2.8 – Real/Non-Ideal Gas Behaviours

Thus far, the ideal gas law, PV = nRT, has been applied to a variety of different types of problems, ranging from reaction stoichiometry and empirical and molecular formula problems to determining the density and molar mass of a gas. As mentioned in the previous modules of this chapter, however, the behavior of a gas is often non-ideal, meaning that the observed relationships between its pressure, volume, and temperature are not accurately described by the gas laws. In this section, the reasons for these deviations from ideal gas behavior are considered.

One way in which the accuracy of PV = nRT can be judged is by comparing the actual volume of 1 mole of gas (its molar volume, V_m) to the molar volume of an ideal gas at the same temperature and pressure. This ratio is called the compressibility factor (Z) with:

Equation 2.8.1 Compressibility Factor

Ideal gas behavior is therefore indicated when this ratio is equal to 1, and any deviation from 1 is an indication of non-ideal behavior. Figure 2.8.1. shows plots of Z over a large pressure range for several common gases.

Figure 2.8.1. A graph of the compressibility factor (Z) vs. pressure shows that gases can exhibit significant deviations from the behavior predicted by the ideal gas law. To examine this behaviour closer to the origin, check out page 3 of the following lecture notes.

As is apparent from Figure 2.8.1., the ideal gas law does not describe gas behavior well at relatively high pressures. To determine why this is, consider the differences between real gas properties and what is expected of a hypothetical ideal gas.

Particles of a hypothetical ideal gas have no significant volume and do not attract or repel each other. In general, real gases approximate this behavior at relatively low pressures and high temperatures. However, at high pressures, the particles of a gas are crowded closer together, and the amount of empty space between the particles is reduced. At these higher pressures, the volume of the gas particles themselves becomes appreciable relative to the total volume occupied by the gas (Figure 2.8.2.). The gas therefore becomes less compressible at these high pressures, and although its volume continues to decrease with increasing pressure, this decrease is not proportional as predicted by Boyle’s law.

Figure 2.8.2. Raising the pressure of a gas increases the fraction of its volume that is occupied by the gas particles and makes the gas less compressible. Here, the increase in pressure is achieved by either (b) decreasing the volume of the container or (c) increasing the amount of gas in the container. In both cases, deviations from ideal behaviour may appear.

At relatively low pressures, gas particles have practically no attraction for one another because they are (on average) so far apart, and they tend to behave like particles of an ideal gas. At higher pressures, however, the force of attraction is also no longer insignificant. This force pulls the particles a little closer together, slightly decreasing the pressure (if the volume is constant) or decreasing the volume (if the pressure is constant) (Figure 2.8.3.). This change is more pronounced at low temperatures because the particles have lower E_k relative to the attractive forces, and so they are less effective in overcoming these attractions after colliding with one another.

Figure 2.8.3. (a) Attractions between gas particles serve to decrease the gas volume at constant pressure compared to an ideal gas whose particles experience no attractive forces. (b) These attractive forces will decrease the force of collisions between the particles and container walls, therefore reducing the pressure exerted compared to an ideal gas.

There are several different equations that better approximate gas behavior than does the ideal gas law. The first, and simplest, of these was developed by the Dutch scientist Johannes van der Waals in 1879. The van der Waals equation improves upon the ideal gas law by adding two terms: one correction factor to account for the volume of the gas particles and another for the attractive forces between them.

Equation 2.8.2 Van der Waals

The constant a corresponds to the strength of the attraction between particles of a particular gas, and the constant b corresponds to the size of the particles of a particular gas. The correction to the pressure term in the ideal gas law is , and the correction to the volume is nb. Note that when V is relatively large and n is relatively small, both of these correction terms become negligible, and the van der Waals equation reduces to the ideal gas law, PV = nRT. Such a condition corresponds to a gas in which a relatively low number of particles is occupying a relatively large volume, that is, a gas at a relatively low pressure. Experimental values for the van der Waals constants of some common gases are given in Table 2.8.1.

Table 2.8.1 Van der Waals constants. For more van der Waals constants, follow the following link.

At low pressures, the correction for intermolecular attraction, a, is more important than the one for molecular volume, b. At high pressures and small volumes, the correction for the volume of the particles becomes important because the particles themselves are incompressible and constitute an appreciable fraction of the total volume. At some intermediate pressure, the two corrections have opposing influences and the gas appears to follow the relationship given by PV = nRT over a small range of pressures. This behavior is reflected by the “dips” in several of the compressibility curves shown in Figure 2.8.4. The attractive force between particles initially makes the gas more compressible than an ideal gas, as pressure is raised (Z decreases with increasing P). At very high pressures, the gas becomes less compressible (Z increases with P), as the gas particles begin to occupy an increasingly significant fraction of the total gas volume.

