The van der Waal's equation of state for some gases can be expressed as :
`(P + (a)/( V^(2))) ( V - b) = RT`
Where `P` is the pressure , `V` is the molar volume , and `T` is the absolute temperature of the given sample of gas and `a, b , and R` are constants.
The dimensions of ` a` are

Options

A. `ML^(5)T^(-2)`
B. ` ML^(-1)T^(-2)`
C. `L^(3)`
D. ` L^(6)`

Explanation

To find the dimensions of the constant \( a \) in the van der Waals equation of state, we start with the equation:

\[ (P + \frac{a}{V^2})(V - b) = RT \]

### Step 1: Understand the equation
The equation consists of pressure \( P \), volume \( V \), and temperature \( T \). The constants \( a \) and \( b \) are specific to the gas being studied, and \( R \) is the universal gas constant.

### Step 2: Identify the dimensions of each variable
- The dimension of pressure \( P \) is given by:
\[ [P] = M L^{-1} T^{-2} \]
- The dimension of molar volume \( V \) is:
\[ [V] = L^3 \]
- The dimension of temperature \( T \) is:
\[ [T] = \Theta \quad \text{(where } \Theta \text{ is the dimension of temperature)} \]
- The dimension of the gas constant \( R \) is:
\[ [R] = \frac{[P][V]}{[T]} = \frac{M L^{-1} T^{-2} \cdot L^3}{\Theta} = M L^{2} T^{-2} \Theta^{-1} \]

### Step 3: Apply the principle of homogeneity
According to the principle of homogeneity, the dimensions of the terms being added or subtracted must be the same. Therefore, we can set the dimensions of \( P \) equal to the dimensions of \( \frac{a}{V^2} \):

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

### Step 4: Substitute the dimensions
Substituting the dimensions we have:

\[ M L^{-1} T^{-2} = \frac{[a]}{(L^3)^2} \]

This simplifies to:

\[ M L^{-1} T^{-2} = \frac{[a]}{L^6} \]

### Step 5: Solve for the dimensions of \( a \)
Rearranging the equation to solve for \( [a] \):

\[ [a] = M L^{-1} T^{-2} \cdot L^6 = M L^{5} T^{-2} \]

### Conclusion
Thus, the dimensions of \( a \) are:

\[ [a] = M L^{5} T^{-2} \]