# TBR vs. Everyone else on: Real vs. Ideal Gases

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#### ilovemcat

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This is a concept that his been troubling me for the past hour.

EK, Kaplan, and most websites all seem to agree that for Real Gases:

1. Preal < Pideal - because molecules DO have attractive forces
2. Vreal > Videal - because molecules DO have volume

It's this second statement that's troubling me, and one that I can't seem to fully understand. There logic is something like this:

Videal = Vempty
Vreal = Vparticles + Vempty = Vparticles + Videal
Vreal > Videal

Website: http://www.mpcfaculty.net/mark_bishop/real_gases.htm

Crayton from EK
The fact that Vreal > Videal is just that the real volume of the gas must also include the amount of volume that the actual gas molecules take up themselves (the size of the atoms or molecules, which is dictated by the size of their electron clouds).

TBR and TPR on the other hand states the exact opposite:

Pg. 10, Chapter 6 Gen Chem
Here's a quote taken from TBR Gen Chem:
Because they have volume (molecules) on the microscopic level, the actual free space (space not occupied by molecules) is less than the volume of the container. The bigger the molecules, the greater the volume they occupy, thus the greater the deviation. The more molecules, the greater the volume they occupy, thus the greater the deviation. The free space (ideal volume) is found by taking the container volume and subtracting the volume of the molecules. This means that the ideal volume (volume of the empty space) is equal to the difference between the volume of the container and the volume of the particles.

Therefore: Videal = Vcontainer - Vparticles

Here's a statement taken from The Princeton Review:

TPR - Pg. 524, Physics
And Vreal < Videal because moles of real gas do have volume that reduces the effective volume of the container (since the moles take up space, there is less space in the container for all the other particles to occupy.)

In my opinion TBR & TPR's logic makes sense to me. How exactly can the ideal volume be GREATER for a real gas. If we assumed the volume for an ideal gas was the volume of it's container... you can't have "GREATER" volume. It just didn't make sense.

Anyways, who the heck is right? Normally I'd be like whatever and move on. But I find myself stuck in a position where I either get multiple choice questions WRONG for EK and Kaplan (if I used TBR and TPR's logic) or multiple choice questions WRONG for TPR and TPR (if I use EK and Kaplans logic).

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Here's a perfect example of why the perspective matters:

Question taken from TBR (hope they don't mind):

For an inert gas, if you were to reduce the pressure to half of its original value, then what is the final volume (Vf) relative to the initial volume (Vi)?

A. 1/2Vi - a little bit
B. 1/2Vi + a little bit
C. 2Vi - a little bit
D. 2Vi + a little bit

Hey,
According to tpr, gases are most ideal at low pressure, high temp.
In the Kaplan General Chemistry Review ebook (p175), it says that at low temp (reduced to condensation/boiling point), Vreal<Videal because of intermolecular attractions making volume smaller. This is the same for mederately high pressure. However, at extremely high P, size of molecules>>distance between them, so Vreal>Videal. What's the answer to your TBR question? According to above, it would C. Is that right?

It seems to depend on how your interpreting how much space the actual molecule takes up (like what EK said about the molecule's volume). However, for the purposes of the mcat, if TPR, TBR and Kaplan all agree, I would probably listen to the majority.

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Hey,
According to tpr, gases are most ideal at low pressure, high temp.
In the Kaplan General Chemistry Review ebook (p175), it says that at low temp (reduced to condensation/boiling point), Vreal<Videal because of intermolecular attractions making volume smaller. This is the same for mederately high pressure. However, at extremely high P, size of molecules>>distance between them, so Vreal>Videal. What's the answer to your TBR question? According to above, it would C. Is that right?

Yes, the answer was C. But that statement in bold (from Kaplan) doesn't make much sense to me either... The size of the molecules themselves doesn't change with temperature or pressure, so there's no reason to expect the relationship to depend on whether we're at "moderately high" or "extremely high" pressures. Incidentally, what are "moderately high" or "extremely high" even supposed to be?

Oh, I paraphrased a little bit. They had moderately high as "a few hundred atmospheres". They didn't say define extremely high was though. Also, which Kaplan version do you have? The one I have was from the ebook giveaway back in January, so it should be pretty recent. I also checked the errata, so it shouldn't be a mistake.

At high pressure, there is very little space for the molecules to move around and the normally miniscule volume of the molecule becomes relevant. More volume is taken up than expected. At low pressure, the volume has increased to a size that greatly overshadows the volume of the molecules. For molecules with significant attractive intermolecular forces, these intermolecular forces now outweigh the effect of the size of the molecules, causing the volume to be smaller than expected.

In either situation, I think the answer should be C.

Also, which Kaplan version do you have? The one I have was from the ebook giveaway back in January, so it should be pretty recent. I also checked the errata, so it shouldn't be a mistake.
I'm not entirely sure what edition it is. They have a copy at my local library. I think it's the 2008 edition!

