Step 1: Understand what is asked. We are told the mole fraction of the solute is $0.02$. We have to find the molality of the solution. Molality means moles of solute present in every $1$ kg of solvent.
Step 2: Use the mole fraction to fix the moles. Mole fraction of solute is $x_{solute} = \dfrac{n_{solute}}{n_{solute}+n_{solvent}}$. The easiest trick is to take a total of $1$ mole of the mixture. Then moles of solute become $0.02$ and moles of water become $1-0.02 = 0.98$.
Step 3: Find the mass of the solvent. The solvent is water, whose molar mass is $18$ g per mole. So the mass of water is $0.98 \times 18 = 17.64$ g.
Step 4: Change grams into kilograms. Molality needs the solvent mass in kilograms. So $17.64$ g $= \dfrac{17.64}{1000} = 0.01764$ kg.
Step 5: Apply the molality formula. \[ m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} = \frac{0.02}{0.01764} \]
Step 6: Do the division and conclude. $\dfrac{0.02}{0.01764} \approx 1.133$. So the molality is about $1.133$ m. \[ \boxed{1.133 \text{ m}} \]