The enthalpy of vaporisation of water at `100^(@)C` is `40.63 KJ mol^(-1)`. The value `Delta E` for the process would be :-

Options

A. `"37.53 kJ mol"^(-1)`
B. `"39.08 kJ mol"^(-1)`
C. `"42.19 kJ mol"^(-1)`
D. `"43.73 kJ mol"^(-1)`

Answer

To find the value of ΔE (change in internal energy) for the vaporization of water at 100°C, we can use the relationship between enthalpy change (ΔH) and internal energy change (ΔE). The equation we will use is:

\[ \Delta H = \Delta E + \Delta N_g RT \]

Where:
- ΔH is the enthalpy of vaporization,
- ΔE is the change in internal energy,
- ΔN_g is the change in the number of moles of gas,
- R is the universal gas constant (8.314 J/(K·mol)),
- T is the temperature in Kelvin.

### Step-by-step Solution:

1. **Identify the given values**:
- ΔH (enthalpy of vaporization of water at 100°C) = 40.63 kJ/mol
- Convert ΔH to J/mol for consistency:
\[ 40.63 \text{ kJ/mol} = 40.63 \times 10^3 \text{ J/mol} = 40630 \text{ J/mol} \]
- R = 8.314 J/(K·mol)
- Temperature, T = 100°C = 373 K (since T(K) = T(°C) + 273)

2. **Calculate ΔN_g**:
- For the vaporization of water (H2O), the reaction can be represented as:
\[ H_2O_{(l)} \rightarrow H_2O_{(g)} \]
- In this reaction, the change in the number of moles of gas (ΔN_g) is:
- Products: 1 mole of H2O(g)
- Reactants: 0 moles of H2O(g)
- Therefore, ΔN_g = 1 - 0 = 1.

3. **Substitute values into the equation**:
- Rearranging the equation to solve for ΔE:
\[ \Delta E = \Delta H - \Delta N_g RT \]
- Substitute the known values:
\[ \Delta E = 40630 \text{ J/mol} - (1) \times (8.314 \text{ J/(K·mol)}) \times (373 \text{ K}) \]

4. **Calculate the second term**:
- Calculate \(RT\):
\[ RT = 8.314 \times 373 = 3100.622 \text{ J/mol} \]

5. **Final calculation**:
- Now substitute back into the equation:
\[ \Delta E = 40630 \text{ J/mol} - 3100.622 \text{ J/mol} \]
\[ \Delta E = 37529.378 \text{ J/mol} \]
- Convert back to kJ/mol:
\[ \Delta E = 37.53 \text{ kJ/mol} \]

### Conclusion:
The value of ΔE for the vaporization of water at 100°C is **37.53 kJ/mol**.