10.4 Real Gases & the Van der Waals Equation
PRE-HEALTH PREP
Real Gases vs Ideal Gas Law
The Van der Waals Equation
General Chemistry 10.4
Real gases deviate from the predictions of the ideal gas law due to intermolecular attractive forces and the finite volume of gas molecules—behaviors quantified by the compressibility factor (PV/RT). The Van der Waals equation corrects for these real gas deviations by introducing two empirical constants: a for intermolecular attractions and b for molecular volume. Worked examples with methane demonstrate that while the ideal gas law and Van der Waals equation yield similar results at low pressures, the predictions diverge dramatically under high-pressure conditions.
The Compressibility Factor: How Real Gases Deviate from Ideal Behavior
To understand how real gases deviate from ideal behavior, consider a graph with pressure on the x-axis and the ratio PV/RT on the y-axis. This ratio is known as the compressibility factor (also called the compression factor). For a single mole of gas, this ratio equals exactly one under ideal conditions.
For a perfectly ideal gas, the compressibility factor remains constant at one regardless of pressure—producing a flat horizontal line across the graph. Real gases, however, tell a more interesting story: their compressibility factors rise and fall depending on pressure and the strength of their intermolecular forces.
Remember: Ideal gases don't actually exist. They are a theoretical model that works well under certain conditions but breaks down at extreme pressures and temperatures. Real gases are what you encounter in the laboratory—and why we need equations like Van der Waals to describe them accurately.
When Real Gases Fall Below the Ideal Line: Intermolecular Attractive Forces
At low pressures, most real gases hover near the ideal line. However, as pressure increases, gases like N₂ and CH₄ dip below the ideal value. What causes this downward deviation from the ideal gas law?
The answer lies in intermolecular attractive forces. These attractions pull molecules toward the interior of the container, reducing how often—and how forcefully—they collide with the walls. Fewer and weaker wall collisions mean the real gas generates less pressure than an ideal gas would under identical conditions. With a lower-than-expected pressure, the compressibility factor drops below one.
Example: Attractive Forces and Real Gas Behavior
Imagine gas molecules heading toward the container wall. In an ideal gas, nothing slows them down. In a real gas, neighboring molecules exert attractive forces that slightly pull each molecule back toward the center. The result: softer collisions with the walls and a measured pressure lower than the ideal gas law prediction.
When Real Gases Rise Above the Ideal Line: Molecular Volume and Repulsive Forces
As pressure increases further, molecules are forced closer and closer together. Eventually, there is so little empty space that the volume occupied by the molecules themselves becomes significant. The ideal gas law treats molecules as having zero volume—but real molecules are not infinitely small.
Once molecules are packed tightly enough, repulsive forces between them become dominant. The molecules resist being compressed further, effectively pushing back against each other. This drives the compressibility factor above one. In the case of H₂, the curve rises above the ideal line even at relatively modest pressures because hydrogen molecules have very weak attractive forces, so the molecular volume effect dominates quickly.
Key Pattern: On a compressibility factor graph, values below the ideal line indicate intermolecular attractive forces dominate real gas behavior. Values above the ideal line indicate repulsive forces (or significant molecular volume) dominate.
| Region on Graph | Compressibility Factor | Dominant Effect | Physical Cause |
|---|---|---|---|
| Below ideal line | PV/RT < 1 | Attractive forces | Molecules pulled inward, fewer wall collisions |
| On ideal line | PV/RT = 1 | Ideal behavior | Forces balanced or negligible |
| Above ideal line | PV/RT > 1 | Repulsive forces / molecular volume | Molecular volume significant, molecules resist compression |
The Van der Waals Equation of State
The Van der Waals Equation and Its 2 Corrections to the Ideal Gas Law
The Van der Waals equation provides a mathematical framework for describing real gas behavior. While more sophisticated equations of state exist—including computer-fitted models with variable coefficients—the Van der Waals equation stands out for its elegant, physically intuitive approach to correcting the Ideal Gas Law for real gas deviations.
The equation takes the form:
Van der Waals Equation of State:
(P + an²/V²)(V − nb) = nRT
If you remove the an²/V² correction and the nb correction, you recover the familiar Ideal Gas Law: PV = nRT. Van der Waals inserted these two correction terms to account for the two types of non-ideal behavior in real gases: intermolecular attractive forces and finite molecular volume.
What Each Correction Term Does:
+an2 / V2 corrects for attractive forces
-nb corrects for repulsive forces (as molecules do have volume)
The "a" Constant: Correcting for Intermolecular Attractive Forces
The term an²/V² corrects for intermolecular attractive forces in real gases. The constant a is one of the two Van der Waals constants, and its value is unique to each gas. When a equals zero, this correction term vanishes entirely—meaning no attractive forces are accounted for, and the gas behaves ideally in that respect.
A gas with a small value of a has weak intermolecular attractions and therefore behaves more like an ideal gas. Larger values of a indicate stronger attractions and greater deviation from the ideal gas law.
The "b" Constant: Correcting for Molecular Volume
The term nb corrects for the finite volume of real gas molecules. The constant b is the second Van der Waals constant, also unique to each gas. When b equals zero, this term disappears—meaning the molecules are treated as point particles with no volume, just as in the ideal model.
Both Van der Waals constants are determined empirically through laboratory measurements and can be found in reference tables for common gases. When asked which gas from a table behaves most ideally, look for the gas with both the smallest a and the smallest b.
