The relation between molarity (M) and molality (m) is given by : (p=density of solution (g/mL), `M_1`= molecular mass of solute)

Options

A. `m=(1000M)/(1000 rho-M_(1))`
B. `m=(1000rhoM)/(1000 rho-MM_(1))`
C. `m=(1000MM)/(1000 rho-MM_(1))`
D. `m=(1000M)/(1000 rho-MM_(1))`

Answer

To derive the relationship between molarity (M) and molality (m) using the given parameters, we can follow these steps:

### Step-by-Step Solution:

1. **Define Molarity (M)**:
Molarity (M) is defined as the number of moles of solute per liter of solution. The formula for molarity is:
\[ M = \frac{\text{moles of solute}}{\text{volume of solution in liters}} = \frac{n}{V} \]
where \( n \) is the number of moles of solute and \( V \) is the volume of the solution in liters.

2. **Convert Volume to Milliliters**:
Since we often use milliliters in calculations, we can express the volume in milliliters:
\[ M = \frac{n}{\frac{V}{1000}} \quad \Rightarrow \quad n = M \times \frac{V}{1000} \]

3. **Define Molality (m)**:
Molality (m) is defined as the number of moles of solute per kilogram of solvent. The formula for molality is:
\[ m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} = \frac{n}{W} \]
where \( W \) is the mass of the solvent in kilograms.

4. **Relate Weight of Solvent to Weight of Solution**:
The weight of the solution can be expressed as the sum of the weight of the solvent and the weight of the solute:
\[ \text{Weight of solution} = \text{Weight of solvent} + \text{Weight of solute} \]
Therefore, the weight of the solvent can be expressed as:
\[ \text{Weight of solvent} = \text{Weight of solution} - \text{Weight of solute} \]

5. **Express Weight of Solution in Terms of Density**:
The density (\( \rho \)) of the solution is defined as:
\[ \rho = \frac{\text{mass of solution}}{\text{volume of solution}} \]
Thus, the weight of the solution can be expressed as:
\[ \text{Weight of solution} = \rho \times V \]

6. **Substituting into the Molality Formula**:
Now, substituting the expression for the weight of solvent into the molality formula:
\[ m = \frac{n}{\text{Weight of solution} - \text{Weight of solute}} = \frac{M \times \frac{V}{1000}}{\rho \times V - \text{Weight of solute}} \]

7. **Express Weight of Solute**:
The weight of the solute can be expressed in terms of its moles and molecular mass (\( M_1 \)):
\[ \text{Weight of solute} = n \times M_1 = M \times \frac{V}{1000} \times M_1 \]

8. **Final Relationship**:
By substituting this into the equation for molality, we can derive the relationship:
\[ m = \frac{M \times \frac{V}{1000}}{\rho \times V - M \times \frac{V}{1000} \times M_1} \]
Simplifying this expression gives us:
\[ m = \frac{M \times 1000}{1000 \times \rho - M \times M_1} \]

### Final Result:
The relationship between molarity (M) and molality (m) is:
\[ m = \frac{M \times 1000}{1000 \times \rho - M \times M_1} \]