The Van der Waals Equation: Derivation and Interpretation

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Hello! Welcome to the first lesson in our second module.

In the previous module, we developed a general thermodynamic language to describe phase transitions, culminating in the introduction of critical exponents and the remarkable concept of universality. We saw that simple mean-field theories, like the one for the Ising model and the Van der Waals model for a fluid, surprisingly belong to the same universality class, even though they describe completely different physical systems.

Now, we will transition from that general framework to our first detailed case study: the Van der Waals Model of a Liquid-Gas Transition. Our goal today is to derive its famous equation of state and to develop a deep physical understanding of its constituent parts. This model, while approximate, is a cornerstone of statistical mechanics because it's one of the simplest theories to capture the essential physics of a liquid-gas phase transition and a critical point.

The Physical Basis: Two Corrections to the Ideal Gas Law

The ideal gas law, $P V = N k_{B} T$ , is built on two key simplifying assumptions: gas particles are sizeless points, and they do not interact with each other. Johannes Diderik van der Waals proposed two corrections to these assumptions to better describe real gases.

Introduction to the Van der Waals Gas Model and Intermolecular Forces — This diagram illustrates the physical basis of the Van der Waals model. It connects the macroscopic behavior of a real gas to the underlying intermolecular forces, often modeled by the Lennard-Jones potential, which features strong short-range repulsion and weak long-range attraction.

These two features of the intermolecular potential lead to the two famous corrections:

Finite Molecular Size (Repulsion): Real molecules are not points; they have a finite size and cannot overlap. This strong short-range repulsion means that the volume available for a molecule to move in is not the total volume $V$ of the container, but something smaller. This is the "excluded volume" effect.
Intermolecular Attraction: At larger separations, molecules attract each other. A molecule deep inside the gas feels these attractions from all directions, resulting in no net force. However, a molecule near the container wall is pulled back by the molecules in the bulk. This reduces the force of its impact on the wall, leading to a measured pressure that is lower than what it would be without attractions.

To see how these physical ideas are mathematically incorporated into the ideal gas law, let's watch a short video that explains each correction in detail.

The van der Waals Equation | Physical Chemistry I | 014

This video provides a clear, step-by-step explanation of how the ideal gas law is modified to account for the excluded volume and intermolecular attractions, leading to the Van der Waals equation.

Please watch from 0:54 to 8:47. Pay close attention to: The 'excluded volume' correction and the physical meaning of the parameter 'b'. The 'pressure correction' due to attractive forces and the justification for the parameter 'a' and its dependence on the square of the density (n/V^2).

As the video explains, these two corrections modify the ideal gas law $P_{i d e a l} V_{i d e a l} = N k_{B} T$ as follows:

The available volume is reduced: $V_{i d e a l} \to V - N b$ , where $b$ is the excluded volume per particle.
The effective pressure is increased: $P_{i d e a l} \to P + a (N / V)^{2}$ , where $P$ is the measured pressure and $a (N / V)^{2}$ is the correction term accounting for the internal pressure from attractive forces.

Combining these gives the celebrated Van der Waals equation of state:
$(P + a \frac{N ^{2}}{V ^{2}}) (V - N b) = N k_{B} T$

The parameters $a$ and $b$ are empirical constants specific to each gas. Given your background in condensed matter physics, we can connect these phenomenological parameters to a more fundamental picture of intermolecular forces, such as the Lennard-Jones potential.

Thermodynamic Properties of the van der Waals Fluid

For a deeper physical interpretation, this document connects the parameters 'a' and 'b' to the microscopic Lennard-Jones potential, which you have likely encountered before. This provides a more fundamental justification for the form of the Van der Waals corrections.

Please read Section III, 'van der Waals INTERMOLECULAR INTERACTION PARAMETERS' (pages 4-5). Focus on how the repulsive part of the Lennard-Jones potential justifies the hard-core excluded volume 'b', and how the attractive part, through a mean-field average over the fluid, gives rise to the parameter 'a'. You don't need to memorize the integral derivation for 'a', but grasp the concept behind it.

Test your understanding!

Consider two real gases, Helium (He) and Carbon Dioxide (CO₂). Based on their molecular properties, which gas do you expect to have a larger 'a' parameter? Which would have a larger 'b' parameter? Justify your reasoning.

