If water vapour is assumed to be a perfect gas, molar enthalpy change for vaporization of 1 mol of water at 1 bar and $100^{\circ}C$ is 41 $kJ/mol$, then the internal energy change for conversion of 1 mol of water into ice is:

Question

Accepted Answer

To determine the internal energy change for the conversion of 1 mol of water into ice, we need to consider the relationship between enthalpy change ($ \Delta H $) and internal energy change ($ \Delta U $) for a process. The relationship is given by the equation:

$$ \Delta H = \Delta U + \Delta nRT $$

where:
- $ \Delta H $ is the enthalpy change,
- $ \Delta U $ is the internal energy change,
- $ \Delta n $ is the change in the number of moles of gas,
- $ R $ is the universal gas constant (8.314 J/mol·K),
- $ T $ is the temperature in Kelvin.

For the vaporization of water at 1 bar and $ 100^{\circ}C $, the process involves converting liquid water to water vapor. The given molar enthalpy change for this process is 41 kJ/mol. Since the process involves a phase change from liquid to gas, $ \Delta n = 1 $ (as 1 mole of liquid water becomes 1 mole of water vapor).

However, the question asks for the internal energy change for the conversion of 1 mol of water into ice, which is a different process. In this case, the phase change is from liquid to solid, and there is no change in the number of moles of gas ($ \Delta n = 0 $) because both water and ice are in condensed phases.

Therefore, for the conversion of water to ice, the equation simplifies to:

$$ \Delta H = \Delta U $$

Given that the enthalpy change for vaporization is 41 kJ/mol, and considering the process of freezing (which is essentially the reverse of vaporization in terms of energy change), the internal energy change for freezing 1 mol of water into ice is also 41 kJ/mol. This is because the energy required to vaporize water is released when water freezes, assuming ideal conditions and no heat loss.

Now, let's address why the other options are incorrect:

1. **Option 1: $0~kJ~mol^{-1}$** - This option suggests no change in internal energy, which is incorrect because phase changes involve energy changes due to the breaking or forming of intermolecular forces.

2. **Option 3: $3.096~kJ~mol^{-1}$** - This value does not correspond to the typical energy changes associated with phase transitions of water. It is significantly lower than the known enthalpy change for vaporization or freezing.

3. **Option 4: $37.904~kJ/mol$** - This value is close to the given enthalpy change but does not match the exact value of 41 kJ/mol, which is the known enthalpy change for the vaporization of water at the specified conditions.

In conclusion, Option 2 is correct because it accurately reflects the internal energy change for the conversion of 1 mol of water into ice, based on the given enthalpy change for vaporization and the relationship between enthalpy and internal energy.