Real Gases

 

Contents of this page   Introduction Effects of intermolecular forces The van der Waals equation of state Condensation and the critical point Supercritical fluids Summary

The "ideal gas laws" as we know them do a remarkably good job of describing the behavior of a huge number chemically diverse substances as they exist in the gaseous state under ordinary environmental conditions, roughly around 1 atm pressure and a temperature of 300 K. This is especially so when we consider that some of the basic tenets of the ideal gas model have to be abandoned in order to explain such properties as

• the average distance between collisions (the molecules really do take up space)
• the viscosity of a gas flowing through a pipe (the molecules do get temporarily "stuck" on the pipe surface, and are therefore affected by intermolecular attractive forces.)

Even so, many of the common laws such as Boyle's and Charles' continue to describe these gases quite well even under conditions where these phenomena are evident.

Under ordinary environmental condtions (moderate pressures and above 0°C), the isotherms of substances we normally think of as gases don't appear to differ very greatly from the hyperbolic form (PV/RT) = constant.

... but over a wider range of conditions, things begin to get more complicated. Thus Isopentane, shown here, behaves in a reasonably ideal manner above 210 K, but below this temperature the isotherms become somewhat distorted, and at 185K and below they cease to be continuous, showing peculiar horizontal segments in which reducing the volume does not change the pressure. It turns out that real gases eventually begin to follow their own unique equations of state, and ultimately even cease to be gases.

 

In this unit we will see why this occurs, what the consequences are, and how we might modify the ideal gas equation of state to extend its usefulness over a greater range of temperatures and pressures.

 

 

 

 

According to Boyle's law, the product PV is a constant at any given temperature, so a plot of PV as a function of the pressure of an ideal gas yields a horizontal straight line. This implies that any increase in the pressure of the gas is exactly counteracted by a decrease in the volume as the molecules are crowded closer together. But we know that the molecules themselves are finite objects having volumes of their own, and this must place a lower limit on the volume into which they can be squeezed. So we must reformulate the ideal gas equation of state as a relation that is true only in the limiting case of zero pressure:

So what happens when a real gas is subjected to a very high pressure? The outcome varies with both the molar mass of the gas and its temperature, but in general we can see the the effects of both repulsive and attractive intermolecular forces:

Repulsive forces: As a gas is compressed, the individual molecules begin to get in each other's way, giving rise to a very strong repulsive force acts to oppose any further volume decrease. We would therefore expect the PV vs P line to curve upward at high pressures, a nd this is in fact what is observed for all gases at sufficiently high pressures.

Attractive forces: At very close distances, all molecules repel each other as their electron clouds come into contact. At greater distances, however, brief statistical fluctuations in the distribution these electron clouds give rise to a universal attractive force between all molecules. The more electrons in the molecule (and thus the greater the molecular weight), the greater is this attractive force. As long as the energy of thermal motion dominates this attractive force, the substance remains in the gaseous state, but at sufficiently low temperatures the attractions dominate and the substance condenses to a liquid or solid.

The universal attractive force described above is known as the dispersion, or London force. There may also be additional (and usually stronger) attractive forces related to charge imbalance in the molecule or to hydrogen bonding. These various attractive forces are often referred to collectively as van der Waals forces.

A plot of PV/RT as a function of pressure is a very sensitive indicator of deviations from ideal behavior, since such a plot is just a horizonal line for an ideal gas. The two illustrations below show how these plots vary with the nature of the gas, and with temperature.

Intermolecular attractions, which generally increase with molecular weight, cause the PV product to decrease as higher pressures bring the molecules closer together and thus within the range of these attractive forces; the effect is to cause the volume to decrease more rapidly than it otherwise would. But as the molecules begin to intrude on each others' territory, repulsive forces always eventually win out.

 

The temperature makes a big difference! At higher temperatures, increased thermal motions overcome the effects of intermolecular attractions which normally dominate at lower pressures. So all gases behave more ideally at higher temperatures. For any gas, there is a special temperature (the Boyle temperature) at which attractive and repulsive forces exactly balance each other at zero pressure. As you can see in this plot for methane, this balance does remain as the pressure is incrased.

