Calculate the number of moles of methanol in `4` litres in its `2m` solution, if the density of the solution `0.981kgL^(-1)` (molar mass of methanol `=32g "mol"^(-1)`)

Explanation

To calculate the number of moles of methanol in a 4-liter solution with a concentration of 2 molarity (2m), we can follow these steps:

### Step 1: Understand the Definition of Molarity
Molarity (M) is defined as the number of moles of solute (in this case, methanol) per liter of solution. A 2m solution means there are 2 moles of methanol in 1 liter of solution.

### Step 2: Calculate the Total Moles of Methanol in 4 Liters
Since the solution is 2m, we can calculate the total number of moles of methanol in 4 liters using the formula:
\[ \text{Total moles of methanol} = \text{Molarity} \times \text{Volume of solution (in liters)} \]
Substituting the values:
\[ \text{Total moles of methanol} = 2 \, \text{moles/L} \times 4 \, \text{L} = 8 \, \text{moles} \]

### Step 3: Verify the Density and Mass of the Solution
Given the density of the solution is 0.981 kg/L, we can find the total mass of the solution:
\[ \text{Mass of solution} = \text{Density} \times \text{Volume} \]
Substituting the values:
\[ \text{Mass of solution} = 0.981 \, \text{kg/L} \times 4 \, \text{L} = 3.924 \, \text{kg} \]

### Step 4: Calculate the Mass of Methanol
We know that in a 2m solution, there are 2 moles of methanol in 1 kg of solution. Therefore, for 3.924 kg of solution:
\[ \text{Mass of methanol} = 2 \, \text{moles} \times 32 \, \text{g/mol} = 64 \, \text{g} \]
Since 1 kg of solution contains 64 g of methanol, we can calculate the total mass of methanol in 3.924 kg of solution:
\[ \text{Total mass of methanol} = \left(\frac{64 \, \text{g}}{1 \, \text{kg}}\right) \times 3.924 \, \text{kg} = 64 \times 3.924 = 250.176 \, \text{g} \]

### Step 5: Convert Mass of Methanol to Moles
To find the number of moles of methanol, we convert grams to moles using the molar mass:
\[ \text{Number of moles of methanol} = \frac{\text{Mass of methanol}}{\text{Molar mass of methanol}} = \frac{250.176 \, \text{g}}{32 \, \text{g/mol}} \approx 7.37 \, \text{moles} \]

### Final Answer
The number of moles of methanol in 4 liters of a 2m solution is approximately **7.37 moles**.