Molarity and Molality of a solution of a liquid (molecular weight = 50) in aqueous solution is 9 and 18 respectively. What is the density of solution?

Options

A. 1 g/cc
B. `0.95` g/cc
C. `1.05` g/cc
D. `0.66` g/cc

Answer

To find the density of the solution, we will use the definitions of molarity and molality, along with the information provided in the question.

### Step 1: Understand the definitions
- **Molarity (M)** is defined as the number of moles of solute per liter of solution.
- **Molality (m)** is defined as the number of moles of solute per kilogram of solvent.

### Step 2: Given data
- Molarity (M) = 9 mol/L
- Molality (m) = 18 mol/kg
- Molecular weight of the solute = 50 g/mol

### Step 3: Calculate the number of moles of solute
From the molarity, we can find the number of moles in 1 liter (1000 mL) of solution:
\[ \text{Moles of solute} = \text{Molarity} \times \text{Volume (L)} = 9 \, \text{mol/L} \times 1 \, \text{L} = 9 \, \text{moles} \]

### Step 4: Calculate the mass of the solute
Using the molecular weight, we can calculate the mass of the solute:
\[ \text{Mass of solute} = \text{Moles} \times \text{Molecular weight} = 9 \, \text{moles} \times 50 \, \text{g/mol} = 450 \, \text{g} \]

### Step 5: Calculate the mass of the solvent
From the molality, we know that:
\[ \text{Molality} = \frac{\text{Moles of solute}}{\text{Mass of solvent (kg)}} \]
Rearranging this gives us:
\[ \text{Mass of solvent} = \frac{\text{Moles of solute}}{\text{Molality}} = \frac{9 \, \text{moles}}{18 \, \text{mol/kg}} = 0.5 \, \text{kg} = 500 \, \text{g} \]

### Step 6: Calculate the total mass of the solution
The total mass of the solution is the sum of the mass of the solute and the mass of the solvent:
\[ \text{Total mass of solution} = \text{Mass of solute} + \text{Mass of solvent} = 450 \, \text{g} + 500 \, \text{g} = 950 \, \text{g} \]

### Step 7: Calculate the volume of the solution
Since we know the molarity (9 mol/L), we can find the volume of the solution:
\[ \text{Volume of solution} = 1 \, \text{L} = 1000 \, \text{mL} \]

### Step 8: Calculate the density of the solution
Density is defined as mass per unit volume:
\[ \text{Density} = \frac{\text{Total mass of solution}}{\text{Volume of solution}} = \frac{950 \, \text{g}}{1000 \, \text{mL}} = 0.95 \, \text{g/mL} \]

### Final Answer
The density of the solution is **0.95 g/mL**.

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