The correct relationship between molarity (M) and molality (m) is ( d = density of the solution, in `"kg L"^(-1),M_(2)`= molar mass of the solute in `"kg mol"^(-1)`)

Options

A. `M=(md)/(1+nM_(2))`
B. `M=(m)/(1+mM_(2)d)`
C. `M=(1+mM_(2))/(md)`
D. `M=(1+md)/(mM_(2))`

Explanation

To find the correct relationship between molarity (M) and molality (m), we can follow these steps:

### Step 1: Define Molarity and Molality
- **Molarity (M)** is defined as the number of moles of solute per liter of solution.
\[ M = \frac{n}{V} \]
where \( n \) is the number of moles of solute and \( V \) is the volume of the solution in liters.

- **Molality (m)** is defined as the number of moles of solute per kilogram of solvent.
\[ m = \frac{n}{W} \]
where \( W \) is the mass of the solvent in kilograms.

### Step 2: Relate Molarity to Mass of Solution
- The mass of the solution can be expressed using its density (d):
\[ \text{Mass of solution} = \text{Density} \times \text{Volume} = d \times 1 \text{ L} = d \text{ kg} \]

### Step 3: Calculate the Mass of Solute
- The mass of the solute can be calculated using its molar mass (M2):
\[ \text{Mass of solute} = n \times M2 = M \times M2 \]
where \( M \) is the number of moles of solute (since we have 1 L of solution).

### Step 4: Calculate the Mass of Solvent
- The mass of the solvent can be found by subtracting the mass of the solute from the mass of the solution:
\[ \text{Mass of solvent} = \text{Mass of solution} - \text{Mass of solute} = d - M \times M2 \]

### Step 5: Substitute into the Molality Formula
- Now, substituting the mass of the solvent into the molality formula:
\[ m = \frac{n}{W} = \frac{M}{d - M \times M2} \]

### Step 6: Rearranging for Molarity
- Rearranging the equation to express M in terms of m:
\[ M = m \times (d - M \times M2) \]
- Expanding this gives:
\[ M = m \times d - m \times M \times M2 \]
- Rearranging further:
\[ M(1 + m \times M2) = m \times d \]
- Finally, we can express M as:
\[ M = \frac{m \times d}{1 + m \times M2} \]

### Conclusion
Thus, the relationship between molarity (M) and molality (m) is given by:
\[ M = \frac{m \times d}{1 + m \times M2} \]