Step-by-Step Derivation of Van der Waals Equation with Examples
Real gases do not always behave exactly as predicted by the ideal gas law. To address this, the Van der Waals equation of state was introduced. This equation modifies the assumptions of the ideal gas law to account for molecular size and intermolecular forces, providing a much more accurate description of real gas behavior, especially at high pressures and low temperatures. Understanding the Van der Waals equation is essential for solving advanced problems in thermodynamics and competitive examinations.
Van der Waals Equation: Introduction and Formula
The classic ideal gas law, given as PV = nRT, assumes that gas molecules have negligible volume and experience no intermolecular forces. However, these assumptions break down for real gases under certain conditions. The Van der Waals equation incorporates two corrections:
- Volume correction (b): Accounts for the finite volume occupied by gas molecules.
- Pressure correction (a): Corrects for the attractive forces between gas molecules.
The Van der Waals equation for n moles of a real gas is:
Where:
Derivation of the Van der Waals Equation
The ideal gas law is modified by considering the actual volume available (excluded by molecules) and the intermolecular attractions. The derivation steps are as follows:
-
Volume Correction:
The actual volume available for movement is less than container volume due to molecule size.Available Volume = V - nb
(b = volume occupied per mole × number of moles) -
Pressure Correction:
Intermolecular attractions decrease observed pressure. The pressure is increased by a correction term:Corrected Pressure = P + a(n/V)2 -
Combined in Ideal Gas Law:
Substituting these corrections into the ideal gas law:[P + a(n/V)2] · [V - nb] = nRT
For one mole (n=1), the equation is:
Constants a and b: Meaning and Importance
The constants 'a' and 'b' are unique for each gas:
- a: Measures the magnitude of intermolecular attractive forces. Higher values of 'a' indicate stronger attractions, which makes gases easier to liquefy.
- b: Represents the effective volume occupied by gas molecules. Larger molecules have higher b values.
| Gas | a (L2 atm mol−2) | b (L mol−1) |
|---|---|---|
| Hydrogen (H2) | 0.244 | 0.0266 |
| Oxygen (O2) | 1.36 | 0.0318 |
| Nitrogen (N2) | 1.39 | 0.0391 |
| Carbon Dioxide (CO2) | 3.59 | 0.0427 |
Comparison: Ideal Gas Law vs. Van der Waals Equation
| Feature | Ideal Gas Law | Van der Waals Equation |
|---|---|---|
| Volume of molecules | Neglected | Corrected by 'b' |
| Intermolecular forces | Ignored | Corrected by 'a' |
| Accuracy (high P/low T) | Poor | Higher |
| Applicability | Theoretical/ideal gases | Real gases, fluids |
Step-by-Step Example: Van der Waals Equation Application
Let's calculate the pressure of 1 mole of nitrogen gas at 300 K in a 10 L vessel using the Van der Waals equation.
Given: a = 1.39 L2 atm mol−2, b = 0.0391 L mol−1, R = 0.0821 L atm K−1 mol−1.
- Calculate volume correction: V - nb = 10 - 0.0391 = 9.9609 L
- Calculate pressure correction: a(n/V)2 = 1.39 × (1/10)2 = 1.39 × 0.01 = 0.0139 atm
- Calculate nRT: 1 × 0.0821 × 300 = 24.63
- Setup equation: [P + 0.0139] × 9.9609 = 24.63
- Solve: P + 0.0139 = 24.63 / 9.9609 = 2.473
P = 2.473 - 0.0139 = 2.459 atm
Therefore, the corrected pressure is 2.46 atm (rounded to two decimals).
Advantages and Limitations of the Van der Waals Equation
- Advantages:
- More accurately describes real gas behavior compared to ideal gas law.
- Applicable to both gases and fluids.
- Can determine behavior near and above the critical temperature. - Limitations:
- Only accurate for real gases above or at the critical temperature.
- Less accurate during phase transitions.
- At low temperatures and high pressures, the equation has shortcomings.
Key Formulas and Quick Reference Table
| Equation | Formula | Description |
|---|---|---|
| Ideal Gas Law | PV = nRT | Valid for ideal gases, ignores interactions and molecular size. |
| Van der Waals | [P + a(n/V)2][V - nb] = nRT | Applies for real gases, includes factors 'a' and 'b'. |
Further Learning and Practice
- Kinetic Theory of Gases
- Ideal Gas Law and Absolute Zero
- Thermodynamics - Laws and Concepts
- Adiabatic Process
- Properties of Gases
To master problem-solving with the Van der Waals equation, regularly practice example problems. Understand the meaning of each correction term and when to apply the ideal versus the real gas law. This fundamental concept appears in many areas of advanced Physics. Explore more with the links above for a comprehensive learning experience.