Show answer

Parameter 'b' (Excluded Volume): The CO₂ molecule is significantly larger than the He atom. Therefore, CO₂ will have a larger excluded volume, meaning $b_{CO_{2}} > b_{He}$ .
Parameter 'a' (Attraction Strength): The attractive forces between neutral, non-polar molecules are primarily London dispersion forces, which increase with the size and polarizability of the electron cloud. CO₂ is a much larger and more polarizable molecule than Helium. Thus, the attractive forces between CO₂ molecules are stronger, meaning $a_{CO_{2}} > a_{He}$ .

A More Rigorous Derivation from Statistical Mechanics

The heuristic argument is physically intuitive, but we can put it on a more solid theoretical footing by deriving the equation from the canonical partition function, $Q (N, V, T)$ . This approach directly demonstrates how the microscopic assumptions translate into a macroscopic equation of state.

Recall that for a monatomic ideal gas of $N$ particles, the partition function is:
$Q_{I G} (N, V, T) = \frac{1}{N ! Λ ^{3 N}} \int d^{3} r_{1} \dots d^{3} r_{N} = \frac{1}{N ! Λ ^{3 N}} V^{N}$
where $Λ = h / 2 πm k_{B} T$ is the thermal de Broglie wavelength.

To construct the Van der Waals partition function, we modify this expression to include our two physical corrections.

The Statistical Mechanical Derivation of the van der Waals Equation of State

Now, let's derive the Van der Waals equation from first principles. This document provides a concise statistical mechanical derivation by modifying the canonical partition function of an ideal gas.

Read the first page of this document, up to and including Equation (8). Follow the logic step-by-step: Note the ideal gas partition function (Eq. 1). See how the volume integral is modified from V^N to (V-Nb)^N to account for the excluded volume 'b' (Eq. 3). This is an approximation, but it captures the essential physics. Understand how the mean-field attractive energy E_{vdw} = -aN^2/V (Eq. 4) is incorporated as a global energy shift, leading to an additional Boltzmann factor \exp(-E_{vdw}/k_BT) (Eq. 5). Follow the final step where the pressure is calculated from the Helmholtz free energy (A = -k_B T \ln Q) using the thermodynamic relation P = -(\partial A / \partial V)_{T,N} to arrive at the Van der Waals equation (Eq. 8).

This derivation beautifully formalizes our physical intuition. By modifying the volume term and adding a mean-field potential energy term to the partition function, we directly obtain the Van der Waals equation of state.

Let's write it down one more time. The equation relates the pressure $P$ , volume $V$ , and temperature $T$ for $N$ particles:
$(P + a \frac{N ^{2}}{V ^{2}}) (V - N b) = N k_{B} T$
Here, the parameters $a$ and $b$ are understood to be on a per-particle basis, consistent with our statistical mechanics derivation. In chemistry contexts, you will often see this equation written for $n$ moles of gas, using the molar volume $V_{m} = V / n$ and the gas constant $R = N_{A} k_{B}$ , with rescaled parameters $a^{'} = N_{A}^{2} a$ and $b^{'} = N_{A} b$ . For our purposes, we will stick to the per-particle formulation.

Conclusion

In this lesson, we have formally introduced the Van der Waals model, which will be our primary tool for understanding the liquid-gas transition in the next few lessons.

Key Takeaways:

The Van der Waals equation modifies the ideal gas law to account for two features of real molecules: finite volume (short-range repulsion) and intermolecular attraction (long-range attraction).
The parameter 'b' represents the excluded volume per particle, reducing the effective volume available for motion to $(V - N b)$ .
The parameter 'a' quantifies the strength of the mean-field attractive forces, which reduce the pressure at the container wall. The correction term is $a (N / V)^{2}$ .
We derived the equation of state both heuristically and more rigorously by modifying the canonical partition function and using the fundamental thermodynamic relation $P = - (\partial A / \partial V)_{T, N}$ .

Preview of the Next Lesson:

Now that we have derived the equation, the next step is to analyze its predictions. In the next lesson, we will plot the Van der Waals isotherms (P-V curves at constant T) and discover that below a certain critical temperature, they contain an intriguing, seemingly unphysical region. We will learn that this feature is the model's prediction of a first-order phase transition between a liquid and a gas.