AP Chemistry: Real Gases and Van der Waals Equation

The ideal gas law is a powerful tool, but it’s a model built on assumptions that sometimes break down. Understanding where and why real gases deviate from ideal behavior is crucial for accurate predictions in chemistry, engineering, and even respiratory medicine.

The Limits of the Ideal Gas Model

The ideal gas law, $PV=nRT$, relies on two key assumptions: that gas particles have zero volume and that there are no intermolecular forces of attraction or repulsion between them. These assumptions work remarkably well under standard conditions—low pressure and high temperature—where gas particles are far apart and moving quickly. However, under conditions of high pressure or low temperature, these assumptions fail, and gases exhibit non-ideal or real behavior.

At high pressure, gas particles are forced closer together. Their finite, non-zero volume becomes a significant fraction of the container's total volume. This means the space available for the particles to move in is less than the measured volume of the container ($V$). At low temperature, the particles move much more slowly. Their lower kinetic energy makes them more susceptible to the weak attractive forces that exist between all molecules, known as London dispersion forces or, more broadly, Van der Waals forces. This attraction pulls particles slightly together, effectively reducing the force of their collisions with the container walls.

The Two Corrections: Attraction and Volume

The Van der Waals equation intelligently corrects the ideal gas law by accounting for these two factors. The equation is written as: $$\left(P+\frac{a{n}^2}{V^2}\right)(V-nb)=nRT$$

Let's break down each correction term.

1. The Pressure Correction ($+\frac{a{n}^2}{V^2}$): This term accounts for intermolecular attraction. The variable $a$ is a constant specific to each gas that quantifies the strength of its intermolecular forces. Gases with stronger attractions, like $N{H}_3$ or $H_{2}O$(vapor), have larger $a$ values. Attraction between particles reduces the observed pressure ($P$). Think of it as if the particles are slightly "sticky"; they pull inward on each other, so they don't hit the container walls as hard. To get the pressure we would observe if there were no attractions, we must add the correction term $\frac{a{n}^2}{V^2}$ to the measured pressure $P$. The term depends on $n^2/V^2$ because attractions depend on the concentration of particles—how close they are on average.

2. The Volume Correction ($V-nb$): This term accounts for the finite volume of the gas particles themselves. The variable $b$ is a constant specific to each gas that relates to the size of its molecules. Larger molecules, like xenon, have larger $b$ values. The term $nb$ represents the total volume occupied by the gas molecules. Therefore, $V-nb$ is the actual free volume available for the gas particles to move in. This corrected volume is what should be used in the gas law calculation, not the full container volume $V$.

Predicting Significant Deviations

Not all gases deviate from ideality to the same degree, and not all conditions produce significant deviations. You can predict when deviations will be most pronounced by considering the properties of the gas and the experimental conditions.

First, consider the gas itself. Gases with large, polar molecules (large $a$ and $b$ constants) will deviate more strongly from ideal behavior even at moderate conditions. For example, $S{O}_2$ or $C{H}_{3}OH$ will show greater deviation than helium or neon, which have very small, nonpolar atoms with weak intermolecular forces.

Second, examine the conditions. High pressure increases the effect of the volume correction (the $b$ term). When particles are squeezed together, their physical size matters more. Low temperature increases the effect of the attraction correction (the $a$ term). At low kinetic energies, attractive forces have a greater relative impact on particle motion. The most extreme deviations occur under simultaneous conditions of very high pressure and very low temperature, where both correction factors are large.

Applying the Van der Waals Equation: A Worked Example

Suppose you have 1.00 mol of nitrogen gas ($N_2$) confined in a 22.4 L container at 0°C (273 K). For $N_2$, $a=1.39{\text{ L}}^2\cdot \text{atm}/{\text{mol}}^2$ and $b=0.0391\text{ L/mol}$.

Step 1: Calculate the pressure using the ideal gas law. $$P_{ideal}=\frac{nRT}{V}=\frac{(1.00\text{ mol})(0.08206\text{ Lcdotpatm/molcdotpK})(273\text{ K})}{22.4\text{ L}}\approx 1.00\text{ atm}$$

Step 2: Calculate the corrected pressure using the Van der Waals equation. We rearrange to solve for $P$: $$P=\frac{nRT}{V-nb}-\frac{a{n}^2}{V^2}$$ Plug in the values: $$P=\frac{(1.00)(0.08206)(273)}{22.4-(1.00\times 0.0391)}-\frac{(1.39)(1.00{)}^2}{(22.4{)}^2}$$ $$P=\frac{22.40}{22.3609}-\frac{1.39}{501.76}$$ $$P\approx 1.0017-0.00277\approx 0.9989\text{ atm}$$

Interpretation: Under these near-standard conditions, the predicted pressure from the Van der Waals equation (0.9989 atm) is very close to the ideal gas prediction (1.00 atm). The small difference shows that nitrogen behaves nearly ideally here. The attractive force correction slightly lowers the pressure (the -0.00277 term), while the volume correction slightly increases it (by reducing the effective volume in the denominator). In this case, they almost cancel out.

Common Pitfalls

1. Misapplying the corrections to conditions. A common mistake is to assume the Van der Waals corrections are always significant. As the example shows, under many conditions the ideal gas law is sufficient. Always assess the conditions (pressure and temperature) and the nature of the gas before deciding which model to use.

2. Confusing the meaning of $a$ and $b$. Remember that $a$ relates to attraction between particles. A larger $a$ means stronger intermolecular forces. The constant $b$ relates to the size or volume of the particles themselves. A larger $b$ means bigger molecules. Mixing these up will lead to incorrect predictions about how a gas will deviate.

3. Forgetting that deviations are not absolute. Gases do not simply "switch" from ideal to real at a specific point. Deviation is a continuum. The ideal gas law becomes progressively less accurate as pressure increases or temperature decreases. The Van der Waals equation provides a continuous mathematical correction across this entire range.

4. Algebraic errors when solving. The Van der Waals equation is more complex to rearrange and solve than $PV=nRT$. Pay meticulous attention to order of operations, especially when calculating the correction terms. For pressure calculations, remember you subtract the attraction term from the modified pressure term.

Summary

• The ideal gas law assumes particles have no volume and no intermolecular forces. It fails under high pressure, where particle volume matters, and low temperature, where intermolecular attractions matter.
• The Van der Waals equation corrects the ideal gas law by adding a pressure term ($+a{n}^2/V^2$) to account for intermolecular attraction and subtracting a volume term ($nb$) from the container volume to account for particle size.
• The constants $a$ and $b$ are specific to each gas; large $a$ indicates strong intermolecular forces, and large $b$ indicates large molecular size.
• Significant deviations from ideal behavior are most pronounced for gases with large, polar molecules under conditions of simultaneously high pressure and low temperature.
• For many common gases under standard conditions, the ideal gas law remains an excellent and simpler approximation, as the Van der Waals corrections are often very small.