Real Gases and Deviations from Ideality
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You should be comfortable with the ideal gas law, \(PV = nRT\), and understand the kinetic molecular theory assumptions regarding point masses and negligible intermolecular forces.
Why this matters
The ideal gas law is a powerful approximation, but it fails under conditions where molecular size and attraction cannot be ignored.
In earlier lectures, we explored how the kinetic theory leads to the ideal gas law by assuming gas particles are point masses that do not interact except during elastic collisions. While this model works remarkably well for dilute gases at moderate temperatures, it breaks down when gases are compressed or cooled. Under these conditions, the finite volume of the molecules and the attractive forces between them become significant, leading to deviations from ideality. Understanding these deviations is crucial for predicting the behaviour of real substances in industrial processes, atmospheric science, and chemical engineering, where precise pressure and volume relationships are required.
The Breakdown of the Ideal Model
Imagine a container filled with gas molecules. In the ideal model, we pretend these molecules are like tiny, hard billiard balls that take up no space themselves and do not pull on each other. They bounce around freely, and the pressure they exert is solely due to their collisions with the walls of the container. This is a useful simplification because it allows us to calculate pressure, volume, and temperature with a single, simple equation. However, reality is more complex. Real molecules have a definite physical size. If you pack a large number of them into a small volume, the space they occupy becomes a significant fraction of the total container volume. This means the actual space available for the molecules to move around is less than the container volume. Furthermore, real molecules are not completely indifferent to one another. When a molecule approaches the wall, it is being pulled back by the attractive forces of its neighbors. This attraction reduces the force with which it strikes the wall, resulting in a lower measured pressure than the ideal model predicts. These two factors—finite volume and intermolecular attraction—are the primary sources of deviation from ideal behaviour. The ideal gas law assumes both effects are zero, which is only true when the gas is very dilute and the temperature is high enough that the kinetic energy of the molecules overcomes any weak attractive forces.
Deviations in Pressure and Volume
To quantify how real gases deviate from ideality, chemists often use the compressibility factor, \(Z\), defined as \(Z = \frac{PV}{nRT}\). For an ideal gas, \(Z\) is exactly 1 at all temperatures and pressures. For real gases, \(Z\) varies. At low pressures, where the gas is dilute, \(Z\) is close to 1 because the molecules are far apart, minimizing both volume exclusion and intermolecular attraction. As pressure increases, the behaviour depends on the temperature. At high temperatures, the kinetic energy of the molecules is high, and they move so fast that attractive forces have little effect. In this regime, the finite size of the molecules dominates, causing the actual volume to be larger than predicted, so \(Z\) is slightly greater than 1. At lower temperatures, the molecules move more slowly, and intermolecular attractions become significant. These attractions pull molecules together, reducing the pressure exerted on the walls. Consequently, the measured pressure is lower than the ideal prediction, leading to \(Z\) values less than 1. This dip in \(Z\) is a hallmark of real gas behaviour and indicates that the gas is more compressible than an ideal gas under those specific conditions.
The Role of Intermolecular Forces
The strength of intermolecular forces varies significantly between different gases. Gases with larger, more polarizable molecules, such as carbon dioxide or ammonia, exhibit stronger attractive forces than small, non-polar molecules like helium or hydrogen. This difference is reflected in the magnitude of their deviations from ideality. For example, at a given temperature and pressure, ammonia will show a larger deviation from the ideal gas law than helium because the dipole-dipole interactions and London dispersion forces in ammonia are much stronger. The temperature also plays a critical role. As temperature increases, the average kinetic energy of the molecules increases, making it harder for attractive forces to slow them down or pull them together. Therefore, at very high temperatures, even gases with strong intermolecular forces behave more ideally. This is why the ideal gas law is often a good approximation for gases at room temperature and atmospheric pressure, but fails for liquids or high-pressure systems where intermolecular forces are dominant.