FAQs on Van der Waals Equation: Derivation, Constants, and Real Gas Insights
1. What is the Van der Waals equation?
The Van der Waals equation is an adjusted gas law that more accurately describes real gas behavior at various temperatures and pressures compared to the ideal gas law. The formula is:
[P + a(n/V)2] × [V - nb] = nRT,
where:
• P = pressure
• V = volume
• n = number of moles
• T = temperature
• R = gas constant
• a = attraction constant
• b = volume correction constant
The equation introduces corrections for intermolecular attractions and finite molecular size.
2. What do the 'a' and 'b' constants in the Van der Waals equation represent?
The 'a' and 'b' constants are Van der Waals constants that account for non-ideal behavior of real gases.
• 'a': Corrects for intermolecular attractive forces. Larger 'a' means stronger attractions.
• 'b': Corrects for the actual, finite volume occupied by gas molecules. It represents the excluded volume per mole.
Bigger values for 'a' and 'b' indicate gases deviate more from ideality.
3. How is the Van der Waals equation derived from the ideal gas law?
Derivation involves adding two corrections to the ideal gas law (PV = nRT):
1. Pressure Correction (a): Accounts for intermolecular attractions, reducing observed pressure.
2. Volume Correction (b): Subtracts the excluded volume of gas molecules from total volume.
Combined, the corrected formula becomes:
[P + a(n/V)2] × [V - nb] = nRT
4. Why do real gases deviate from ideal gas behavior?
Real gases deviate from ideal behavior due to:
• Intermolecular forces: Attractions or repulsions are ignored in ideal gas law but exist in real gases.
• Finite molecular size: Real gas particles occupy space; ideal gas assumes point masses.
These deviations are significant at high pressures and low temperatures.
5. What are the main limitations of the Van der Waals equation?
The Van der Waals equation improves upon the ideal gas law but has these limitations:
• Does not accurately predict behavior near gases' critical points or during phase transitions.
• Fails for gases under extremely high pressure or very low temperature.
• Assumes uniform intermolecular forces and molecular shapes for all gases, which may not always apply.
6. How are Van der Waals constants 'a' and 'b' determined?
'a' and 'b' are determined experimentally for each gas.
They are usually calculated by fitting the Van der Waals equation to experimental pressure-volume-temperature (P-V-T) data, especially near the critical point. Their values depend on the nature and size of the gas molecules.
7. How does the Van der Waals equation explain liquefaction of gases?
Van der Waals equation predicts gas condensation:
• At low temperatures or high pressures, molecules are closer, and attractive forces ('a') become significant.
• The equation helps identify critical temperature and pressure at which gases condense to liquids.
This explains real gases transitioning to the liquid phase, which the ideal gas law cannot explain.
8. What types of questions are frequently asked on Van der Waals equation in JEE, NEET or CBSE exams?
Common exam questions include:
• Numerical problems calculating pressure, volume, or temperature using the Van der Waals equation.
• Derivations of the equation or explanation of corrections made.
• Interpretation/meaning and calculation of the 'a' and 'b' constants.
• Comparison between ideal and real gases.
• Questions on limitations and significance in physical situations.
9. When does the Van der Waals equation reduce to the ideal gas law?
Van der Waals equation reduces to the ideal gas law when:
• Value of 'a' = 0 (no intermolecular attraction)
• Value of 'b' = 0 (negligible molecular size)
In such cases, the equation simplifies to PV = nRT, treating gases as ideal.
10. How is the Van der Waals equation applied to solve numerical problems?
To solve numericals using the Van der Waals equation:
1. Identify all given values (n, V, T, a, b, R).
2. Plug values into [P + a(n/V)2] × [V - nb] = nRT.
3. Calculate each term step-by-step, following corrections for pressure and volume.
4. Solve for the unknown quantity (P, V, or T).
11. What is the physical meaning of the 'nb' term in the Van der Waals equation?
The 'nb' term corrects for the volume occupied by gas molecules themselves:
• 'b' is the excluded volume per mole.
• 'n' is the number of moles.
Thus, (V - nb) represents the actual free space available for molecular movement in the vessel.
12. Why does the Van der Waals equation not work well near the critical point?
The Van der Waals equation fails near the critical point because:
• Real gas behavior becomes highly non-linear and fluctuations increase.
• The assumptions of uniform molecular size and attraction no longer hold well.
Thus, predictions for phase transition or properties at the critical state are inaccurate.