 

 

The effect of intermolecular attractions on a PV-vs.-P plot would be to hold the molecules slightly closer together, so that the volume would decrease more rapidly than the pressure increases. The resulting curve would dip downward as the pressure increases, and this dip would be greater at lower temperatures and for heavier molecules. At higher pressures, however, the stronger repulsive forces would begin to dominate, and the curve will eventually bend upward as before. The effects of intermolecular interactions are most evident at low temperatures and high pressures; that is, at high densities.

 

 

 

 

 

How might we modify the ideal gas equation of state to take into account the effects of intermolecular interactions?

The first and most well known answer to this question was offered by the Dutch scientist J.D. van der Waals (1837-1923) in 1873.

van der Waals recognized that the molecules themselves take up space that subtracts from the volume of the container, so that the “volume of the gas” V in the ideal gas equation should be replaced by the term (V-b) which is called the excluded volume, typically of the order of 20-100 cm³ mol^–1. The excluded volume (indicated by the gray circle below) surrounding any moleculedefines the closest possible approach of any two molecules during collision.

 

The intermolecular attractive forces act to slightly diminish the frequency and intensity of encounters between the molecules and the walls of the container; the effect is the same as if the pressure of the gas were slightly higher than it actually is. This imaginary increase is called the internal pressure, and we can write

P_effective = P_ideal + P_internal

Thus we should replace the P in the ideal gas equation by

P_ideal = P_effective – P_internal

Since the attractions are between pairs of molecules, the total attractive force is proportional to the square of the number of molecules per volume of space, and thus for a fixed number of molecules such as one mole, the force is inversely proportional to the square of the volume of the gas; the smaller the volume, the closer are the molecules and the greater the attractions between pairs (hence the square term) of molecules. The pressure that goes into the corrected ideal gas equation is

in which the constants a and b depend respectively on the magnitudes of the attractive and repulsive forces in a particular gas. The complete van der Waals equation of state thus becomes

 

Although you do not have to memorize this equation, you are expected to understand it and to explain the significance of the terms it contains. You should also understand that the van der Waals constants a and b must be determined empirically for every gas. This can be done by plotting the P-V behavior of the gas and adjusting the values of a and b until the van der Waals equation results in an identical plot. The constant a is related in a simple way to the molecular radius; thus the determination of a constitutes an indirect measurment of an important microscopic quantity.

Below: van der Waals constants for some gases

Substance	molar mass g	a (L2-atm mole–2)	b (L mol–1)
hydrogen  H2	2	0.244	.0266
helium  He	4	0.034	.0237
methane CH4	16	2.25	.0428
water  H2O	18	5.46	.0305
nitrogen N2	28	1.39	.0391
carbon dioxide  CO2	44	3.59	.0427
carbon tetrachloride  CCl4	154	20.4	.1383

 

The van der Waals equation is only one of many equations of state for real gases. More elaborate equations are required to describe the behavior of gases over wider pressure ranges. These generally take account of higher-order nonlinear attractive forces, and require the use of more empirical constants. Although we will make no use of them in this course, they are widely employed in chemical engineering work in which the behavior of gases at high pressures must be accurately predicted.

More on the van der Waals equation and the significance of the constants a and b
from the Illinois State U and Purdue U Chemistry sites.

 

Condensation and the critical point

 

 

 

The most striking feature of real gases is that they cease to remain gases as the temperature is lowered and the pressure is increased. The plot below illustrates this behavior; as the volume is decreased, the lower-temperature isotherms suddenly change into straight lines. Under these conditions, the pressure remains constant as the volume is reduced. This can only mean that the gas is “disappearing" as we squeeze the system down to a smaller volume. In its place, we obtain a new state of matter, the liquid. In the shaded region, two phases, liquid, and gas, are simultaneously present. Finally, at very small volume all the gas has disappeared and only the liquid phase remains. At this point the isotherms bend strongly upward, reflecting our common experience that a liquid is practically incompressible.