At high pressure, there is very little space for the molecules to move around and the normally miniscule volume of the molecule becomes relevant. More volume is taken up than expected.
Exactly, but then the question becomes: Is Vreal > Videal or Vreal < Videal. It's this perspective that I'm having trouble with. If a question asks, "Is the volume of the real gas LARGER than the predicted ideal Volume or SMALLER than the predicted ideal Volume" which answer would be correct? It seems much more intuitive to agree with the latter choice - that Vreal is less than the ideal Volume because: Vreal = Videal - Vparticles

For molecules with significant attractive intermolecular forces, these intermolecular forces now outweigh the effect of the size of the molecules, causing the volume to be smaller than expected.
You confused me here. Because real gases do have attractive forces, they would pull molecules closer to each other and hit the wall less. This inturn would result in a reduced pressure as compared to the ideal pressure (Preal < Pideal). But, I'm having a hard time following what you meant when you said the volume would be smaller than expected. The volume of the container wouldn't change - and the size of the molecules themselves are fixed.

Hello again. I see the conceptual problem you're having, and with the help of my PChem textbook, I hope I can help clear it up. Prepare for a wall of text.

First of all, let's define what we mean by Vreal, Videal, Preal, Pideal, and so on. Videal is the volume that a given mole of ideal gas would occupy at a given pressure and temperature. Pideal is the pressure that a given mole of ideal gas would exert when it occupies a given volume at a given temperature. Vreal is the same as Videal, but for a real gas, and same goes for Preal. To compare Vreal to Videal, you have to hold everything else constant except for the identity of the gas (the factor that would make a gas not behave ideally).

Preal < Pideal and Vreal > Videal are both true statements only in strict circumstances. For the first one, you assume that the only thing that separates real from ideal gases is intermolecular forces. For the second statement, you assume that the only thing that separates real from ideal is that a real gas molecule occupies some volume, whereas an ideal gas molecule does not. Those assumptions have to be made for the statements to be true. Also, you might see that the two assumptions are mutually exclusive. If you have a gas that has significant intermolecular attractions, but insignificant volume taken up by the particles (i.e. the molecules are very small compared to the Videal so that Vreal = Videal), then you can say that Preal < Pideal. Similarly, if you have a gas that has significant volume taken up by the particles, but insignificant IMFs, then you can say that Vreal > Videal. If both factors are significant, then... we'll discuss that later. If both are insignificant, then the gas behaves ideally, of course.

The reason TBR and TPR state what appears to be opposite has to do with different terminology, and Van der Waal's equation. Forgetting about IMFs entirely and only considering the physical volume of the particles, it is true that the volume available will have to be less than if the particles behaved ideally and thus did not take up any volume (volume of the container). But here's the kicker. When TBR says Videal, they don't mean Videal in the sense I defined it earlier. They mean the volume that you would have to now plug into the ideal gas equation PV = nRT to get it to work out. The ideal gas equation becomes P(Vcontainer - Vparticles) = nRT. They define Vcontainer - Vparticles as Videal. TPR, however, defines Vcontainer as Videal, and (Vcontainer - Vparticles) as Vreal, so Vreal < Videal. Both of their approaches are not how I defined Videal and Vreal. To me, Vreal is the volume of the container of a real gas, because Vreal has to be measurable, and the only volume that is measurable is the volume of the container. To me, Videal is the volume of the container of a theoretical ideal gas under the same temperature and temperature as the real gas. The way Kaplan defines Vreal and Videal is the way I defined it, which is why tttgo's question makes no sense to you. By the way, (Vcontainer - Vparticles) is the same as V - nb in the Van der Waal's equation.

Now I will introduce a concept called compressibility factor. Z = pVm/RT, where Vm is molar volume (volume per mole). If a gas behaves ideally, then Z should be 1 at all times. Since R is constant, this also means that knowing two of the three variables (P, Vm, and T) is enough. The third variable is entirely determined by the other two. (A fancy, physical chemist's way of saying this is that the ideal gas law is an equation of state, but this is unimportant) For real gases, however, Z does not always equal 1. It turns out at low pressure and high temperature, Z approaches 1, and we can say that a real gas approaches ideal behavior as its pressure approaches 0 and its temperature approaches infinity. For most real gases, Z is below 1 if pressure is moderately high, but above 1 if pressure is extremely high. Going back to the definition of Z, this means that at moderately high pressure, pVm/RT is less than what would be expected by the ideal gas law. Since we've already prescribed a value for p, and R is a constant, that really means that Vm/T is less than what would be expected. At a given temperature, then this means that Vm is less than expected, or that Vreal < Videal. Conversely, at extremely high pressure, Vreal > Videal. What constitutes moderately high or extremely high is unique for each gas, and also dependent on temperature.

