Calculate the moles of methanol in 5 litres of its 2m solution, if the density of the solution is 0.981 kg `L^(-1)`. (Molar mass = 32.0 g `"mol"^(-1)`).

Explanation

To calculate the moles of methanol in a 5-liter solution of 2m (2 molal) concentration, we can follow these steps:

### Step 1: Calculate the mass of the solution
We know the volume of the solution and its density. The mass of the solution can be calculated using the formula:
\[ \text{Mass of solution} = \text{Volume} \times \text{Density} \]
Given:
- Volume = 5 L
- Density = 0.981 kg/L

Calculating the mass:
\[ \text{Mass of solution} = 5 \, \text{L} \times 0.981 \, \text{kg/L} = 4.905 \, \text{kg} \]

### Step 2: Convert the mass of the solution to grams
Since we need the mass in grams, we convert kilograms to grams:
\[ \text{Mass of solution in grams} = 4.905 \, \text{kg} \times 1000 \, \text{g/kg} = 4905 \, \text{g} \]

### Step 3: Calculate the mass of methanol in the solution
In a 2m (2 molal) solution, the number of moles of solute (methanol) is defined as:
\[ \text{moles of solute} = \frac{\text{mass of solute (g)}}{\text{molar mass of solute (g/mol)}} \]
For a 2m solution, this means there are 2 moles of methanol per 1000 g of solvent.

First, we need to find the mass of the solvent in the solution. The total mass of the solution is the mass of the solute plus the mass of the solvent.

Let’s denote:
- Mass of solvent = \( m_s \)
- Mass of solute (methanol) = \( m_{methanol} \)

From the definition of molality (m):
\[ m = \frac{\text{moles of solute}}{\text{mass of solvent (kg)}} \]
Since we have 2 moles of methanol in 1000 g (1 kg) of solvent:
\[ 2 = \frac{m_{methanol}}{1} \]
Thus, for 2 moles of methanol, the mass of methanol is:
\[ m_{methanol} = 2 \, \text{moles} \times 32.0 \, \text{g/mol} = 64.0 \, \text{g} \]

### Step 4: Calculate the total mass of the solvent in the solution
The total mass of the solution is:
\[ \text{Mass of solution} = \text{mass of solute} + \text{mass of solvent} \]
We can rearrange this to find the mass of the solvent:
\[ \text{mass of solvent} = \text{mass of solution} - \text{mass of solute} \]
Using the values we calculated:
\[ \text{mass of solvent} = 4905 \, \text{g} - 64.0 \, \text{g} = 4841 \, \text{g} \]

### Step 5: Calculate the number of moles of methanol in the solution
Now we can find the number of moles of methanol in the 5-liter solution:
\[ \text{moles of methanol} = \frac{\text{mass of methanol}}{\text{molar mass of methanol}} = \frac{64.0 \, \text{g}}{32.0 \, \text{g/mol}} = 2 \, \text{moles} \]

### Conclusion
Thus, the total number of moles of methanol in the 5-liter solution is:
\[ \text{Total moles of methanol} = 2 \, \text{moles} \]