To calculate the entropy change ($\Delta S$) for a phase transformation occurring at equilibrium (like boiling at 373 K), we use the relationship between enthalpy and temperature: $\Delta S = \Delta H_{vap} / T$ .

1. Identify the molar mass of water ($H_2O$): The molar mass is the sum of the atomic masses of hydrogen and oxygen, which is approximately $18.02\text{ g/mol}$ .
2. Convert enthalpy of vaporization to molar enthalpy: The given value is $2.257\text{ kJ/g}$. To find the molar enthalpy ($\Delta H_{vap}$ per mole), multiply by the molar mass: $\Delta H_{vap} = 2.257\text{ kJ/g} \times 18.02\text{ g/mol} \approx 40.67\text{ kJ/mol}$ .
3. Calculate the entropy change: Using the boiling point temperature ($T = 373\text{ K}$): $\Delta S = 40.67\text{ kJ/mol} / 373\text{ K} \approx 0.10903\text{ kJ/(mol\cdot K)}$.

For one mole of water, the change in entropy is approximately $0.109\text{ kJ/K}$, which matches Option B .