If `DeltaH` for a reaction has a positive value, how would you know the sign requirement of `DeltaS` for it so that the reaction is spontaneous?

Answer

To determine the sign requirement of ΔS (entropy change) for a reaction with a positive ΔH (enthalpy change) to be spontaneous, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Gibbs Free Energy Equation**:
The Gibbs free energy (ΔG) is defined by the equation:
\[ \Delta G = \Delta H - T \Delta S \]
where:
- ΔG = change in Gibbs free energy
- ΔH = change in enthalpy
- T = absolute temperature (in Kelvin)
- ΔS = change in entropy

2. **Identify the Given Information**:
We are given that ΔH is positive (ΔH > 0). This means that the reaction is endothermic, absorbing heat from the surroundings.

3. **Determine the Condition for Spontaneity**:
For a reaction to be spontaneous, ΔG must be negative (ΔG < 0). Therefore, we need to analyze the equation:
\[ \Delta G = \Delta H - T \Delta S < 0 \]

4. **Rearranging the Inequality**:
Rearranging the inequality gives us:
\[ T \Delta S > \Delta H \]
This means that the term \(T \Delta S\) must be greater than ΔH for ΔG to be negative.

5. **Analyzing the Sign of ΔS**:
Since ΔH is positive, for the inequality \(T \Delta S > \Delta H\) to hold true, ΔS must also be positive (ΔS > 0). This is because if ΔS were negative, then \(T \Delta S\) would also be negative, making it impossible for \(T \Delta S\) to exceed a positive ΔH.

6. **Conclusion**:
Therefore, for a reaction with a positive ΔH to be spontaneous, ΔS must be positive.

### Final Answer:
For a reaction with a positive ΔH, ΔS must be positive for the reaction to be spontaneous.

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