The expression relating mole fraction of solute `(chi_(2))` and molarity `(M)` of the solution is: (where `d` is the density of the solution in `g L^(-1)` and `Mw_(1)` and `Mw_(2)` are the molar masses of solvent and solute, respectively

Options

A. `x_(2) = (M xx Mw_(1))/(M (Mw_(1) xx Mw_(2)) + 1000d)`
B. `x_(2) = (M xx Mw_(1))/(M (Mw_(1) xx Mw_(2)) + d)`
C. `x_(2) = (M xx Mw_(1))/(M (Mw_(1) xx Mw_(2)) - 1000d)`
D. `x_(2) = (M xx Mw_(1))/(M (Mw_(1) xx Mw_(2)) - d)`

Explanation

To derive the expression relating the mole fraction of solute (χ₂) and molarity (M) of the solution, we will follow a systematic approach.

### Step-by-Step Solution:

1. **Assume the Volume of the Solution**:
Let's assume the volume of the solution is 1 liter.

2. **Calculate the Weight of the Solution**:
The weight of the solution can be calculated using the density (d) of the solution:
\[ \text{Weight of the solution} = \text{Density} \times \text{Volume} = d \times 1 \text{ L} = d \text{ g} \]

3. **Define Molarity and Calculate Moles of Solute**:
Molarity (M) is defined as the number of moles of solute per liter of solution. Therefore, the number of moles of solute (n₂) can be expressed as:
\[ n_2 = M \text{ (moles of solute)} \]

4. **Calculate the Weight of the Solute**:
The weight of the solute (w₂) can be calculated using its molar mass (Mw₂):
\[ w_2 = n_2 \times Mw_2 = M \times Mw_2 \]

5. **Calculate the Weight of the Solvent**:
The weight of the solvent (w₁) can be found by subtracting the weight of the solute from the weight of the solution:
\[ w_1 = \text{Weight of solution} - \text{Weight of solute} = d - w_2 = d - M \times Mw_2 \]

6. **Calculate the Number of Moles of Solvent**:
The number of moles of solvent (n₁) can be calculated using its weight and molar mass (Mw₁):
\[ n_1 = \frac{w_1}{Mw_1} = \frac{d - M \times Mw_2}{Mw_1} \]

7. **Calculate the Mole Fraction of the Solute (χ₂)**:
The mole fraction of the solute (χ₂) is defined as the number of moles of solute divided by the total number of moles (solute + solvent):
\[ \chi_2 = \frac{n_2}{n_1 + n_2} = \frac{M}{\frac{d - M \times Mw_2}{Mw_1} + M} \]

8. **Simplify the Expression**:
To simplify the expression, we can combine the terms in the denominator:
\[ \chi_2 = \frac{M \times Mw_1}{(d - M \times Mw_2) + M \times Mw_1} \]

9. **Final Expression**:
The final expression relating the mole fraction of solute (χ₂) and molarity (M) of the solution is:
\[ \chi_2 = \frac{M \times Mw_1}{d + M \times Mw_1 - M \times Mw_2} \]