Introducing Corrections for Reality
To account for these deviations, we need to modify the ideal gas equation. The first correction addresses the finite size of the molecules. Since the molecules occupy some volume, the free volume available for motion is \(V - nb\), where \(b\) is a constant related to the size of the molecules. The second correction addresses the attractive forces. The pressure measured is lower than the ideal pressure due to these attractions. We can correct for this by adding a term \(a(n/V)^2\) to the measured pressure, where \(a\) is a constant representing the strength of the intermolecular attractions. The term \((n/V)^2\) appears because the attraction depends on the density of the gas squared, as it involves interactions between pairs of molecules. By incorporating these two corrections, we arrive at a more accurate equation of state for real gases. This equation allows us to predict the behaviour of gases under a wider range of conditions, bridging the gap between the simple ideal model and the complex reality of molecular interactions.
The result
The van der Waals equation
Real gas molecules have finite size and attract one another — both directly contradicting the idealisations kinetic-theory-pressure used to derive the ideal gas law. Correcting for finite molecular volume (subtracting an excluded volume \(nb\) from the container volume) and for intermolecular attraction (adding a pressure correction \(a(n/V)^2\) to the measured pressure) gives the van der Waals equation, \(\left(P+a\dfrac{n^2}{V^2}\right)(V-nb)=nRT\), with \(a\) and \(b\) empirically determined constants specific to each gas, unlike the universal \(R\).
A common misconception is that the constants \(a\) and \(b\) are universal for all gases. In reality, they are specific to each substance and must be determined experimentally. Using the wrong values for a particular gas will lead to significant errors in calculating pressure or volume, especially under non-ideal conditions. Always verify the specific constants for the gas in question before applying the corrected equation. C
Worked example
Problem. Calculate the pressure exerted by 2.00 mol of carbon dioxide gas in a 10.0 L container at 300 K, assuming it behaves as an ideal gas. Then, calculate the pressure using the van der Waals equation, given that for carbon dioxide, \(a = 3.59\ \mathrm{L^2\,atm\,mol^{-2}}\) and \(b = 0.0427\ \mathrm{L\,mol^{-1}}\). Use \(R = 0.08206\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}\).
- Calculate the ideal pressure using \(P_{\text{ideal}} = \frac{nRT}{V}\). Substituting the values: \(P_{\text{ideal}} = \frac{2.00 \times 0.08206 \times 300}{10.0} = 4.9236\ \mathrm{atm}\).
- Calculate the volume correction term \(nb\). \(nb = 2.00 \times 0.0427 = 0.0854\ \mathrm{L}\).
- Determine the effective volume \(V_{\text{eff}} = V - nb = 10.0 - 0.0854 = 9.9146\ \mathrm{L}\).
- Calculate the pressure correction term \(a(n/V)^2\). \((n/V) = 2.00/10.0 = 0.200\ \mathrm{mol\,L^{-1}}\). So, \(a(n/V)^2 = 3.59 \times (0.200)^2 = 3.59 \times 0.0400 = 0.1436\ \mathrm{atm}\).
- Solve for the real pressure \(P\) using the rearranged van der Waals equation: \(P = \frac{nRT}{V-nb} - a\left(\frac{n}{V}\right)^2\). \(P = \frac{2.00 \times 0.08206 \times 300}{9.9146} - 0.1436\). \(P = \frac{49.236}{9.9146} - 0.1436 = 4.9660 - 0.1436 = 4.8224\ \mathrm{atm}\).
Units check. The term \(\frac{nRT}{V-nb}\) has units of \(\frac{\mathrm{mol \cdot L \cdot atm \cdot mol^{-1} \cdot K^{-1} \cdot K}}{\mathrm{L}} = \mathrm{atm}\), and the correction term \(a(n/V)^2\) has units of \(\mathrm{L^2 \cdot atm \cdot mol^{-2} \cdot (mol \cdot L^{-1})^2} = \mathrm{atm}\), so the subtraction is dimensionally consistent.
Three quick tests
- Why is the compressibility factor \(Z\) less than 1 for many gases at low temperatures and moderate pressures?
Answer
At low temperatures, intermolecular attractive forces are significant relative to kinetic energy, reducing the pressure exerted on the container walls compared to the ideal prediction. - What does the constant \(b\) in the van der Waals equation represent?
Answer
The constant \(b\) represents the excluded volume per mole of gas molecules, accounting for the finite size of the molecules. - If a gas has a very large value of \(a\), what can you infer about its intermolecular forces?
Answer
A large value of \(a\) indicates that the gas molecules experience strong intermolecular attractive forces.
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Lecture notes checked line by line by the editor; every number in the worked example was recomputed.