The unit of a in van der Waal's gas equation is

The correct option is atm L²per mol.

Let's derive the unit of the van der Waals constant $a$ step-by-step:

According to the van der Waals equation for a real gas:

$(P+\frac{a{n}^2}{V^2})(V-nb)=nRT$

Where:
- $P$ is the pressure of the gas
- $V$ is the volume of the gas
- $n$ is the number of moles of the gas
- $a$ and $b$ are van der Waals constants

According to the principle of dimensional homogeneity, only physical quantities with the same units can be added or subtracted. Therefore, the pressure correction term $\frac{a{n}^2}{V^2}$ must have the same unit as pressure ($P$):

$\text{Unit of}P=\text{Unit of}\frac{a{n}^2}{V^2}$

Rearranging the equation to solve for the unit of $a$:

$\text{Unit of}a=\frac{\text{Unit of}P\times {\left(\text{Unit of}V\right)}^2}{{\left(\text{Unit of}n\right)}^2}$

Substituting the standard units where pressure is measured in atmospheres ($\text{atm}$), volume in liters ($\text{L}$), and quantity of substance in moles ($\text{mol}$):

$\text{Unit of}a=\frac{\text{atm}\times {\text{L}}^2}{{\text{mol}}^2}=\text{atm}{\text{L}}^2{\text{mol}}^{:-2}$

When expressing the quantity relative to the molar properties, the unit corresponds to the correct option: atm L²per mol.