The expression relating molarity (M) of a solution with its molarity (m) is

Options

A. `m=(1000 M)/(1000d+ M M_B)`
B. `m=(1000 M)/(1000d-M M_a)`
C. `m= (1000d+MM_a)/(1000 M)`
D. `m=(1000d- MM_B)/(1000 M)`

Answer

To find the expression relating molarity (M) of a solution with its 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.
\[ M = \frac{\text{moles of solute}}{\text{volume of solution in liters}} \]

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

### Step 2: Relate Molarity to Volume
Assuming we have a solution with a volume of 1 liter (1000 mL), the number of moles of solute can be expressed as:
\[ \text{moles of solute} = M \times 1 \text{ L} = M \]

### Step 3: Calculate the Mass of the Solution
To find the mass of the solution, we need to consider the density (ρ) of the solution. The mass of 1 liter of solution can be expressed as:
\[ \text{mass of solution} = \rho \times 1000 \text{ mL} = 1000 \rho \text{ g} \]

### Step 4: Calculate the Mass of the Solute
The mass of the solute can be calculated using the number of moles and the molar mass (M1) of the solute:
\[ \text{mass of solute} = \text{moles of solute} \times \text{molar mass} = M \times M1 \text{ g} \]

### Step 5: Calculate the Mass of the Solvent
The mass of the solvent can be derived from the mass of the solution and the mass of the solute:
\[ \text{mass of solvent} = \text{mass of solution} - \text{mass of solute} \]
\[ \text{mass of solvent} = (1000 \rho - M \times M1) \text{ g} \]

### Step 6: Convert Mass of Solvent to Kilograms
Since molality is defined per kilogram of solvent, we convert the mass of the solvent from grams to kilograms:
\[ \text{mass of solvent in kg} = \frac{1000 \rho - M \times M1}{1000} \]

### Step 7: Relate Molarity and Molality
Now we can express molality in terms of molarity:
\[ m = \frac{M}{\frac{1000 \rho - M \times M1}{1000}} \]
\[ m = \frac{M \times 1000}{1000 \rho - M \times M1} \]

### Final Expression
Thus, the expression relating molarity (M) to molality (m) is:
\[ m = \frac{M \times 1000}{1000 \rho - M \times M1} \]

---