enough, the ideal gas law breaks down.

And that is the answer to your question. The ideal gas law works at high temperatures and low pressures, with gases that have only small particles... and fails otherwise. Absolute zero is the lowest temperature you can go.

--------------------------------------------------

What happens physically is that the molecules of the gas at (that is, near) absolute zero are moving so slow that the energy exchange by things like gravity are significant.

Consider gravity for example. At room temperature (~300 K), gas molecules of air move at an average of hundreds of metres per second. That's a couple thousand kilometres an hour. Even in a container the size of a bathtub (or a whole room!), how much can gravity possibly deflect the particle even if manages to travel the entire distance across the container? Not a whole lot, compared to the size of the container. When the temperature drops low enough so that the average particle is moving at just a few metres per hour, gravity makes a huge difference on the path of the particle.

Consider electrostatic effects - every atom and molecule has positive and negative charges, and they're not always in balance (the electrons may all happen to be on one side of the molecule at a given moment, for example). When two atoms shoot past each other at a thousand kilometres a second, they don't really effect each others' path much. When they mosey on by slowly, they can actually capture and orbit each other.

And finally, consider quantum effects. At high energies, the change in energy levels is pretty much continuous - which is the way we're used to things. But at really low energies, energy levels are quantized. That means you can have particles moving at 10 m/s and 20 m/s (for example), but not 15 m/s. So particle speeds will leap up and down erratically. Also, at high energies, the distribution of energies can be Gaussian (a smooth bell curve) about the average energy. At low energies, it can't be smooth because of quantum levels - there must be steps - and it can't get to zero so the bell curve will be skewed to one side. In other words, Gaussian statistics just won't work.

Each of these effects combines as you approach absolute zero, making the ideal gas law totally useless.

Incidentally:
Xanatos wrote:
Besides the fact that at some point before zero the gas would condense into a liquid and then a solid...

Not necessarily! ^_^
Xanatos
^^ You always have the best and most informative posts. I both hate and love you at the same time.
Afaceinthematrix
Thanks for the replies. I guess the gas laws do not qualify at absolute zero. I was confused because my chemistry book specifically said that theoretically, at absolute zero, the volume would be zero and I couldn't understand how that would work.
Indi
Afaceinthematrix wrote:
Thanks for the replies. I guess the gas laws do not qualify at absolute zero. I was confused because my chemistry book specifically said that theoretically, at absolute zero, the volume would be zero and I couldn't understand how that would work.

Yeah, you're book is just wrong there, unless the wording is something like: "theoretically according to the ideal gas law..." but even then that's just really sloppy "theory", because the ideal gas law just doesn't apply there, and that is implicit in the gas law itself.

You were right to suspect foul play though. A non-zero amount of anything that takes up space cannot have zero volume, no matter how hot or cold it gets. As you intuited, that just makes no sense.

Also, you should bear in mind that there is more to matter than solid-liquid-gas. ^_^ Most substances do go from gas to liquid to solid at normal pressures - as Xanatos said. But some skip the liquid phase (dry ice, for example), and some... do something else. ^_^; For example, scientists managed to take a sample of rubidium gas down to a hundred or so nanokelvins, and it went from gas to something called a Bose-Einstein condensate.
Afaceinthematrix
Indi wrote:
Yeah, you're book is just wrong there, unless the wording is something like: "theoretically according to the ideal gas law..." but even then that's just really sloppy "theory", because the ideal gas law just doesn't apply there, and that is implicit in the gas law itself.


Unfortunately, the book did not say that. It said:
Chemistry: The Molecular Nature of Matter and Change by Silberberg wrote:
For any amount of an ideal gas at any pressure, the volume is theoretically zero at -273.15C (0K).


I think the reason why the incorrectly chose this wording is because the text is accompanied by a graph, and they were describing the graph (temperature vs. volume) in words. They should have said something like "The limit of the volume as the temperature approaches absolute zero will be zero." I think that would have been correct.

Quote:
Also, you should bear in mind that there is more to matter than solid-liquid-gas. ^_^ Most substances do go from gas to liquid to solid at normal pressures - as Xanatos said. But some skip the liquid phase (dry ice, for example), and some... do something else. ^_^; For example, scientists managed to take a sample of rubidium gas down to a hundred or so nanokelvins, and it went from gas to something called a Bose-Einstein condensate.


