The Compression Factor and Corresponding States
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You should be comfortable with the ideal gas law, the definition of molar volume, and the basic structure of the van der Waals equation as a correction to ideal behaviour.
Why this matters
The compression factor quantifies how far a real gas deviates from ideality, providing a universal language for comparing different substances.
In the previous lecture, we established that real gases possess finite molecular volume and intermolecular attractions, leading to the van der Waals equation. However, solving that cubic equation for every condition is cumbersome. The compression factor, or compressibility factor, offers a dimensionless metric that normalises gas behaviour against the ideal standard. By plotting this factor against reduced pressure and temperature, we can observe that all gases converge to similar behaviour at high temperatures and low pressures. This principle of corresponding states allows us to predict the properties of one gas based on the data of another, simplifying the thermodynamic analysis of industrial processes where precise gas behaviour is critical.
What is the Compression Factor?
Imagine you have a container with a fixed volume and a fixed amount of gas. If the gas were ideal, its pressure would be exactly determined by the ideal gas law. In reality, the pressure you measure might be slightly higher or lower than this prediction. The compression factor, denoted as \( Z \), is simply the ratio of the actual molar volume of the gas to the molar volume it would have if it were ideal, at the same temperature and pressure. Alternatively, it is the ratio of the actual pressure to the ideal pressure. If \( Z \) equals 1, the gas behaves ideally. If \( Z \) is less than 1, the gas is more compressible than an ideal gas, typically because attractive forces are pulling molecules together, reducing the volume. If \( Z \) is greater than 1, the gas is less compressible, usually because the finite size of the molecules dominates, effectively reducing the available space. This single number encapsulates the net effect of all intermolecular interactions and molecular size, making it a powerful diagnostic tool for gas behaviour.
Defining the Compression Factor
The compression factor \( Z \) is defined as the ratio of the molar volume \( V_m \) of a real gas to the molar volume \( V_{m,\text{ideal}} \) of an ideal gas at the same temperature \( T \) and pressure \( P \). For an ideal gas, \( V_{m,\text{ideal}} = \frac{RT}{P} \). Therefore, the definition is \( Z = \frac{V_m}{V_{m,\text{ideal}}} = \frac{V_m P}{RT} \). Since \( V_m = \frac{V}{n} \), we can also write \( Z = \frac{PV}{nRT} \). This equation shows that \( Z \) is a dimensionless quantity. When \( Z = 1 \), the gas obeys the ideal gas law. Deviations from unity indicate non-ideality. For example, if \( Z = 0.9 \), the real gas occupies 10% less volume than predicted by the ideal gas law at the same \( P \) and \( T \). This is often observed at moderate pressures where intermolecular attractions are significant. Conversely, if \( Z = 1.1 \), the gas occupies 10% more volume, which occurs at high pressures where the excluded volume effect of finite molecular size becomes dominant. The value of \( Z \) is not constant; it varies with pressure and temperature, tracing a curve that reflects the balance between attractive and repulsive forces.
Corresponding States and Reduced Properties
The principle of corresponding states suggests that all gases exhibit similar behaviour when their properties are expressed in terms of reduced variables. These reduced variables are defined by scaling the actual properties by the critical constants of the substance. The reduced temperature \( T_r \) is the ratio of the absolute temperature \( T \) to the critical temperature \( T_c \), so \( T_r = \frac{T}{T_c} \). Similarly, the reduced pressure \( P_r \) is the ratio of the pressure \( P \) to the critical pressure \( P_c \), so \( P_r = \frac{P}{P_c} \). The reduced volume \( V_r \) is the ratio of the molar volume \( V_m \) to the critical molar volume \( V_c \), so \( V_r = \frac{V_m}{V_c} \). The critical point represents the unique state where the distinction between liquid and gas phases disappears. By using these reduced variables, we can plot \( Z \) as a function of \( P_r \) and \( T_r \). Remarkably, for many gases, these plots collapse onto a single universal curve. This means that if we know the critical constants of a gas, we can predict its compression factor at any condition by referring to this universal chart. This universality arises because the intermolecular forces and molecular sizes scale proportionally with the critical constants, allowing for a generalised description of gas behaviour.
Interpreting the Z vs P_r Curve
When we plot the compression factor \( Z \) against the reduced pressure \( P_r \) for a fixed reduced temperature \( T_r \), we observe characteristic trends. At very low pressures, \( Z \) approaches 1 for all gases, confirming that all gases behave ideally under dilute conditions. As pressure increases, the curve initially dips below 1, reaching a minimum, and then rises above 1. The region where \( Z < 1 \) indicates that attractive forces dominate, causing the gas to be more compressible than ideal. The minimum value of \( Z \) and the pressure at which it occurs depend on the temperature. At higher temperatures, the dip is shallower and occurs at higher pressures. At lower temperatures, the dip is deeper and occurs at lower pressures. If the temperature is below the critical temperature, the curve may show a region where \( Z \) is significantly less than 1, corresponding to the liquid phase. Above the critical temperature, the curve remains continuous, reflecting the supercritical fluid state. The point where \( Z = 1 \) at high pressure indicates that the repulsive forces due to finite molecular size have become the dominant factor, making the gas less compressible than ideal. Understanding these trends allows chemists to estimate the non-ideality of gases without performing complex calculations.
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 \( Z \) is always less than 1 for real gases. In fact, \( Z \) can be greater than 1, especially at high pressures or high temperatures, where the finite size of molecules (repulsive forces) dominates over attractive forces. Always consider the balance between attraction and repulsion when interpreting \( Z \). C
Worked example
Problem. At 300 K and 10.0 atm, the measured molar volume of carbon dioxide is 2.34 L mol-1. Calculate its compression factor \(Z\), and its reduced temperature and pressure (\(T_c = 304.1\ \mathrm{K}\), \(P_c = 72.8\ \mathrm{atm}\)). Use \(R = 0.08206\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}\).
- Ideal molar volume at these conditions: \(V_{m,\mathrm{ideal}} = RT/P = 0.08206 \times 300 / 10.0 = 2.462\ \mathrm{L\,mol^{-1}}\).
- Compression factor: \(Z = \dfrac{PV_m}{RT} = \dfrac{10.0 \times 2.34}{24.618} = \dfrac{23.4}{24.618}\).
- So \(Z = 0.951\): the gas occupies about 5% less volume than an ideal gas would, because attractions dominate at this pressure.
- Reduced variables: \(T_r = 300/304.1 = 0.987\) and \(P_r = 10.0/72.8 = 0.137\).
- On a generalised compressibility chart, \(T_r \approx 1\) and \(P_r \approx 0.14\) indeed give \(Z\) a little below 1, consistent with the measurement: any gas at these reduced conditions would show a similar \(Z\).
Units check. \(\mathrm{atm} \times \mathrm{L\,mol^{-1}}\) divided by \(\mathrm{L\,atm\,mol^{-1}\,K^{-1}} \times \mathrm{K}\) is dimensionless, as are the reduced variables.
Three quick tests
- What does a compression factor \( Z \) of 1.0 indicate about a gas?
Answer
It indicates that the gas behaves ideally, meaning its molar volume is exactly as predicted by the ideal gas law. - How is the reduced temperature \( T_r \) defined?
Answer
It is the ratio of the absolute temperature \( T \) to the critical temperature \( T_c \), i.e., \( T_r = T/T_c \). - Why do all gases have \( Z \) approaching 1 at low pressures?
Answer
At low pressures, the distance between molecules is large, so intermolecular forces are negligible, and the finite size of molecules is insignificant compared to the container volume.
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Lecture notes checked line by line by the editor; every number in the worked example was recomputed.