Figure 2.8.4. A graph of the compressibility factor (Z) vs. pressure shows that gases can exhibit significant deviations from the behavior predicted by the ideal gas law. a) shows where intermolecular attraction (a) most impacts the ideal gas equation, and b) shows where particle volume (b) most impacts the ideal gas equation

Strictly speaking, the ideal gas equation functions well when intermolecular attractions between gas particles are negligible and the gas particles themselves do not occupy an appreciable part of the whole volume. These criteria are satisfied under conditions of low pressure and high temperature. Under such conditions, the gas is said to behave ideally, and deviations from the gas laws are small enough that they may be disregarded—this is, however, very often not the case.

Example 2.8.1 – Comparison of Ideal Gas Law and van der Waals Equation

A 4.25-L flask contains 3.46 mol CO₂ at 229 °C. Calculate the pressure of this sample of CO₂:

(a) from the ideal gas law

(b) from the van der Waals equation

(c) Explain the reason(s) for the difference.

Solution

(a) From the ideal gas law:

(b) From the van der Waals equation:

This finally yields P = 32.4 atm.

(c) This is not very different from the value from the ideal gas law because the pressure is not very high and the temperature is not very low. The value is somewhat different because CO₂ particles do have some volume and attractions between particles, and the ideal gas law assumes they do not have volume or attractions.

Check your Learning 2.8.1 – Comparison of Ideal Gas Law and van der Waals

Equation

A 560-mL flask contains 21.3 g N₂ at 145 °C. Calculate the pressure of N₂:

(a) from the ideal gas law

(b) from the van der Waals equation

(c) Explain the reason(s) for the difference.

Answer

(a) 46.562 atm; (b) 46.594 atm; (c) The van der Waals equation takes into account the volume of the gas particles themselves as well as intermolecular attractions.

Questions

★ Questions

1. Graphs showing the behavior of several different gases follow. Which of these gases exhibit behavior significantly different from that expected for ideal gases?

Explain why the plot of PV for CO₂ differs from that of an ideal gas.

3. Under which of the following sets of conditions does a real gas behave most like an ideal gas, and for which conditions is a real gas expected to deviate from ideal behavior? Explain.

a. high pressure, small volume

b. high temperature, low pressure

c. low temperature, high pressure

4. Describe the factors responsible for the deviation of the behavior of real gases from that of an ideal gas.

5. For which of the following gases should the correction for the molecular volume be largest: CO, CO₂, H₂, He, NH₃, SF₆?

6. A 0.245-L flask contains 0.467 mol CO2 at 159 °C. Calculate the pressure:

a. using the ideal gas law

b. using the van der Waals equation

c. Explain the reason for the difference.

d. Identify which correction (that for P or V) is dominant and why.

★★ Questions

7. Answer the following questions:

a. If XX behaved as an ideal gas, what would its graph of Z vs. P look like?

b. For most of this chapter, we performed calculations treating gases as ideal. Was this justified?

c. What is the effect of the volume of gas molecules on Z? Under what conditions is this effect small? When is it large? Explain using an appropriate diagram.

d. What is the effect of intermolecular attractions on the value of Z? Under what conditions is this effect small? When is it large? Explain using an appropriate diagram.

e. In general, under what temperature conditions would you expect Z to have the largest deviations from the Z for an ideal gas?

Answers

1. Gases C, E, and F

2. CO₂ interacts intermolecularly with other molecules, and occupies a volume in space. Hence the CO₂ molecules repel other molecules and create pressure through collisions, leading to a non linear relationship between PV and P.

3. The gas behavior most like an ideal gas will occur under the conditions in (b). Molecules have high speeds and move through greater distances between collisions; they also have shorter contact times and interactions are less likely. Deviations occur with the conditions described in (a) and (c). Under conditions of (a), some gases may liquefy. Under conditions of (c), most gases will liquefy.

4. An ideal gas is assumed to have no volume or intermolecular interaction. Molecules that portray real gas behaviour show attraction between other molecules resulting in a deviation in ideal gas behaviour. Real gases only show ideal behaviour at high temperatures and low pressures.

5. SF₆

6. (a) 66.2 atm, (b) 60.5 atm, (c) van der Waals accounts for non ideal gas factors (repulsive forces and volume occupation) that the ideal gas law does not account for, (d) The factor for pressure correction used in the van der Waals is dominant in low pressure cases. At higher volumes, the pressure is higher as well.

7. (a) A straight horizontal line at 1.0; (b) When real gases are at low pressures and high temperatures they behave close enough to ideal gases that they are approximated as such, however, in some cases, we see that at a high pressure and temperature, the ideal gas approximation breaks down and is significantly different from the pressure calculated by the ideal gas equation (c) The greater the compressibility, the more the volume matters. At low pressures, the correction factor for intermolecular attractions is more significant, and the effect of the volume of the gas molecules on Z would be a small lowering compressibility. At higher pressures, the effect of the volume of the gas molecules themselves on Z would increase compressibility (d) Once again, at low pressures, the effect of intermolecular attractions on Z would be more important than the correction factor for the volume of the gas molecules themselves, though perhaps still small. At higher pressures and low temperatures, the effect of intermolecular attractions would be larger. (e) low temperatures