I knew about dry ice but I did not know about the rubidium gas. I'm going to research into that because it sounds interesting.
Bikerman
BECs (Bose Einstein Condensates) are weird and wonderful 'phases of matter'.
http://en.wikipedia.org/wiki/Bose-Einstein_condensation
http://www.colorado.edu/physics/2000/bec/
http://physicsworld.com/cws/article/print/2242
http://www.jupiterscientific.org/sciinfo/boseeinstein.html
Flakky
TomS wrote:
The ideal gas is only ideal, if it follows this mathematical rule (that's the definition of an ideal gas).



Dividing it by the pressure, you get: Volume = Mass * Some Constant * Temperature / Pressure

If the Temperture is zero you get: Volume = 0 / Pressure = 0

So the volume is 0, if the temperture is 0.

The ideal gas exists only theoreticaly.
Logically the volume can't be 0 and therefor no gas is ideal.


To support this you should know that any formula is just an attempt to explain real life behaviours so as cool as the theory may sound, it's still a theory which many think is untrue.
victornumber
hmm....I read an article before that talked about super atoms at low temperature where the atoms all come together. Starts with B something.
Bikerman
victornumber wrote:
hmm....I read an article before that talked about super atoms at low temperature where the atoms all come together. Starts with B something.
Bose Einstein Condensate.
http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate
Afaceinthematrix
victornumber wrote:
hmm....I read an article before that talked about super atoms at low temperature where the atoms all come together. Starts with B something.


Bose Einstein Condensates
liljp617
Good to see Razz I'm currently using the same book.
Afaceinthematrix
liljp617 wrote:
Good to see :P I'm currently using the same book.


You're using the Silberberg chemistry book? Well if you're using the fifth edition, the mistake is on page 195 underneath the graph.
PatTheGreat42
The volume of the gas would be zero because it should all be a liquid or a solid, and therefore it is the liquid or solid that would have the volume.
Xanatos
PatTheGreat42 wrote:
The volume of the gas would be zero because it should all be a liquid or a solid, and therefore it is the liquid or solid that would have the volume.


Not necessarily at low pressures, gases stay gases even at extremely low temperatures.
Indi
Afaceinthematrix wrote:
Indi wrote:
Yeah, you're book is just wrong there, unless the wording is something like: "theoretically according to the ideal gas law..." but even then that's just really sloppy "theory", because the ideal gas law just doesn't apply there, and that is implicit in the gas law itself.


Unfortunately, the book did not say that. It said:
Chemistry: The Molecular Nature of Matter and Change by Silberberg wrote:
For any amount of an ideal gas at any pressure, the volume is theoretically zero at -273.15C (0K).


I think the reason why the incorrectly chose this wording is because the text is accompanied by a graph, and they were describing the graph (temperature vs. volume) in words. They should have said something like "The limit of the volume as the temperature approaches absolute zero will be zero." I think that would have been correct.

Eeeeh... i would be inclined to let this slide. Their wording is technically correct, once you highlight the key words:
Quote:
For any amount of an ideal gas at any pressure, the volume is theoretically zero at -273.15C (0K).
It's true: the volume of an ideal gas at 0 K is 0, theoretically speaking. It's just that an ideal gas is just a theoretical construct - there is no such thing in reality (although some fluids behave like ideal gases under certain conditions)... but they do use the word "theoretically".

i wouldn't call this an error, personally. If it were me, i would add a note that the ideal gas law is only a theoretical model, and certainly will never apply under those conditions, but otherwise i wouldn't change it.
ocalhoun
Indi wrote:

i wouldn't call this an error, personally. If it were me, i would add a note that the ideal gas law is only a theoretical model, and certainly will never apply under those conditions, but otherwise i wouldn't change it.

Wouldn't it actually apply, though, with an ideal gas?
That might be why there's no such thing as an ideal gas, because if there was, then there would have to be some way for it to have 0 volume.

