2.4.1 Behavior of Real Gases

Experimentally all gases become ideal in the limit of low density. At low density the average distance between gas molecules becomes large and what forces do exist between real gas molecules become attenuated. This is because non-electrostatic forces^2.8 between molecules fall off very rapidly with distance.^2.9

As a practical matter, instead of using the limit of low density, we often use the limit of low pressure, which is almost the same thing. Also sometimes used is the limit of large volume, which for fixed number of moles also means low density.

A useful quantity in discussing ideality of gases is the compression factor Z. This is defined as:

$$Z = \frac{pV}{nRT}$$ (2.4.2)

Of course, this is exactly 1 for an ideal gas. Indeed it is found experimentally for all gases that

$$\lim_{p \rightarrow 0 } Z = 1$$ (2.4.3)

and, what is essentially the same thing:

$$\lim_{1/V \rightarrow 0 } Z = 1$$ (2.4.4)

A plot of Z vs p for real gases shows that Z can be less than 1 in some ranges and greater than 1 in others. Generally Z becomes larger than 1 as the pressure is increased. This is because under those conditions the gas molecules are forced to be close enough together that their repulsive forces dominate. At lower pressures the attractive forces predominate, producing the typical dip and rise of these plots.

Figure 2.2: Compression factor Z vs pressure

Examination of p-V plots for real gases also show non-ideal behavior as has been hinted before. At high enough temperatures the isotherms of real gases closely resemble the hyperbolas one gets when plotting p against V for an ideal gas. At lower temperatures a "wiggle" develops in the isotherm. At a still lower temperature a zero-width flat spot occurs. This point is called the critical point, and the temperature at which it occurs the critical temperature. The corresponding coordinates are the critical pressure and the critical volume.

The zero-width flat spot is clearly an inflection point. Not only is the derivative zero here, but the second derivative is zero as well, which is the definition of an inflection point.

Figure 2.3: Isotherms of a Real Gas

At this critical temperature and above, isotherms are not only continuous, but have continuous derivatives as well.

But below the critical isotherm, isotherms have flat line portions and, while the isotherm remains continuous, their slopes are no longer continuous.

The meaning of the critical point is that below this temperature the gas can exist both as gas and as liquid. Indeed, that is what exists in the region of flat lines at temperatures below the critical isotherm. Above the critical isotherm a liquid phase never forms.

This seems contradictory, but it is not. As one approaches the critical point from the gas (high volume) side of the isotherm, the density of the gas gets greater and greater. And as one approaches from the low volume side, the density gets less and less. At the critical point the two densities are equal; indeed a plot of density vs. volume would show a continuous curve at the critical temperature and above.

Figure 2.4: Density vs Temperature Near the Critical Point

Below the critical temperature gas and liquid are distinct and have distinctly different densities. Of course, as one approaches the critical temperature from below, the gas and liquid densities come closer and closer to each other, finally becoming identical at the critical point.