The relation between molarity `M` and molality `m` is given by : {p= density of solution (mg/mL)}
`m_1` = Molecular weight of solute)

Options

A. `m=(1000M)/(1000p-M_1)`
B. `m=(1000pM)/(1000p-MM_1)`
C. `m=(1000MM_1)/(1000p-MM_1)`
D. `m=(1000M)/(1000p-MM_1)`

Answer

To derive the relationship between molarity (M) and molality (m), we can start with the definitions of both terms and the given parameters.

### Step-by-Step Solution:

**Step 1: Define Molarity (M) and Molality (m)**
- 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 and Molality**
To relate molarity and molality, we need to express the number of moles of solute in terms of mass and density.

**Step 3: Use Density (ρ)**
The density of the solution (ρ) is given in mg/mL. To convert this to g/L, we use the conversion factor:
\[ \rho \, (\text{g/L}) = \rho \, (\text{mg/mL}) \times 1000 \]
Thus, if \( \rho \) is given in mg/mL, we can express it as:
\[ \rho = \frac{mass \, of \, solution}{volume \, of \, solution} \]

**Step 4: Calculate Mass of Solvent**
Let’s denote the mass of solute as \( m_1 \) (molecular weight of solute) and the mass of the solvent as \( W \). The total mass of the solution is:
\[ \text{Mass of solution} = m_1 + W \]

**Step 5: Express Molarity in Terms of Molality**
Using the definitions and the relationship between mass and volume, we can express molarity in terms of molality:
\[ M = \frac{n}{V} = \frac{m \cdot W}{V} \]
Substituting for \( W \) using the density relation:
\[ W = \frac{mass \, of \, solution - m_1}{\rho} \]

**Step 6: Final Relationship**
Combining all the equations, we can derive a final expression that relates molarity and molality, factoring in the density of the solution and the molecular weight of the solute.

### Final Formula:
The relationship can be summarized as:
\[ M = \frac{m \cdot 1000 \cdot \rho}{1000 + m \cdot m_1} \]