The equation of state for real gas is given by $\left(P+\frac{a}{V^{2}}\right)(V-b)=RT$. The dimensions of the constant $a$ is ______.

Step 1: State the given equation and identify the variables.

The equation of state for a real gas is:

$$\left(P+\frac{a}{V^{2}}\right)(V-b)=RT$$

Here, $P$ represents pressure, $V$ represents volume, $T$ represents temperature, and $R$ is the universal gas constant.

Step 2: Apply the principle of dimensional homogeneity.

Physical quantities can only be added or subtracted if they have the same dimensions.

$$[P] = \left[\frac{a}{V^{2}}\right]$$

Rearranging this relation gives the dimensional formula for the constant $a$.

$$[a] = [P][V^{2}]$$

Step 3: Determine the dimensions of pressure and volume.

Pressure is defined as force per unit area.

$$[P] = \frac{[F]}{[A]}$$ $$[P] = \frac{\text{M}\text{L}\text{T}^{-2}}{\text{L}^{2}}$$ $$[P] = \text{M}\text{L}^{-1}\text{T}^{-2}$$

Volume has the dimension of length cubed.

$$[V] = \text{L}^{3}$$ $$[V^{2}] = \text{L}^{6}$$

Step 4: Calculate the dimensions of the constant $a$.

Substitute the dimensions of pressure and volume squared into the expression for $[a]$.

$$[a] = (\text{M}\text{L}^{-1}\text{T}^{-2})(\text{L}^{6})$$ $$[a] = \text{M}\text{L}^{5}\text{T}^{-2}$$

Answer: The dimensions of the constant $a$ are $\text{M}\text{L}^{5}\text{T}^{-2}$.