# 10.4 Graham's Law of Effusion and Real Gases

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###### In this lesson you will learn:

-An explanation of Graham's Law of Effusion

-How to perform calculations involving Graham's Law of Effusion

## What is Effusion?

Before we can talk about Graham's Law of Effusion it would be helpful to know what effusion is. Simply put, effusion is the passing of a gas through a narrow slit or hole. An example that is easy to visualize often occurs when you get a flat tire. Let's say you run over a nail and it punctures your tire. When you remove the nail from your tire it will begin to deflate as the gases inside the tire exit through the puncture hole...that's effusion.

## Graham's Law of Effusion

Graham's Law of Effusion gives the mathematical relationship between the rates of effusion of two gases based upon their molecular weights.

Ultimately, the rate at which a gas effuses is inversely proportional to the square root of its molecular weight; the lighter the gas the faster it effuses. This makes intuitive sense. One of the tenets of Kinetic Molecular Theory is that two gases at the same temperature have the same average kinetic energy. But this does NOT mean that the two gases have the same rms speed. With the same average kinetic energy the lighter gas will have be moving faster and have a higher rms speed, and therefore pass through a narrow hole or slit faster. Take for example a balloon filled with an equal number of moles of H_{2} and O_{2} gases having a small hole.

Being in the same balloon for any significant amount of time will result in H_{2} and O_{2} being at the same temperature and having the same average kinetic energy. But H_{2} is significantly lighter than O_{2} (16 times lighter) and therefore will have a higher rms speed and effuse out of the balloon faster. As Graham's law relates the rate of effusion to the square root of the molecular weight, a molecule that is 16 times lighter will have a rate of effusion that is 4 times greater.

## van der Waals Equation for Real Gases

Two of the major assumptions that are made for ideal gases are that there are no attractive forces between molecules and that the molecules themselves have negligible volume. van der Waals attempted to correct for the inaccuracies in both of these assumptions in what is now referred to as the van der Waals equation of state.

You'll note that this has similarities with the Ideal Gas Law but with the addition of two terms into the equation. These two terms contain the van der Waals constants, 'a' and 'b,' which are determined empirically for a gas and are published in tables for a variety of gases.

The first additional term to appear in the van der Waals equation accounts for the attractive intermolecular forces that all gases have which are more significant at lower temperatures and for gases with larger attractive forces.

'a' is the first van der Waals constant. The greater the attractive forces between atoms/molecules the greater this correction will need to be leading to a greater magnitude of this van der Waals constant. A gas that has lower attractive forces is said to behave more ideally and would be expected to have a smaller value for this constant.

The second additional term to appear in the van der Waals equation accounts for the volume occupied by the atoms/molecules themselves. While the ideal gas law assumes that this volume is negligible, this is less and less true as pressure increases and the molecules are pressed closer and closer together.

'b' is the second van der Waals constant. The greater the volume of the atoms/molecules the greater this correction will need to be leading to a greater magnitude of this van der Waals constant. A gas with a smaller volume is said to behave more ideally and would be expected to have a smaller value for this constant.

More accurate equations of state such as the virial equation of state are more commonly used today than the van der Waals equation. But the van der Waals equation was one of the earliest attempts to model real gases and generally gives more accurate calculations of pressure, volume, or temperature than the ideal gas law.