Van der Waals Equation
Equation of state for real gases accounting for molecular size and intermolecular forces
The Van der Waals Equation
For one mole of gas
Variables:
- P = Pressure (atm)
- V = Molar volume (L/mol)
- R = Gas constant (0.08206 L·atm/mol·K)
- T = Temperature (K)
Constants:
- a = Intermolecular attraction constant
- b = Molecular volume constant
- Both are substance-specific
- Found in reference tables
Understanding the Corrections
Pressure Correction (a/V²):
- • Accounts for intermolecular attractions
- • Reduces effective pressure on container walls
- • Larger 'a' = stronger intermolecular forces
- • More significant at high density (low V)
Volume Correction (b):
- • Accounts for finite molecular size
- • Reduces available space for gas movement
- • Larger 'b' = larger molecular volume
- • More significant at high pressure (low V)
Common Van der Waals Constants
| Gas | a (L²·atm/mol²) | b (L/mol) |
|---|---|---|
| Helium (He) | 0.03412 | 0.02370 |
| Nitrogen (N₂) | 1.390 | 0.03913 |
| Carbon dioxide (CO₂) | 3.640 | 0.04267 |
| Water (H₂O) | 5.536 | 0.03049 |
Step-by-Step Example
Problem:
Calculate the pressure of 2.0 mol of CO₂ gas in a 1.0 L container at 300 K using the Van der Waals equation.
Step 1: Identify known values
- n = 2.0 mol
- V = 1.0 L
- T = 300 K
- For CO₂: a = 3.640 L²·atm/mol², b = 0.04267 L/mol
- R = 0.08206 L·atm/mol·K
Step 2: Calculate molar volume
V_molar = V_total/n = 1.0 L / 2.0 mol = 0.50 L/mol
Step 3: Apply Van der Waals equation
(P + a/V²)(V - b) = RT
P = RT/(V - b) - a/V²
Step 4: Substitute values
P = (0.08206 × 300)/(0.50 - 0.04267) - 3.640/(0.50)²
P = 24.618/0.45733 - 3.640/0.25
P = 53.84 - 14.56
P = 39.28 atm
Answer:
The pressure is 39.3 atm (vs. 49.2 atm predicted by ideal gas law)
Common Mistakes to Avoid
Molar Volume
Use molar volume (V/n), not total volume, in the equation
Constants
Use correct Van der Waals constants for the specific gas
Temperature Units
Always convert temperature to Kelvin
Algebra Complexity
Be careful with order of operations when solving for P, V, or T
Related Calculators
Frequently Asked Questions
When should I use Van der Waals instead of ideal gas law?
Use Van der Waals at high pressures, low temperatures, or when high accuracy is needed. Real gas behavior deviates most under these conditions.
How are Van der Waals constants determined?
They're determined experimentally by fitting the equation to real gas data, often derived from critical temperature and pressure values.
Why do both corrections matter?
The 'a' term becomes significant when molecules are close (high density), while the 'b' term matters when available space is limited (high pressure).
Can I use this for gas mixtures?
Yes, but you need mixing rules to determine effective 'a' and 'b' values for the mixture, which can be quite complex.