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 solution, `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)-y_(M))lt(x_(N)-y_(N))`
C. `(x_(M))/(x_(N))lt(y_(M))/(y_(N))`
D. `(x_(M))/(x_(N))gt(y_(M))/(y_(N))`

Answer

To solve the problem, we need to analyze the relationship between the mole fractions of the components in the liquid phase (solution) and the vapor phase, given that the liquids M and N form an ideal solution.

### Step-by-Step Solution:

1. **Identify Given Information:**
- Vapor pressure of pure liquid M, \( P^0_M = 450 \, \text{mmHg} \)
- Vapor pressure of pure liquid N, \( P^0_N = 700 \, \text{mmHg} \)
- Mole fraction of M in solution, \( x_M \)
- Mole fraction of N in solution, \( x_N \)
- Mole fraction of M in vapor phase, \( y_M \)
- Mole fraction of N in vapor phase, \( y_N \)

2. **Use Raoult's Law:**
For an ideal solution, Raoult's Law states:
\[ y_M = \frac{P^0_M \cdot x_M}{P_{solution}} \]
\[ y_N = \frac{P^0_N \cdot x_N}{P_{solution}} \]
where \( P_{solution} \) is the total vapor pressure of the solution.

3. **Calculate Total Vapor Pressure:**
The total vapor pressure of the solution can be expressed as:
\[ P_{solution} = P^0_M \cdot x_M + P^0_N \cdot x_N \]

4. **Express \( y_M \) and \( y_N \):**
Substitute \( P_{solution} \) into the equations for \( y_M \) and \( y_N \):
\[ y_M = \frac{P^0_M \cdot x_M}{P^0_M \cdot x_M + P^0_N \cdot x_N} \]
\[ y_N = \frac{P^0_N \cdot x_N}{P^0_M \cdot x_M + P^0_N \cdot x_N} \]

5. **Form the Ratio of Mole Fractions:**
To find the relationship between \( \frac{x_M}{x_N} \) and \( \frac{y_M}{y_N} \), we can divide the equations:
\[ \frac{y_M}{y_N} = \frac{P^0_M \cdot x_M}{P^0_N \cdot x_N} \]

6. **Substitute Known Values:**
Substitute the values of \( P^0_M \) and \( P^0_N \):
\[ \frac{y_M}{y_N} = \frac{450 \cdot x_M}{700 \cdot x_N} \]

7. **Rearranging the Equation:**
Rearranging gives:
\[ \frac{x_M}{x_N} = \frac{y_M}{y_N} \cdot \frac{700}{450} \]

8. **Conclusion:**
Since \( \frac{700}{450} > 1 \), we can conclude:
\[ \frac{x_M}{x_N} > \frac{y_M}{y_N} \]
This means that the mole fraction of M in the solution is greater than that in the vapor phase, which leads us to the correct statement.

### Final Answer:
The correct statement is:
\[ \frac{x_M}{x_N} > \frac{y_M}{y_N} \]