Exam Tip: If given a table of Van der Waals constants and asked which real gas behaves most ideally, choose the gas with both the smallest a value (weakest intermolecular attractive forces) and the smallest b value (smallest molecular volume).
| Constant | Corrects For | Effect When Large | When Zero |
|---|---|---|---|
| a | Intermolecular attractive forces | Strong attractions, large deviation from ideal gas law | No attractive correction; ideal behavior in this respect |
| b | Molecular volume | Large molecules, significant excluded volume | No volume correction; ideal behavior in this respect |
| Van der Waals Constants | a | b |
|---|---|---|
| H2 | 0.244 | 0.0266 |
| N2 | 1.39 | 0.0391 |
| CH4 | 2.25 | 0.0428 |
Solving for Pressure Using the Van der Waals Equation
If you need to perform a calculation with the Van der Waals equation, you will most likely be solving for pressure. Solving for volume requires tackling a cubic equation—not a quadratic, a cubic—which is extremely difficult to do by hand and requires a numerical solver.
Rearranging the Van der Waals equation to isolate pressure gives:
Van der Waals Equation Solved for Pressure:
P = nRT / (V − nb) − an² / V²
You need to know: the number of moles (n), the temperature (T), the volume (V), and the Van der Waals constants (a and b) for the specific real gas in question.
Common Mistake: Attempting to solve the Van der Waals equation for volume by hand. The rearranged equation is cubic in V, making analytical solutions impractical without a computer. Stick to solving for pressure unless explicitly given a numerical solver.
Worked Example: Real vs Ideal Gas Pressure at Low Pressure (V = 8.2 L)
Consider 1 mole of CH₄ (methane) at a temperature of 100 K and a volume of 8.2 L. Calculate the pressure using both the Ideal Gas Law and the Van der Waals equation to compare real gas vs ideal gas predictions.
Van der Waals constants for CH₄ (methane): a = 2.25 atm·L²/mol², b = 0.0428 L/mol
Ideal Gas Law Calculation
Ideal Gas Law: P = nRT / V
P = (1 mol)(0.08206 L·atm/mol·K)(100 K) / (8.2 L)
P = 8.206 / 8.2
P = 1.00 atm
Van der Waals Calculation
Van der Waals Equation: P = nRT / (V − nb) − an² / V²
P = (1)(0.08206)(100) / (8.2 − (1)(0.0428)) − (2.25)(1)² / (8.2)²
P = 8.206 / 8.1572 − 2.25 / 67.24
P = 1.006 − 0.0335
P = 0.973 atm
At this relatively low pressure of approximately 1 atm, the Ideal Gas Law and Van der Waals equation produce similar results. The Van der Waals prediction is expected to be slightly more accurate for this real gas, but the ideal approximation works reasonably well under these low-pressure conditions.
Worked Example: Real vs Ideal Gas Pressure at High Pressure (V = 0.5 L)
Now consider the same system—1 mole of CH₄ at 100 K—but compressed into only 0.5 L. This much smaller volume produces significantly higher pressure, and the real gas deviations from ideal behavior become dramatic.
Ideal Gas Law Calculation
Ideal Gas Law: P = nRT / V
P = (1)(0.08206)(100) / (0.5)
P = 16.4 atm
Van der Waals Calculation
Van der Waals Equation: P = nRT / (V − nb) − an² / V²
P = (1)(0.08206)(100) / (0.5 − 0.0428) − (2.25)(1)² / (0.5)²
P = 8.206 / 0.4572 − 2.25 / 0.25
P = 17.95 − 9.00
P = 8.95 atm
The difference is now enormous: the Ideal Gas Law predicts 16.4 atm while the Van der Waals equation predicts only 8.95 atm. Under these elevated-pressure conditions, the real gas behaves far less ideally. The Van der Waals equation provides a much more accurate description because it accounts for the significant intermolecular attractive forces and molecular volume effects that the Ideal Gas Law ignores entirely.
Key Insight: The greater the pressure (or the smaller the volume), the larger the deviation between ideal and Van der Waals predictions for real gases. This is precisely why we need equations of state like Van der Waals—the Ideal Gas Law simply cannot capture real gas behavior under extreme conditions.
A Caution About Units in Van der Waals Calculations
When performing Van der Waals calculations for real gases, pay careful attention to the units of the constants a and b. The constant b is typically straightforward—its units are usually liters per mole, and you simply multiply by the number of moles. However, the units for a can require conversions depending on the reference table you are using.
Tip: If your Van der Waals calculation produces a nonsensical answer, the most likely culprit is a unit mismatch in the constants a and b. Always verify that your units are consistent before plugging values into the Van der Waals equation.
7 Key Takeaways: Real Gases and the Van der Waals Equation
- The compressibility factor (PV/RT) equals 1 for an ideal gas; deviations from 1 indicate non-ideal behavior in real gases caused by intermolecular forces and molecular volume.
- When the compressibility factor falls below 1, intermolecular attractive forces are responsible for the real gas deviation; when it rises above 1, repulsive forces and molecular volume dominate.
- The Van der Waals equation modifies the Ideal Gas Law with 2 correction terms: an²/V² for intermolecular attractive forces and nb for molecular volume.
- The Van der Waals constants a and b are unique to each real gas and determined empirically—smaller values indicate more ideal gas behavior.
- At low pressures, the Ideal Gas Law and Van der Waals equation produce similar results; at high pressures, real gas deviations become significant.
- When calculating with the Van der Waals equation, you will almost always solve for pressure since solving for volume yields a cubic equation.
- Always verify that the units of your Van der Waals constants are consistent with the other quantities in your calculation.