(Unless I misunderstand the situation, which is likely.)
Afaceinthematrix
Indi wrote:
Afaceinthematrix wrote:
Indi wrote:
Yeah, you're book is just wrong there, unless the wording is something like: "theoretically according to the ideal gas law..." but even then that's just really sloppy "theory", because the ideal gas law just doesn't apply there, and that is implicit in the gas law itself.


Unfortunately, the book did not say that. It said:
Chemistry: The Molecular Nature of Matter and Change by Silberberg wrote:
For any amount of an ideal gas at any pressure, the volume is theoretically zero at -273.15C (0K).


I think the reason why the incorrectly chose this wording is because the text is accompanied by a graph, and they were describing the graph (temperature vs. volume) in words. They should have said something like "The limit of the volume as the temperature approaches absolute zero will be zero." I think that would have been correct.

Eeeeh... i would be inclined to let this slide. Their wording is technically correct, once you highlight the key words:
Quote:
For any amount of an ideal gas at any pressure, the volume is theoretically zero at -273.15C (0K).
It's true: the volume of an ideal gas at 0 K is 0, theoretically speaking. It's just that an ideal gas is just a theoretical construct - there is no such thing in reality (although some fluids behave like ideal gases under certain conditions)... but they do use the word "theoretically".

i wouldn't call this an error, personally. If it were me, i would add a note that the ideal gas law is only a theoretical model, and certainly will never apply under those conditions, but otherwise i wouldn't change it.


Well it may not be an error in technical senses, but it is an error in teaching style (in my opinion) because it leaves a lot of room for confusion among the students (which is why I started this thread).
Indi
ocalhoun wrote:
Indi wrote:

i wouldn't call this an error, personally. If it were me, i would add a note that the ideal gas law is only a theoretical model, and certainly will never apply under those conditions, but otherwise i wouldn't change it.

Wouldn't it actually apply, though, with an ideal gas?
That might be why there's no such thing as an ideal gas, because if there was, then there would have to be some way for it to have 0 volume.

(Unless I misunderstand the situation, which is likely.)

Well, it is possible, in theory, for there to be a truly ideal gas. Not "literally", of course - you would have to allow for something exotic, like non-baryonic fluids, to count as "gases", but the situation that the ideal gas law describes is not completely beyond the realm of possibility if you're willing to be flexible with what you call a gas.

But you have to understand what that would mean to have a truly ideal gas. Among other things, it would have to be made up of particles that are not only literally of zero size (point particles), but are also capable of coexisting in the same place. In other words, you could literally have an infinite number of particles in zero volume, because each particle takes up absolutely no space and it is possible to have absolutely no space between them.

So it is technically possible for a truly ideal "gas" to exist, and for it to have zero volume.

But of course, that only applies for "gases" that aren't made up of any kind of matter i can think of off the top of my head. For normal matter - baryonic matter - the exclusion principle applies, which causes atoms to take up non-zero volumes (and prevents more than one atom from occupying the same position). Even for bosonic matter (such as the Bose-Einstein condensates that a couple people have mentioned), although the exclusion principle doesn't apply, zero point energy does: bosons can't hold still, even at 0 K, so they can't occupy a single point in space (they occupy a "probability cloud" that they dance around in).

So, basically it boils down to this: if you forgo the idea of a gas being regular matter, you can have an ideal "gas", and it can occupy zero volume.
Arnie
A better approximation to gas behaviour is the van der Waals equation. This is still an approximation with unphysical artefacts: mathematically it allows negative pressure.
Flakky
I've been thinking about this again and came to the short conclusion that this only applies to molecules which liquify or change to solid state under 0K and I am not aware of any substance of that kind.
Indi
Flakky wrote:
I've been thinking about this again and came to the short conclusion that this only applies to molecules which liquify or change to solid state under 0K and I am not aware of any substance of that kind.

No, it does not apply to any molecules, anywhere, ever. It just doesn't apply. Period.

In some cases, though, it is close enough that it works as a very good estimator.

But do not confuse being a good estimator with actually applying. It doesn't apply. Even in theory, it doesn't apply.

(Incidentally, don't assume that all matter follows that simple high school model of solid/liquid/gas. For the vast majority of cases, matter does not go from solid to liquid to gas. It just happens that within the range of cases that we usually deal with, that usually happens.)
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