# Why is there a subtractive volume correction in the van der Waals' equation for real gases?

Why there is a subtractive volume correction in van der Waals' equation for real gases, both real and ideal gases occupy volume of container in which they are kept, so there shouldn't be any correction?

• Your post is rather unclear at the moment: please edit it for grammar and clarity and be sure to add your thoughts concerning the answer. If this is homework, please be sure to read the site policy on such. Sep 8, 2015 at 13:21
• This is a wholly different kind of volume. Sure, any gas occupies the volume of container, but here we are talking about the volume of molecules themselves. Ideal gas has none; real gas has some. Sep 8, 2015 at 13:52
• I don't understand whose volume is talked about of in ideal gas and real gas Sep 8, 2015 at 13:54
• Is the volume is of container or gas molecules or something else Sep 8, 2015 at 13:56
• You have two volumes here. The volume of the container V and the volume of the gas molecule, b. Ideal gases are thought of as being points ie zero volume, b=0. Therefore they move about the total volume of the container. Real gases have a volume, b>0. Each gas molecule takes up some space in the container, meaning there is less effective volume for them to move about in. Effective volume is V-nb where n is no of molecules ( or moles depending on units of b). Sep 8, 2015 at 17:11

The van der Waals equation is an attempt to explain the pressure-volume relationship for real gases. For example, what will the pressure be of a given amount of gas in a vessel of a fixed volume.

In ideal gases the size and interactions between the atoms or molecules of the gas don't matter at all. We think of them as point-like entities that don't interact except to bounce elastically off each other and off the walls of the vessel. Their behaviour is described by the ideal gas equation and doesn't depend at all on the nature of the particles making up the gas.

Real molecules and atoms deviate from this ideal behaviour because the particles of the gas both have finite volume and attractive interactions which mean they don't behave exactly as the ideal gas equation describes. The van der Waals equation is a simple attempt to explain those deviations from ideality. The volume correction is one of those factors. It takes into account the fact that real gases are made from particles that actually have a finite size and are not mathematical points. The factor is about the volume of the particles of the gas not about the vessel the gas is in. The result of the correction is that the pressure of a given amount of gas in a vessel of fixed size is slightly different than would be expected from the ideal gas equation.

• The factor nb is about the particle of gas but the V in PV=nRT is volume of the vessel. So, the correction is on the volume of the vessel (gas)? Mar 11, 2022 at 15:49

One thing that this discussion missed is that the volume occupied by a gas is the volume available for molecular motion. So the volume of a real gas can be given as:

Volume of gas = Volume of the container - Volume not available for molecular motion

While studying ideal gas the volume not available for molecular motion turns out to be zero (due to the assumption of negligible molecular volume).

But for real gasses that is not the case and hence you apply that correction term.

We often frame ideal gas interactions as "particles do not attract or repel each other" and "particles have no volume." Often, it is more useful to call an ideal gas a gas where particles pass through each other. This framing implies both principles. Real gases are any gases where this behavior is not exhibited (i.e. all actual gases).

Let's say that particles in a closed container do not interact except they collide with each other and bounce off elastically like billiard balls. The volume term in the ideal gas law $$V$$ stands for the volume of the entire container; however, if two molecules collide elastically with each other, then for every molecule added to the container, there is a molecule's volume less space for every other molecule to be. So, while we still see the volume of the container as $$V$$, from the molecule's perspective there isn't a full $$V$$ volume worth of space to go, because other molecules are in the way.

For an analogy, if you walk into an empty room, you can navigate every square foot of the room easily. However, if you invite a bunch of friends over, suddenly there is much less space to navigate even though the room did not literally shrink in size—you effectively have less space to move. We'll call the amount of space that the molecule can actually move "effective volume" or $$V_\text{eff}$$.

We expect for every additional molecule we add that the volume $$V$$ stays the same, but the effective volume $$V_\text{eff}$$ decreases a little bit. So, our effective volume should be 1) linearly related to the number of molecules proportional to the molecule's volume and 2) smaller than the original volume. We can use this to make an equation, where $$b$$ depends on the volume of the molecule:

$$V_\text{eff} = V - nb$$

The properties of the ideal gas law come from conditions in the particle's perspective. This is because gas laws are derivable from the behaviors of individual particles through the application of statistical mechanics, a set of mathematical methods that can connect particle states to bulk-system states. In the previous analogy, each person represents a particle. Your behavior in a large room, like how fast you can run around in it, is a function of the room size and of other people in the room. The ideal gas law does not take this latter consideration into account; it assumes each person is a ghost that can phase through any other ghost in the room. In reality, you are limited to the unoccupied space available to you. This is also the case for every other person in the room—your individual behaviors are all linked to the room size and the number of people present.