Real Gases vs Ideal Gases
The Ideal Gas Law ($PV=nRT$) is incredibly useful, but it fundamentally lies to you. It assumes gas molecules have exactly zero physical volume and that they exert zero attractive force on one another. In reality, atoms take up physical space, and they do pull on each other with slight intermolecular forces.
In 1873, Johannes Diderik van der Waals formulated this equation to correct those lies. He introduced two highly specific constants: '$a$' to correct for the attractive forces between molecules (which slightly reduces pressure), and '$b$' to correct for the actual physical volume of the molecules themselves (which slightly reduces the available volume in the container).
Why Engineers Use It
Under normal room conditions, the Ideal Gas Law is perfectly fine. However, engineers absolutely must use the Van der Waals equation when dealing with extreme conditions:
- High Pressure Systems: Inside a highly compressed SCUBA tank or a liquid nitrogen dewar, the gas molecules are squished so close together that their physical size (the '$b$' constant) suddenly takes up a significant percentage of the tank, invalidating the Ideal Gas Law.
- Low Temperatures: When a gas is chilled near its condensation point (like cooling Propane or Butane), the molecules slow down enough that their attractive forces (the '$a$' constant) grab hold of each other, drastically reducing the pressure compared to what an ideal gas would predict.
- Chemical Plants: When designing massive industrial reactors that operate at hundreds of atmospheres of pressure, using the Ideal Gas Law would result in catastrophic miscalculations. The Van der Waals equation provides the necessary real-world accuracy.
The Formula
Example Calculation
Let's calculate the real pressure of $1 , \text{mole}$ of Carbon Dioxide ($CO_2$) occupying a tiny volume of $0.001 , \text{m}^3$ at $300 , \text{K}$. For $CO_2$, the constants are $a = 0.364 , \text{Pa}\cdot\text{m}^6\text{/mol}^2$ and $b = 0.00004267 , \text{m}^3\text{/mol}$.
- Ideal Gas Prediction: Using $P = nRT/V$, the pressure would be $(1 \cdot 8.314 \cdot 300) / 0.001 = 2,494,200 , \text{Pa}$.
- Van der Waals Correction: $\left[ \frac{1 \cdot 8.314 \cdot 300}{0.001 - (1 \cdot 0.00004267)} \right] - \left[ 0.364 \cdot \left(\frac{1}{0.001}\right)^2 \right]$.
- Calculate: $2,605,066 - 364,000 = 2,241,066 , \text{Pa}$.
The actual, real pressure is $2.24 , \text{MPa}$, which is nearly 10% lower than the Ideal Gas Law blindly predicted. This massive difference is due to the strong attractive forces of the $CO_2$ molecules pulling on each other, reducing the force with which they strike the walls.