Which of the following conditions should be satisfied for the given reaction to be spontaneous at `0^(@)C` and 1 atm ?
`H_(2)O(s) hArr H_(2)O(l)`

Options

A. `Delta H = Delta G`
B. `Delta H lt T Delta S`
C. `Delta H gt T Delta S`
D. `Delta H = T Delta S`

Answer

To determine the conditions for the reaction \( H_2O(s) \rightleftharpoons H_2O(l) \) to be spontaneous at \( 0^\circ C \) and 1 atm, we can analyze the Gibbs free energy change (\( \Delta G \)) for the reaction.

### Step-by-Step Solution:

1. **Understanding the Reaction**:
The reaction involves the phase change of water from solid (ice) to liquid at \( 0^\circ C \). At this temperature, water can exist in both solid and liquid states.

2. **Gibbs Free Energy Equation**:
The spontaneity of a reaction can be assessed using the Gibbs free energy equation:
\[ \Delta G = \Delta H - T \Delta S \]
where:
- \( \Delta G \) = change in Gibbs free energy
- \( \Delta H \) = change in enthalpy
- \( T \) = temperature in Kelvin
- \( \Delta S \) = change in entropy

3. **Conditions for Spontaneity**:
For a reaction to be spontaneous, \( \Delta G \) must be less than or equal to zero (\( \Delta G \leq 0 \)). At equilibrium, \( \Delta G = 0 \), which implies:
\[ \Delta H = T \Delta S \]

4. **Identifying the Values**:
- At \( 0^\circ C \), the temperature in Kelvin is:
\[ T = 0 + 273.15 = 273.15 \, K \]
- The phase change from solid to liquid generally has a positive \( \Delta S \) (increase in entropy) because the liquid state has more disorder than the solid state.

5. **Analyzing the Enthalpy Change**:
The enthalpy change \( \Delta H \) for the melting of ice is positive (endothermic process). Thus, we can summarize:
- \( \Delta H > 0 \)
- \( \Delta S > 0 \)

6. **Condition for Spontaneity**:
For the reaction to be spontaneous at \( 0^\circ C \):
\[ \Delta H \leq T \Delta S \]
Since \( \Delta S \) is positive, if \( \Delta H \) is not too large compared to \( T \Delta S \), the reaction will be spontaneous.

7. **Conclusion**:
Therefore, the condition that should be satisfied for the reaction \( H_2O(s) \rightleftharpoons H_2O(l) \) to be spontaneous at \( 0^\circ C \) and 1 atm is that the enthalpy change must be less than or equal to the product of temperature and entropy change.