Calculate the number of moles of methyl alcohol in 2 molal solutions in 5 L. The density of the solution is 0.981 kg.m^{-3}. Given that, the molar mass of methyl alcohol is 32 g.mol^{-1}.

To calculate the number of moles of methyl alcohol in a 2 molal solution in 5 L, we first need to understand what a 2 molal solution means and then use the given density of the solution and the molar mass of methyl alcohol to find the number of moles.

### Definitions
1. **Molality (m)**: This is defined as the number of moles of solute (in this case, methyl alcohol) per kilogram of solvent (water).
- \( m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \)

2. **Density**: This is the mass of the solution per unit volume.
- Given density \( = 0.981 \text{ kg/dm}^3 \) (or \( 0.981 \text{ kg/L} \))

### Steps to Calculate Moles of Methyl Alcohol

1. **Calculate the mass of the solution**:
Since we have a volume of 5 L and the density of the solution is \(0.981 \text{ kg/L}\):
\[
\text{Mass of solution} = \text{Volume} \times \text{Density} = 5 \text{ L} \times 0.981 \text{ kg/L} = 4.905 \text{ kg}
\]

2. **Find the mass of the solvent (water)**:
In a 2 molal solution, we define the molality with respect to the solvent (water). The molality (2 mol/kg) indicates that there are 2 moles of methyl alcohol for every 1 kg of water.

We can set up the equation for the mass of the solvent (water) used in this solution:
\[
\text{mass of solvent} = \frac{\text{mass of solution}}{1 + m \times \text{molar mass of solute (in kg)}}
\]
Here, \( m = 2 \) (molal concentration) and the molar mass of methyl alcohol (32 g/mol or 0.032 kg/mol):
\[
\text{mass of solvent} = \frac{4.905 \text{ kg}}{1 + 2 \times 0.032 \text{ kg}} = \frac{4.905}{1 + 0.064} = \frac{4.905}{1.064} \approx 4.61 \text{ kg}
\]

3. **Calculate the mass of methyl alcohol**:
The mass of the solution is the sum of the mass of the solvent and the mass of the solute (methyl alcohol):
\[
\text{mass of methyl alcohol} = \text{mass of solution} - \text{mass of solvent} = 4.905 \text{ kg} - 4.61 \text{ kg} \approx 0.295 \text{ kg}
\]

4. **Convert the mass of methyl alcohol to grams**:
\[
\text{mass of methyl alcohol} = 0.295 \text{ kg} \times 1000 \text{ g/kg} = 295 \text{ g}
\]

5. **Calculate the number of moles of methyl alcohol**:
Using the molar mass of methyl alcohol (32 g/mol):
\[
\text{moles of methyl alcohol} = \frac{\text{mass of methyl alcohol}}{\text{molar mass of methyl alcohol}} = \frac{295 \text{ g}}{32 \text{ g/mol}} \approx 9.22 \text{ mol}
\]

### Conclusion
The number of moles of methyl alcohol in the 2 molal solution in 5 L is approximately **9.22 moles**.