Van der Waals Isotherms

Explore excluded volume, attraction and unstable portions of reduced real-gas isotherms.

Van der Waals Isotherms: modeling real gases

The ideal gas law assumes point particles with no volume and no intermolecular forces. The van der Waals equation introduces two parameters to correct for this: one for the excluded volume of the particles, and one for their attractive forces. In reduced units relative to the critical point, the equation becomes universal for all gases.

Worked example: Critical point behavior

The van der Waals equation in reduced variables is given by p = 8T/(3v - 1) - 3/v². Let's calculate the reduced pressure, slope, and compressibility factor at the critical point, where reduced temperature T = 1 and reduced volume v = 1.

For the pressure, substitute T = 1 and v = 1: p = 8(1)/(3(1) - 1) - 3/(1)².
Simplify the terms: p = 8/2 - 3 = 4 - 3 = 1.0. The reduced critical pressure is indeed 1.0.
Next, consider the slope dp/dv = -24T/(3v - 1)² + 6/v³. Substitute T = 1 and v = 1.
Simplify the slope terms: dp/dv = -24(1)/(3(1) - 1)² + 6/(1)³ = -24/4 + 6 = -6 + 6 = 0.0. The isotherm has a flat marginal slope at the critical point.
Finally, the critical compressibility factor is Z = 3pv / (8T). Substituting the values, Z = 3(1)(1) / (8(1)) = 3/8 = 0.375.

Experiment: Traversing the mechanical regimes

Use the Van der Waals Isotherms module to observe the physical behavior and mechanical stability of the mean-field model under different conditions.

Set the Reduced temperature to 1 and Reduced volume to 1. Note the status "Finite model", pressure of 1.0, and the marginal slope of 0.0 indicating the critical point.
Decrease the Reduced temperature to 0.75 and the Reduced volume to 0.75. Notice the pressure dips into negative values (-0.5333) and the slope becomes positive (+2.7022). A positive slope (dp/dv > 0) indicates mechanical instability; an increase in volume would cause an increase in pressure, causing the system to separate into two phases (liquid and gas).
Increase the Reduced temperature to 1.25 and Reduced volume to 2. The slope is now negative (-0.45), indicating a stable single phase. The curve here resembles that of an ideal gas at large volumes.
Set the Reduced volume to 1/3 (any temperature). Observe that the values become unavailable. This represents the excluded volume limit where the gas is compressed so densely that the particles are packed tightly together, and volume cannot decrease further.

Limitations of the mean-field approach

The van der Waals model is a mean-field approximation. The subcritical loops, which predict a positive slope (dp/dv > 0) and even negative absolute pressures, represent unphysical unstable states. In reality, a gas will not follow this loop but will instead undergo a phase transition along a horizontal line of constant pressure (the Maxwell construction), coexisting as both liquid and gas.

Additionally, while the equation captures the qualitative features of real gases and phase transitions, its quantitative accuracy is limited near the critical point and at very high densities. For example, it predicts a universal critical compressibility factor Z = 0.375 for all gases, whereas real gases typically have Z between 0.27 and 0.30.

References

The foundational concepts of non-ideal gas behavior, the excluded volume, intermolecular attractions, and the van der Waals equation can be explored further in standard chemistry texts, such as OpenStax Chemistry 2e, Section 9.6: Non-Ideal Gas Behavior, which supports the model's physical interpretations.

Van der Waals Isotherms