The van der Waal's equation: $\left(P + \dfrac{a}{4V^2}\right)\left(V - \dfrac{b}{2}\right) = \dfrac{RT}{2}$ is valid for

The standard van der Waals equation for $n$ moles of a real gas is given by:
$\left(P + \dfrac{an^2}{V^2}\right) (V - nb) = nRT$

The given equation is:
$\left(P + \dfrac{a}{4V^2}\right) \left(V - \dfrac{b}{2}\right) = \dfrac{RT}{2}$

Comparing the right-hand side of the given equation with the standard equation:
$\dfrac{RT}{2} = nRT$
Thus, $n = \dfrac{1}{2}$.

Let us verify this by comparing the other terms.
In the volume term: $nb = \dfrac{1}{2}b = \dfrac{b}{2}$, which matches the given equation.
In the pressure term: $\dfrac{an^2}{V^2} = \dfrac{a \left(\dfrac{1}{2}\right)^2}{V^2} = \dfrac{a}{4V^2}$, which also matches the given equation.

The van der Waals equation describes the behavior of real gases by accounting for intermolecular forces and the finite size of gas molecules through the constants $a$ and $b$ respectively. Therefore, the given equation is valid for $\dfrac{1}{2}$ mole of a real gas.

Answer: $\dfrac{1}{2}$ mole of a real gas