Liquid ‘M’ and liquid ‘N’ form an ideal solution. The vapour pressures of pure liquids ‘M’ and ‘N’ are 450 and 700 mmHg, respectively, at the same temperature. Then correct statement is:
( `x_(M)=`Mole fraction of ‘M’ in solutions , `" " x_(N)=` Mole fraction of ‘N’ in solution ,
`y_(M)=`Mole fraction of ‘M’ in vapour phase , `" " y_(N)=` Mole fraction of ‘n’ in vapour phase)

Options

A. `(x_(M))/(x_(N))=(y_(M))/(y_(N))`
B. `(x_(M))/(x_(N)) lt (y_(M))/(y_(N))`
C. `(x_(M)-y_(M)) lt (x_(N)-y_(N))`
D. `(x_(M))/(x_(N)) gt (y_(M))/(y_(N))`

Explanation

To solve the problem, we need to analyze the relationship between the mole fractions of the components in the solution and their corresponding vapor phases using Raoult's Law and Dalton's Law.

### Step-by-Step Solution:

1. **Understanding Raoult's Law**:
According to Raoult's Law, the partial pressure of a component in an ideal solution is directly proportional to its mole fraction in the solution. For liquids M and N, we can express this as:
\[ P_M = X_M \cdot P_{M}^{0} \]
\[ P_N = X_N \cdot P_{N}^{0} \]
where \( P_M \) and \( P_N \) are the partial pressures of M and N, \( X_M \) and \( X_N \) are their mole fractions in the solution, and \( P_{M}^{0} \) and \( P_{N}^{0} \) are the vapor pressures of pure M and N, respectively.

2. **Using Dalton's Law**:
Dalton's Law states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of each gas. We can express the mole fractions in the vapor phase as:
\[ Y_M = \frac{P_M}{P_{total}} \quad \text{and} \quad Y_N = \frac{P_N}{P_{total}} \]
where \( Y_M \) and \( Y_N \) are the mole fractions of M and N in the vapor phase.

3. **Total Pressure**:
The total pressure \( P_{total} \) can be expressed as:
\[ P_{total} = P_M + P_N = X_M \cdot P_{M}^{0} + X_N \cdot P_{N}^{0} \]

4. **Relating Mole Fractions**:
We can now relate the mole fractions in the vapor phase to those in the solution:
\[ Y_M = \frac{X_M \cdot P_{M}^{0}}{P_{total}} \quad \text{and} \quad Y_N = \frac{X_N \cdot P_{N}^{0}}{P_{total}} \]

5. **Finding the Ratio**:
To find the ratio \( \frac{Y_M}{Y_N} \):
\[ \frac{Y_M}{Y_N} = \frac{X_M \cdot P_{M}^{0}}{X_N \cdot P_{N}^{0}} \]

6. **Substituting Values**:
Given \( P_{M}^{0} = 450 \, \text{mmHg} \) and \( P_{N}^{0} = 700 \, \text{mmHg} \), we substitute these values:
\[ \frac{Y_M}{Y_N} = \frac{X_M \cdot 450}{X_N \cdot 700} \]

7. **Simplifying the Ratio**:
Rearranging gives:
\[ \frac{X_M}{X_N} = \frac{Y_M}{Y_N} \cdot \frac{700}{450} \]
Simplifying \( \frac{700}{450} \) gives \( \frac{14}{9} \).

8. **Final Comparison**:
Thus, we have:
\[ \frac{X_M}{X_N} = \frac{Y_M}{Y_N} \cdot \frac{14}{9} \]
Since \( \frac{14}{9} > 1 \), we conclude that:
\[ \frac{X_M}{X_N} > \frac{Y_M}{Y_N} \]

### Conclusion:
The correct statement is:
\[ X_M / X_N > Y_M / Y_N \]