Van der Waals Equation
Correcting the Ideal Gas Law for Real Gas Behavior.
The Ideal Gas Equation ($PV = nRT$) works perfectly at low pressures and high temperatures. But at high pressures or low temperatures, real gases deviate wildly. Dutch physicist Johannes Diderik van der Waals realized that two fundamental postulates of the Kinetic Theory of Gases were flawed for real gases.
1. Gas molecules have negligible volume. (False: Molecules take up physical space).
2. There are no intermolecular forces. (False: Molecules attract each other).
1. Volume Correction (Excluded Volume)
Because real gas molecules have a finite, physical size, they are not "point masses." The entire volume of the container is NOT completely free for molecules to move around in. The "ideal" free volume is the container's volume minus the space occupied by the molecules themselves.
- $n$: Number of moles.
- $b$: The excluded volume (or co-volume) per mole of the gas. It represents the effective volume of the gas molecules.
- Mathematical relation: $b = 4 \times N_A \times (\frac{4}{3}\pi r^3)$. The excluded volume is 4 times the actual volume of the molecules in one mole.
2. Pressure Correction (Attractive Forces)
In a real gas, molecules attract each other. When a molecule is about to hit the container wall, the molecules behind it pull it back slightly. Because of this inward pull, the molecule hits the wall with less force than an ideal gas would.
Therefore, the real measured pressure is lower than the ideal pressure. To find the ideal pressure, we must add a correction term ($p'$).
The drop in pressure ($p'$) depends on the density of molecules striking the wall ($\frac{n}{V}$) and the density of molecules pulling them back ($\frac{n}{V}$).
3. The Van der Waals Equation
By substituting the corrected Ideal Pressure and Ideal Volume into the Ideal Gas Law ($P_{ideal} V_{ideal} = nRT$), we get the famous Van der Waals Equation for $n$ moles of a real gas:
For $1 \text{ mole}$ of gas ($n=1$), it simplifies to: $\left(P + \frac{a}{V^2}\right)(V - b) = RT$
4. Significance and Units of '$a$' and '$b$'
The constants $a$ and $b$ are characteristic properties of each specific gas. They are highly tested in competitive exams.
| Constant | Physical Significance | Unit |
|---|---|---|
| $a$ |
Magnitude of Intermolecular Attractive Forces. $\bullet$ Higher '$a$' = Stronger forces. $\bullet$ Higher '$a$' = Easier to liquefy the gas (e.g., $NH_3, CO_2, H_2O$). |
$\text{atm} \cdot \text{L}^2 \cdot \text{mol}^{-2}$ |
| $b$ |
Effective Size/Volume of the molecules. $\bullet$ Higher '$b$' = Larger molecule size. |
$\text{L} \cdot \text{mol}^{-1}$ |
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