STATEMENT-1 : `1//4^(th)` of the initial mole of the air is expelled, if air present in an open vessel is heated from `27^(@)C` to `127^(@)C`.
STATEMENT-2 : Rate of diffusion of a gas is inversely proportional to the square root of its molecular mass.

Options

A. If both the statement are TRUE and STATEMENT-2 is the correct explanation of STATEMENT-1
B. If both the statement are TRUE but STATEMENT-2 is NOT the correct explanation of STATEMENT-1
C. If STATEMENT-1 is TRUE and STATEMENT-2 is FALSE
D. If STATEMENT-1 is FALSE and STATEMENT-2 is TRUE

Answer

To solve the problem step by step, we will analyze both statements given in the question.

### Step 1: Analyze Statement 1
The first statement claims that one-fourth of the initial moles of air is expelled when the air in an open vessel is heated from \(27^\circ C\) to \(127^\circ C\).

1. **Convert temperatures to Kelvin**:
- Initial temperature: \(T_1 = 27^\circ C = 27 + 273 = 300 \, K\)
- Final temperature: \(T_2 = 127^\circ C = 127 + 273 = 400 \, K\)

2. **Use the Ideal Gas Law**:
The Ideal Gas Law is given by:
\[ PV = nRT \]
Since the vessel is open, the pressure (P) and the gas constant (R) remain constant. Therefore, we can express the relationship between temperature and moles of gas (n):
\[ \frac{T_1}{T_2} = \frac{n_2}{n_1} \]
where \(n_1\) is the initial number of moles and \(n_2\) is the final number of moles.

3. **Substituting the values**:
\[ \frac{300}{400} = \frac{n_2}{n_1} \]
Simplifying this gives:
\[ \frac{3}{4} = \frac{n_2}{n_1} \]
This implies:
\[ n_2 = \frac{3}{4} n_1 \]

4. **Determine the expelled moles**:
If \(n_1\) is the initial number of moles, then the number of moles expelled is:
\[ n_{\text{expelled}} = n_1 - n_2 = n_1 - \frac{3}{4} n_1 = \frac{1}{4} n_1 \]
Thus, one-fourth of the initial moles of air is indeed expelled, confirming that Statement 1 is true.

### Step 2: Analyze Statement 2
The second statement claims that the rate of diffusion of a gas is inversely proportional to the square root of its molecular mass. This is a well-known principle known as Graham's Law of Effusion.

1. **Understanding Graham's Law**:
Graham's Law states:
\[ \text{Rate of diffusion} \propto \frac{1}{\sqrt{M}} \]
where \(M\) is the molecular mass of the gas. This means that lighter gases diffuse faster than heavier gases.

2. **Conclusion for Statement 2**:
Since this relationship is established and widely accepted in chemistry, Statement 2 is also true.

### Final Conclusion
Both statements are true:
- Statement 1 is true because we calculated that one-fourth of the initial moles are expelled when the air is heated.
- Statement 2 is true as it correctly describes the relationship between the rate of diffusion and molecular mass.

However, there is no direct relationship between the two statements. Therefore, the correct answer is that both statements are true, but Statement 2 does not explain Statement 1.