The density of 2.45 M aqueous methanol `(CH_(3)OH)` is `0.976 g/mL`. What is the molatiy of the solution `(CH_(3)OH=32)` ?

Options

A. `27.3 m`
B. `0.273 m`
C. `7.23 m`
D. `2.73 m`

Answer

To find the molality of the aqueous methanol solution, we can follow these steps:

### Step 1: Understand the given data
- Molarity (M) of methanol solution = 2.45 M
- Density of the solution = 0.976 g/mL
- Molar mass of methanol (CH₃OH) = 32 g/mol

### Step 2: Calculate the mass of the solution
Since the density is given in g/mL, we can find the mass of 1 liter (1000 mL) of the solution:
\[ \text{Mass of solution} = \text{Density} \times \text{Volume} = 0.976 \, \text{g/mL} \times 1000 \, \text{mL} = 976 \, \text{g} \]

### Step 3: Calculate the number of moles of methanol
Using the molarity, we can find the number of moles of methanol in 1 liter of solution:
\[ \text{Moles of methanol} = \text{Molarity} \times \text{Volume} = 2.45 \, \text{mol/L} \times 1 \, \text{L} = 2.45 \, \text{mol} \]

### Step 4: Calculate the mass of methanol
Now, we can calculate the mass of methanol using its molar mass:
\[ \text{Mass of methanol} = \text{Moles} \times \text{Molar mass} = 2.45 \, \text{mol} \times 32 \, \text{g/mol} = 78.4 \, \text{g} \]

### Step 5: Calculate the mass of water (solvent)
To find the mass of the solvent (water), we subtract the mass of methanol from the total mass of the solution:
\[ \text{Mass of water} = \text{Mass of solution} - \text{Mass of methanol} = 976 \, \text{g} - 78.4 \, \text{g} = 897.6 \, \text{g} \]

### Step 6: Calculate the molality of the solution
Molality (m) is defined as the number of moles of solute per kilogram of solvent. First, we convert the mass of water from grams to kilograms:
\[ \text{Mass of water in kg} = \frac{897.6 \, \text{g}}{1000} = 0.8976 \, \text{kg} \]
Now, we can calculate the molality:
\[ \text{Molality} = \frac{\text{Moles of methanol}}{\text{Mass of water in kg}} = \frac{2.45 \, \text{mol}}{0.8976 \, \text{kg}} \approx 2.73 \, \text{mol/kg} \]

### Final Answer
The molality of the solution is approximately **2.73 mol/kg**.

---