The van der Waals Equation
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You should be comfortable with the ideal gas law \(PV = nRT\) and understand the two key assumptions of kinetic theory: that gas molecules occupy negligible volume and exert no intermolecular forces on each other.
Why this matters
The ideal gas law is a powerful approximation, but it fails under conditions where molecular size and attraction become significant.
In the previous lecture, we observed that real gases deviate from ideality, particularly at high pressures and low temperatures. These deviations arise because the simple kinetic model ignores the physical reality that molecules have finite dimensions and interact via attractive forces. The van der Waals equation provides a modified form of the ideal gas law that accounts for these two factors, offering a more accurate description of real gas behaviour. By introducing two empirical constants, \(a\) and \(b\), we can correct for the volume occupied by the molecules themselves and the pressure reduction caused by intermolecular attraction. This correction is essential for predicting the behaviour of gases in industrial processes, atmospheric science, and chemical engineering, where precision is required.
The Physical Intuition
To understand why the ideal gas law fails, imagine a container filled with gas molecules. In the ideal model, we treat these molecules as point particles with no volume and no interaction. However, in reality, every molecule occupies a certain amount of space. When the gas is compressed, the total volume available for the molecules to move in is less than the volume of the container because the molecules themselves take up space. This is the concept of excluded volume. If you have a box of tennis balls, the empty space available for movement is the box volume minus the volume of the balls. Similarly, for a gas, the effective volume is the container volume minus the volume occupied by the molecules. This correction is represented by the constant \(b\), which is related to the size of the molecules. A larger molecule, like a heavy hydrocarbon, will have a larger \(b\) value than a small molecule like helium.
The second issue is intermolecular attraction. Molecules in a gas are not completely independent; they attract each other slightly due to van der Waals forces. When a molecule is about to collide with the wall of the container, it is pulled back by the other molecules behind it. This reduces the force of the impact on the wall, leading to a lower measured pressure than what would be expected for an ideal gas. This effect is more pronounced when the gas is dense, as there are more molecules nearby to exert the attractive force. The constant \(a\) quantifies this attractive interaction. Stronger intermolecular forces, such as those in polar molecules or large molecules with significant surface area, result in a larger \(a\) value. By accounting for both the finite size of the molecules and the attractive forces between them, the van der Waals equation provides a more realistic picture of gas behaviour, bridging the gap between the ideal model and the complex reality of real substances.
Correcting for Molecular Volume
The first correction to the ideal gas law addresses the finite size of gas molecules. In the ideal gas model, the volume \(V\) in the equation \(PV = nRT\) is assumed to be the entire volume of the container. However, this ignores the fact that the molecules themselves occupy space. The volume actually available for the molecules to move in is the container volume minus the volume occupied by the molecules. This excluded volume is proportional to the number of moles \(n\), and the proportionality constant is \(b\). Therefore, the effective volume is \(V - nb\). The constant \(b\) is specific to each gas and depends on the size of the molecules. For example, the \(b\) value for argon is approximately \(3.22 \times 10^{-5} \ \mathrm{m^3\,mol^{-1}}\), while for a larger molecule like carbon dioxide, it is \(4.27 \times 10^{-5} \ \mathrm{m^3\,mol^{-1}}\). This correction becomes significant at high pressures, where the gas is compressed and the volume occupied by the molecules is a non-negligible fraction of the total volume. At low pressures, the volume of the molecules is small compared to the container volume, and the correction \(nb\) is negligible, reducing the equation to the ideal gas law.
To see the impact of this correction, consider a gas at high pressure. If the container volume is \(V\) and the number of moles is \(n\), the ideal gas law predicts a pressure \(P_{\text{ideal}} = \frac{nRT}{V}\). However, because the effective volume is smaller, the actual pressure should be higher than the ideal prediction if we only considered volume exclusion. The van der Waals equation modifies the volume term to \(V - nb\), which increases the pressure term \(\frac{nRT}{V - nb}\) compared to \(\frac{nRT}{V}\). This reflects the fact that the molecules are confined to a smaller space, leading to more frequent collisions with the walls and thus a higher pressure. This correction is crucial for accurately modelling the behaviour of gases under compression, such as in high-pressure reactors or storage tanks.
Correcting for Intermolecular Attraction
The second correction accounts for the attractive forces between gas molecules. In the ideal gas model, molecules are assumed to be non-interacting, but in reality, they experience weak attractive forces, primarily van der Waals forces. These forces reduce the pressure exerted by the gas on the container walls. When a molecule approaches the wall, it is attracted by the other molecules in the bulk of the gas, which reduces the force of its collision with the wall. This effect is more significant when the gas is dense, as there are more molecules nearby to exert the attractive force. The pressure correction is proportional to the square of the molar density \(\left(\frac{n}{V}\right)^2\), and the proportionality constant is \(a\). Therefore, the measured pressure \(P\) is less than the ideal pressure by an amount \(a\left(\frac{n}{V}\right)^2\). The constant \(a\) is specific to each gas and depends on the strength of the intermolecular forces. For example, the \(a\) value for nitrogen is approximately \(0.137 \ \mathrm{Pa\,m^6\,mol^{-2}}\), while for a polar molecule like water, it is much larger, around \(0.554 \ \mathrm{Pa\,m^6\,mol^{-2}}\). This correction becomes significant at low temperatures and high pressures, where the molecules are closer together and the attractive forces have a greater impact.
