Van der Waals Constants
The Physical Significance of '$a$' and '$b$'.
In the Van der Waals equation for real gases, $\left( P + \frac{a n^2}{V^2} \right) (V - nb) = nRT$, the terms $a$ and $b$ are empirical constants specific to each gas. Understanding their physical meaning is one of the most frequently tested concepts in chemistry competitive exams.
1. Significance of Constant '$a$'
The constant $a$ is part of the pressure correction term ($P + \frac{an^2}{V^2}$). It corrects the flawed ideal gas assumption that there are no forces between molecules.
Physical Meaning:
- '$a$' is a direct measure of the magnitude of intermolecular attractive forces within the gas.
- A higher value of '$a$' means the gas molecules attract each other very strongly.
Relation to Ease of Liquefaction:
To turn a gas into a liquid, the molecules must be pulled close together. If a gas has strong natural attractive forces (a high '$a$' value), it will "want" to clump together, making it easier to liquefy.
Units of '$a$':
Since we add $\frac{an^2}{V^2}$ to Pressure ($P$), its units must be the same as pressure.
$\frac{a \cdot \text{mol}^2}{\text{L}^2} = \text{atm} \implies a = \text{atm} \cdot \text{L}^2 \cdot \text{mol}^{-2}$
2. Significance of Constant '$b$'
The constant $b$ is part of the volume correction term ($V - nb$). It corrects the flawed ideal gas assumption that molecules are point masses with zero volume.
Physical Meaning:
- '$b$' represents the effective size or excluded volume of the gas molecules.
- It is the volume that is "off-limits" to other molecules because it is already physically occupied.
- A higher value of '$b$' means the gas molecules are physically larger.
The constant '$b$' is NOT just the literal volume of the molecules. Because spherical molecules clash and bounce off each other, the "forbidden" volume is larger than their physical volume.
Formula: $b = 4 \times N_A \times (\frac{4}{3}\pi r^3)$
Meaning: '$b$' is exactly four times the actual volume of the molecules in one mole of gas.
Units of '$b$':
Since we subtract $nb$ from Volume ($V$), its units must be volume.
$n \cdot b = \text{Liters} \implies \text{mol} \cdot b = \text{L} \implies b = \text{L} \cdot \text{mol}^{-1}$
3. Comparing Common Gases
Examiners frequently ask you to arrange gases in order of their '$a$' or '$b$' values. Here is how to analyze them:
| Gas | Nature / Size | '$a$' value (Attraction) | '$b$' value (Size) |
|---|---|---|---|
| $NH_3$ | Polar, Hydrogen Bonding | Very High (4.17) | 0.037 |
| $CO_2$ | Non-polar, but large electron cloud (high dispersion forces) | High (3.59) | Large (0.043) |
| $N_2$ | Non-polar, diatomic | Moderate (1.39) | 0.039 |
| $H_2$ | Tiny, Non-polar | Very Low (0.24) | Small (0.027) |
| $He$ | Smallest atom, Noble gas | Lowest (0.034) | Smallest (0.024) |
• To rank '$a$': Look for Hydrogen bonding ($NH_3, H_2O$) first. Then look for polarity ($HCl, SO_2$). Finally, for non-polar gases, look at molecular mass/size (larger electron cloud = stronger London dispersion forces = higher '$a$').
• To rank '$b$': Simply look at the physical size of the molecule. A massive molecule like $C_4H_{10}$ has a much larger '$b$' than $He$.
Knowledge Check
Test your understanding of constants 'a' and 'b'
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