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 derive the expression relating molarity (M) and molality (m) of a solution, we will follow these steps:

### Step 1: Define Molarity and Molality
- **Molarity (M)** is defined as the number of moles of solute per liter of solution. Mathematically, it can be expressed as:
\[ M = \frac{n}{V} \]
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. It can be expressed as:
\[ m = \frac{n}{m_{solvent}} \]
where \( m_{solvent} \) is the mass of the solvent in kilograms.

### Step 2: Relate Molarity to Moles of Solute
Using the definition of molarity, we know that in 1 liter (1000 mL) of solution, the number of moles of solute can be expressed as:
\[ n = M \times 1 \text{ L} = M \text{ moles} \]

### Step 3: Calculate the Mass of the Solution
To find the mass of the solution, we can use the density (\( d \)) of the solution:
\[ \text{Mass of solution} = \text{Density} \times \text{Volume} = d \times 1000 \text{ mL} = 1000d \text{ grams} \]

### Step 4: Calculate the Mass of the Solute
To find the mass of the solute, we use the molar mass (\( M_a \)):
\[ \text{Mass of solute} = n \times M_a = M \times M_a \text{ grams} \]

### Step 5: Calculate the Mass of the Solvent
The mass of the solvent can be calculated by subtracting the mass of the solute from the mass of the solution:
\[ \text{Mass of solvent} = \text{Mass of solution} - \text{Mass of solute} \]
Substituting the values we found:
\[ \text{Mass of solvent} = 1000d - M \times M_a \text{ grams} \]

### Step 6: Convert Mass of Solvent to Kilograms
To use the definition of molality, we need the mass of the solvent in kilograms:
\[ \text{Mass of solvent in kg} = \frac{1000d - M \times M_a}{1000} \]

### Step 7: Substitute into the Molality Formula
Now, substituting this into the molality formula:
\[ m = \frac{M}{\frac{1000d - M \times M_a}{1000}} \]
Simplifying this gives:
\[ m = \frac{1000M}{1000d - M \times M_a} \]

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