Real Gases

(©2002, François G. Amar, All rights reserved)

Now for 1 mole of gas, n=1, the ideal gas law predicts where is the molar volume

This equation is only true at very low pressures. As we increase the pressure we can measure the volume and temperature and plot the quantity PV/RT (also known as the compressibility). The following plots show how the compressibility deviates from unity for some common gases.

 

Figure 10.22 (page 380 of BLB): 1 mole of 4 different gases at 300K.

 

Figure 10.23 (page 380 of BLB): 1 mole of nitrogen gas at different temperatures.

This behavior is explained by the fact that real gas molecules have a certain size and can also attract each other.

The van der Waals equation of state is an equation that fits the basic behavior of real gases with only two parameters, a and b:

The first term contains the correction for the size of the molecules, nb is called the "excluded volume" and b has units of L/mole. As b gets larger, P is increased.

The second term on the right-hand-side of this equation accounts for the attraction between gas molecules. a has units of L²-atm/mol². As a gets larger, P is decreased.

For a given gas with definite values of a and b, the van der Waals equation tends toward the ideal gas equation as V and T get large.

The following Table of van der Waals constants is from BLB page 382:

Notice that both a and b increase roughly with molar mass (or number of electrons)

 

Finally let us look at plotting the van der Waals equation for CO₂ for small values of V (V a few times larger than b)

For CO₂, a=3.59 L²-atm/mol² and b=0.0427 L/mol.

It turns out that V must be greater than a certain value for the

equation,   NOT to yield negative values of P (unphysical values!)

.

 Dotted line is ideal gas at 500K, temperatures for solid lines are 500, 400, 350, 304, 290, 275 K

Close up of lower temperature range for CO₂ represented by van der Waals equation.

At 290 K and 275 K, we see that as volume decreases, pressure also decreases!

There is a problem here! What is it?

(PHASE TRANSITION region)

The van der Waals equation does not have enough parameters nor is it complicated enough to predict the true P-T curve in a two phase region. For this we need a an infinite series expansion called a virial expansion.

 

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