The physical explanation for this observation is that at moderately high pressure, the IMFs are significant, and thus bring the gas molecules closer together. This results in a reduction of volume (as I defined Vreal earlier) compared to what would be if there were no IMFs (Videal). At extremely high pressure, the gas molecules are so close together that the volume occupied by the particles begin to matter, so expansion of volume has to occur. Now imagine that a particular gas has no IMFs. Then Z would never go < 1, but it would still go > 1 as pressure increases. Since all gases, even the smallest possible (H2) has particles that take up definite volume, Z will eventually increase as pressure increases. For the TBR question, I think inert gas means a gas like helium that has practically no IMFs. In this case, Z is at minimum 1 (at the limit of P -> 0). As P increases, Z has to also increase, however slowly. Halving P means approaching 0, so Z has to decrease to 1, however little. Since Z is Vm/T, and we're presumably keeping T constant, then Vm cannot quite be twice the initial Vm. It has to be a little bit less.

I hope that helped. I really should use my spring break time with better things... Here are some wikipedia links to clear up confusions.

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

Wow, dude! When I saw this, I went from to , haha. Anyways, thanks so much for taking the time to write that. It really clarified a lot of things! I must admit though, I'm still slightly confused. Could you tell me if this sounds right? If this is wrong, I'm gonna die.

Let's say we're told the volume of an ideal gas is placed inside a container of 10 mL. Because we're assuming an ideal gas is essentially volumeless, it is this volume we plug into the ideal gas equation. This is the ideal volume.

However, for the same container - if we're asked to compare a Real Gas (where IMFs can be neglected) to an Ideal Gas - the actual (real) volume occupied by the gas IS the full container, but in this case Videal = (Vreal - Vmolecules) because molecules DO have volume. Therefore, Videal would occupy less of the real volume: Vreal > Videal

Is this what you mean when you say Vreal = Videal?

Now this is what I don't get though.
Because they have volume (molecules) on the microscopic level, the actual free space (space not occupied by molecules) is less than the volume of the container. The bigger the molecules, the greater the volume they occupy, thus the greater the deviation. The more molecules, the greater the volume they occupy, thus the greater the deviation. The free space (ideal volume) is found by taking the container volume and subtracting the volume of the molecules. This means that the ideal volume (volume of the empty space) is equal to the difference between the volume of the container and the volume of the particles.

Therefore: Videal = Vcontainer - Vparticles
When TBR says Videal, they don't mean Videal in the sense I defined it earlier. They mean the volume that you would have to now plug into the ideal gas equation PV = nRT to get it to work out. The ideal gas equation becomes P(Vcontainer - Vparticles) = nRT. They define Vcontainer - Vparticles as Videal.
Based on the example I gave above, where we're asked to consider how a real gas deviates from an ideal gas - this is when the above statement is applied, right? That is to say, for the ideal equation: Videal = (Vcontainer-Vparticles) only when we are considering real gases. Otherwise, Videal = Volume of the container (your definition of Videal). Wouldn't this then mean that you and TBR's perspective is ... sorta the same? I hope I explained this right, because boy ... this sure is confusing to explain.

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You are right. TBR and I pretty much have the same perspective. TPR seems weird to me, though. Also, the point I want to emphasize is that Vreal > Videal when you only have to consider the volume of the gas particles and neglect any IMFs. Otherwise, the whole spiel about the compressibility and moderate/extreme pressure matters. TPR's answer though, makes no sense, unless you define Vreal and Videal differently.

Also, your picture is beautiful and correct. Defining Videal as nRT/P (TBR) and the way I defined it is not actually any different.

You are right. TBR and I pretty much have the same perspective. TPR seems weird to me, though. Also, the point I want to emphasize is that Vreal > Videal when you only have to consider the volume of the gas particles and neglect any IMFs. Otherwise, the whole spiel about the compressibility and moderate/extreme pressure matters. TPR's answer though, makes no sense, unless you define Vreal and Videal differently.

Also, your picture is beautiful and correct. Defining Videal as nRT/P (TBR) and the way I defined it is not actually any different.

Awesome. Thanks so much dude!

TBR and I pretty much have the same perspective.

It figures you P Chemist types stick together.

Ouch my brain hurts a bit from this thread. I've never seen Z = pVm/RT and I have a feeling this is what MCAT likes to do..just confuse me come test day with something I haven't seen.

But, I guess what I'll take away from this is that Videal = volume of empty space.

Also for the picture I think you meant Videal= Vreal-Vmolecule not Videal=Vreal-Videal.

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Ouch my brain hurts a bit from this thread. I've never seen Z = pVm/RT and I have a feeling this is what MCAT likes to do..just confuse me come test day with something I haven't seen.

But, I guess what I'll take away from this is that Videal = volume of empty space.

Also for the picture I think you meant Videal= Vreal-Vmolecule not Videal=Vreal-Videal.

Just imagine how I felt trying to figure out what the heck was going on. Thanks to Rabolisk it all makes sense now.

And yeah, you're right haha. I tried to make an image because for some apparent reason I can remember things I've seen for a very long time as opposed to a concept I've read a few times.