The vander waal's equation for a gas is `(p + a/V^(2))(V - b) = nRT` where p, V, R, T and n represent the Pressure, Volume, universal gas constant, absolute temperature and number of moles of a gas respectively, where a and b are constants. The ratio`b/a` will have the following dimensional formula

Options

A. `M^(-1) L^(-2) T^(2)`
B. `M^(-1) L^(-1)T^(-1)`
C. `ML^(2)T^(2)`
D. `MLT^(-2)`

Answer

To find the dimensional formula for the ratio \( \frac{b}{a} \) from the Van der Waals equation, we can follow these steps:

### Step 1: Write down the Van der Waals equation
The Van der Waals equation is given as:
\[ \left( p + \frac{a}{V^2} \right)(V - b) = nRT \]
where \( p \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, \( T \) is absolute temperature, and \( a \) and \( b \) are constants.

### Step 2: Identify the dimensions of pressure
The dimensions of pressure \( p \) can be expressed as:
\[ \text{Pressure} (p) = \frac{\text{Force}}{\text{Area}} = \frac{M L T^{-2}}{L^2} = M L^{-1} T^{-2} \]

### Step 3: Identify the dimensions of volume
The dimensions of volume \( V \) are:
\[ \text{Volume} (V) = L^3 \]

### Step 4: Determine the dimensions of constant \( a \)
From the equation, we can see that:
\[ \frac{a}{V^2} \text{ must have the same dimensions as pressure } (p) \]
Thus, we can write:
\[ \frac{a}{V^2} \sim p \implies a \sim p \cdot V^2 \]
Substituting the dimensions:
\[ a \sim (M L^{-1} T^{-2}) \cdot (L^3) = M L^{2} T^{-2} \]

### Step 5: Determine the dimensions of constant \( b \)
The constant \( b \) has the same dimensions as volume:
\[ b \sim V \implies b \sim L^3 \]

### Step 6: Find the ratio \( \frac{b}{a} \)
Now we can find the dimensions of the ratio \( \frac{b}{a} \):
\[ \frac{b}{a} = \frac{L^3}{M L^2 T^{-2}} = \frac{L^3}{M L^2 T^{-2}} = \frac{L^{3-2}}{M T^{-2}} = \frac{L^1}{M T^{-2}} = \frac{L}{M} T^2 \]

### Step 7: Write the final dimensional formula
Thus, the dimensional formula for the ratio \( \frac{b}{a} \) is:
\[ \frac{b}{a} \sim M^{-1} L^{1} T^{2} \]

### Conclusion
The correct dimensional formula for the ratio \( \frac{b}{a} \) is:
\[ M^{-1} L^{1} T^{2} \]