To understand the impact of this correction, consider a gas at low temperature. At low temperatures, the kinetic energy of the molecules is lower, and the attractive forces have a more significant effect on their motion. The pressure correction \(a\left(\frac{n}{V}\right)^2\) reduces the effective pressure in the van der Waals equation. This means that the actual pressure is lower than the ideal prediction, reflecting the fact that the molecules are being held back by intermolecular attractions. This correction is crucial for accurately modelling the behaviour of gases near their condensation points, where the attractive forces are strong enough to cause liquefaction. By accounting for both the finite size of the molecules and the attractive forces between them, the van der Waals equation provides a more realistic picture of gas behaviour, bridging the gap between the ideal model and the complex reality of real substances.
The Combined Equation
Combining the two corrections, we obtain the van der Waals equation. The pressure term is corrected by adding \(a\left(\frac{n}{V}\right)^2\) to the measured pressure, and the volume term is corrected by subtracting \(nb\) from the container volume. The resulting equation is \(\left(P + a\frac{n^2}{V^2}\right)(V - nb) = nRT\). This equation reduces to the ideal gas law \(PV = nRT\) when \(a\) and \(b\) are zero, which is the case for an ideal gas. The constants \(a\) and \(b\) are empirically determined for each gas and are tabulated in standard references. For example, for carbon dioxide, \(a = 0.364 \ \mathrm{Pa\,m^6\,mol^{-2}}\) (3.64 L2 bar mol-2) and \(b = 4.27 \times 10^{-5} \ \mathrm{m^3\,mol^{-1}}\). These values allow us to calculate the pressure, volume, or temperature of a real gas with greater accuracy than the ideal gas law. The van der Waals equation is particularly useful for predicting the behaviour of gases under conditions where they deviate significantly from ideality, such as at high pressures and low temperatures. It is also used to estimate the critical constants of a gas, which are important for understanding the phase behaviour of substances.
The van der Waals equation is a significant improvement over the ideal gas law, but it is still an approximation. It assumes that the attractive forces are uniform and that the excluded volume is a simple fraction of the total volume. More complex equations of state, such as the Redlich-Kwong or Peng-Robinson equations, provide even more accurate descriptions of real gas behaviour. However, the van der Waals equation remains a valuable tool for understanding the fundamental principles of gas behaviour and for making quick estimates in situations where high precision is not required. It provides a clear physical picture of the factors that cause deviations from ideality and serves as a foundation for more advanced models of gas behaviour.
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 van der Waals constants \(a\) and \(b\) are universal constants, like the gas constant \(R\). In reality, \(a\) and \(b\) are specific to each gas and depend on the size and intermolecular forces of the molecules. Using the wrong values for \(a\) and \(b\) can lead to significant errors in the calculated pressure or volume. Always ensure that you use the correct constants for the specific gas you are working with. C
Worked example
Problem. 1.00 mol of carbon dioxide is held at 300 K in a 0.500 L (\(5.00 \times 10^{-4}\ \mathrm{m^3}\)) cylinder. Calculate the pressure from the ideal gas law and from the van der Waals equation, with \(a = 0.364\ \mathrm{Pa\,m^6\,mol^{-2}}\) and \(b = 4.27 \times 10^{-5}\ \mathrm{m^3\,mol^{-1}}\).
- Ideal: \(P_{\mathrm{ideal}} = \dfrac{nRT}{V} = \dfrac{1.00 \times 8.314 \times 300}{5.00 \times 10^{-4}} = 4.99 \times 10^{6}\ \mathrm{Pa}\) (about 49 atm).
- Free volume: \(V - nb = 5.00 \times 10^{-4} - 4.27 \times 10^{-5} = 4.573 \times 10^{-4}\ \mathrm{m^3}\).
- Repulsive (size) term: \(\dfrac{nRT}{V - nb} = \dfrac{2494.2}{4.573 \times 10^{-4}} = 5.454 \times 10^{6}\ \mathrm{Pa}\), higher than ideal.
- Attractive term: \(a\dfrac{n^2}{V^2} = \dfrac{0.364 \times 1.00^2}{(5.00 \times 10^{-4})^2} = 1.456 \times 10^{6}\ \mathrm{Pa}\).
- van der Waals pressure: \(P = 5.454 \times 10^{6} - 1.456 \times 10^{6} = 4.00 \times 10^{6}\ \mathrm{Pa}\), which is 20% below the ideal value: at this density attraction outweighs molecular size (\(Z \approx 0.80\)).
Units check. \(\mathrm{J\,m^{-3}} = \mathrm{Pa}\) for the first term and \(\mathrm{Pa\,m^6\,mol^{-2}} \times \mathrm{mol^2\,m^{-6}} = \mathrm{Pa}\) for the second, so they can be subtracted.
Three quick tests
- What is the physical meaning of the constant \(b\) in the van der Waals equation?
Answer
It represents the excluded volume per mole of gas molecules, accounting for their finite size. - How does the constant \(a\) affect the pressure predicted by the van der Waals equation?
Answer
It reduces the pressure by accounting for the attractive forces between molecules, which lower the force of collisions with the container walls. - Under what conditions does the van der Waals equation reduce to the ideal gas law?
Answer
When the constants \(a\) and \(b\) are zero, which corresponds to an ideal gas with no intermolecular forces and negligible molecular volume.
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Lecture notes checked line by line by the editor; every number in the worked example was recomputed.