# Real Gases

Learning Goals

You will understand why real gases do not behave ideally, and learn of another equation relating state variables that is an improvement over the ideal gas law.

Synopsis

Solve the IG EOS for volume and now consider what this equation says happens to the volume as T goes to zero. It says that the volume goes to zero! Now what does the equation say happens to the volume as P goes to infinity? It says that the volume goes to zero! These are not realistic predictions, which is why this EOS is called the Ideal Gas EOS. We can fix this misbehavior very simply. Just add a "b" to the right side of the equation for volume and you have

V = nRT/P + b .....................eq 11

Now as T goes to zero and/or P goes to infinity, the volume goes to "b" which is a parameter to represent the size of the gas molecules. We will need a different "b" for each gas. But we are not through yet. Solve equation 11 for the pressure and you will have

P = nRT/(V- b) ........................eq 12

Remember that pressure is force per unit area and for gases in a container, the pressure is the force per unit area that the molecules make in collisions with the container. The limit behaviors of P with T and V are satisfactory in equation 12. However let's think about what happens if there are attractive forces between the individual gas molecules in the container. Put a rubber band around both hands and imagine that one of your hands is traveling toward an imaginary wall in the opposite direction of the other hand. What is the effect of the rubber band on the force with which your hand hits the imaginary wall? That is right, the attractive force of the rubber band reduces the force that the hand hits the wall - assuming that the moving hand moves with the same effort with and without the rubber band. So what is the rubber band between the hands simulating on the molecular level? Each hand represents a different gas molecule, and the rubber band represents the intermolecular attractive force between the different gas molecules. So to account for this intermolecular force we must subtract something from the right hand of equation 12:

P = nRT/(V - b) - a/(V/n)2 = P = RT/(Vn - b/n) - a/(Vn2)..........eq 13

We can also write this equation as the following if the units of "b" include (per mole). Check the algebra yourself.

P = RT/(Vn - b) - a/(Vn2).......eq 14

where Vn is the molar volume (V/n).

Here "a" is a constant accounting for the intermolecular forces between the gas molecules and will be different for each gas. Equation 11 is the van der Waals EOS. Table 12.3 gives some values of the "a" and "b" constants for some gases. Be careful with the units of the constants to know whether to use equation 13 or 14 (check to see the units of "b"). This equation then accounts for deviations from ideality caused by the size effect and the intermolecular force effect. There are many much more accurate and complicated EOS, but they all just account for the two causes of deviations from ideality.

Review Questions

1. Prove that equation 13 above is the same as Equation 12.11 in your text.
2. Using the information from the last section, explain why Dalton's law is an approximation.
3. The Boyle temperature is the temperature at which a gas behaves close to ideally over a rather broad pressure range. The Boyle temperature according to the VDW EOS is given by Tb = a/(Rb) where "a" and "b" are the VDW constants for a gas. The experimental Boyle temperature for N2 is 332 K and for H2 is 116.4 K.

a.) Calculate Tb = a/(Rb) for N2 and H2.

b.) Calculate the percent error of the calculated Tb for each gas.

c.) Comment on what the different percent errors might mean concerning the behaviors of the gases. Consider the electronic structures of each gas.

4. Go to this site for a streaming video presentation on real gases (you will need the RealVideo plugin to view this movie).
5. We made the following comment above "The limit behaviors of P with T and V are satisfactory in equation 12." Show that indeed equation 12 does yield what you would expect for P as T approach infinity and zero, and as V approaches zero and infinity.

Web Author: Dr. Leon L. Combs