Animated Solution for Chemistry - States of Matter: The unit of the van der Waals' gas equation parameter '$a$' in $\left(p+\frac{a{n}^2}{V^2}\right)(V-nb)=nRT$ is

Have you ever tried adding three apples to four oranges? It doesn't make much sense, does it? In physics and chemistry, we have a similar rule called the Principle of Dimensional Homogeneity. It states that you can only add or subtract physical quantities if they share the exact same dimensions and units.

This simple yet profound rule is our secret weapon for solving this problem involving the famous van der Waals equation.

The ideal gas equation, $pV=nRT$, is a great approximation, but real gases don't always behave perfectly. Johannes Diderik van der Waals introduced two correction factors, $a$ and $b$, to account for intermolecular forces and the actual volume of gas molecules, respectively.

The modified equation looks like this:

Our goal is to find the unit of the parameter $a$.

Let's zoom in on the first bracket: $\left(p+\frac{a{n}^2}{V^2}\right)$.

Here, we are adding the term $\frac{a{n}^2}{V^2}$ to the pressure $p$. According to our golden rule of dimensional homogeneity, this is only legally allowed in the universe of physics if the entire term $\frac{a{n}^2}{V^2}$ has the exact same units as pressure!

So, we can confidently write:

Now, it's just a matter of simple algebra. Let's isolate our target variable, $a$:

Next, we substitute the standard units for each physical quantity. In the context of the given options, pressure $p$ is measured in atmospheres ($\text{atm}$), volume $V$ is measured in liters, which is equivalent to cubic decimeters (${\text{dm}}^3$), and $n$ is measured in moles ($\text{mol}$).

Plugging these in:

Simplifying the exponents, we arrive at our final, elegant answer:

This perfectly matches option (d). By simply understanding that you can't add apples and oranges, we've effortlessly decoded the units of a complex thermodynamic parameter!