PVT surface of a real gas To better understand this plot, look at the isotherm labeled 1. As the gas is compressed from 1 to 2, the pressure rises in much the same way as Boyle's law predicts. Compression beyond 2, however, does not cause any rise in the pressure. What happens instead is that some of the gas condenses to a liquid. At 3, the substance is entirely in its liquid state. The very steep rise to 4 corresponds to our ordinary experience that liquids have very low compressibilities. The range of volumes possible for the liquid diminishes as the critical temperature is approached.

 

 

Critical constants of some gases Note especially • The very low critical pressure and temperature of helium, reflecting the very small intermolecular attractions of this atom. • Tc of the noble gas elements increases with atomic number • Hydrogen gas cannot be liquified above 33 K; this poses a major difficulty in the use of hydrogen as an automotive fuel; storage as a high-pressure gas requires heavy steel containers which add greatly to its effective weight-per-joule of energy storage. • The properties of carbon dioxide (particularly its use as a supercritical fluid) are described above. • The high Tc of H2O is another manifestation of its "anomalous" properties relating to hydrogen-bonding.

gas Pc atm Vc cm3 mol–1 Tc K He 2.3 58 5.2 Ne 27 42 44 Ar 48 75 150 Xe 58 119 290 H2 13 65 33 O2 50 78 155 N2 34 90 126 CO2 73 94 302 H2O 218 59 647 NH3 111 72 405 CH4 46 99 191

 

Supercritical fluids

The maximum temperature at which the two phases can coexist is called the critical temperature. The set of (P,V,T) corresponding to this condition is known as the critical point. Liquid and gas can coexist only within the regions indicated by the wedge-shaped cross section on the left and the shaded area in the diagram above. An important consequence of this is that a liquid phase cannot exist at temperatures above the critical point.

The critical point of a gas is defined by its critical temperature, pressure and volume, denoted by T_c, P_c, and V_c.

The critical temperature of carbon dioxide is 31°C, so you can tell whether the temperature is higher or lower than this by shaking a CO₂ fire extinguisher; on a warm day, you will not hear any liquid sloshing around inside. The critical temperature of water is 374K, and that of hydrogen is only 33K.

Critical behavior of carbon dioxide At temperatures below 31°C (the critical temperature), CO2 acts somewhat like an ideal gas at higher pressures (1). Below this temperature, subjecting the gas to a higher pressure eventually causes condensation to begin. Thus at 21°C, at a pressure of about 62 atm, the volume can be reduced from 200 cm3 to about 55 cm3 without any further rise in the pressure. Instead of the gas being compressed, it is replaced with the far more compact liquid as the gas is essentially being "squeezed" into its liquid phase. After all of the gas has disappeared (2), the pressure rises very rapidly because now all that remains is an almost incompressible liquid. The isotherm 3 that passes through the critical point is called the critical isotherm. Above this isotherm (4), CO2 exists only as a supercritical fluid.

 

One intriguing consequence of the very limited bounds of the liquid state is that you could start with a gas at large volume and low temperature, raise the temperature, reduce the volume, and then reduce the temperature so as to arrive at the liquid region at the lower left, without ever passing through the two-phase region, and thus without undergoing condensation!

Supercritical fluids

The supercritical state of matter, as the fluid above the critical point is often called, possesses the flow properties of a gas and the solvent properties of a liquid. The density of a supercritical fluid can be changed over a wide range by adusting the pressure; this, in turn, changes its solubility, which can thus be optimized for a particular application. The picture at the right shows a commercial laboratory device used for carrying out chemical reactions under supercritical conditions.

Supercritical carbon dioxide is widely used to dissolve the caffeine out of coffee beans and as a dry-cleaning solvent. Supercritical water has recently attracted interest as a medium for chemically decomposing dangerous environmental pollutants such as PCBs.

 

More on the solvation properties of supercritical fluids