In which of the following reactions,standard reaction entropy change`(DeltaS^(@))`is positive and standard Gibb,s energy change`(DeltaG^(@))`decreases sharply with increasing temperature?

Options

A. `Mg(s)+(1)/(2)O_(2)(g)rarrMgO(s)`
B. `(1)/(2)C("graphite")+(1)/(2)O_(2)(g)rarr(1)/(2)CO_(2)(g)`
C. `C("graphite")+(1)/(2)O_(2)(g)rarrCO(g)`
D. `CO(g)+(1)/(2)O_(2)(g)rarrCO_(2)(g)`

Explanation

To determine which reaction has a positive standard reaction entropy change (ΔS°) and a standard Gibbs energy change (ΔG°) that decreases sharply with increasing temperature, we can analyze each option provided.

### Step-by-Step Solution:

1. **Understanding the Terms**:
- **Entropy Change (ΔS°)**: This measures the change in disorder or randomness in a system. A positive ΔS° indicates an increase in disorder.
- **Gibbs Free Energy Change (ΔG°)**: This indicates the spontaneity of a reaction. A negative ΔG° means the reaction is spontaneous. The relationship is given by the equation:
\[ ΔG° = ΔH° - TΔS° \]
- Here, ΔH° is the change in enthalpy, T is the temperature in Kelvin, and ΔS° is the change in entropy.

2. **Analyzing Each Reaction**:
- **Option A**: \( \text{Mg} + \frac{1}{2} \text{O}_2 \rightarrow \text{MgO} \)
- Reactants: 1 mole of Mg (solid) + 0.5 moles of O2 (gas) = 0.5 gaseous moles.
- Products: 1 mole of MgO (solid) = 0 gaseous moles.
- **ΔS°**: Negative (decrease in gaseous moles).
- **Conclusion**: Not the answer.

- **Option B**: \( \text{C (s)} + \text{O}_2 (g) \rightarrow \text{CO}_2 (g) \)
- Reactants: 1 mole of C (solid) + 1 mole of O2 (gas) = 1 gaseous mole.
- Products: 1 mole of CO2 (gas) = 1 gaseous mole.
- **ΔS°**: Zero (no change in gaseous moles).
- **Conclusion**: Not the answer.

- **Option C**: \( \frac{1}{2} \text{N}_2 (g) + \frac{3}{2} \text{H}_2 (g) \rightarrow \text{NH}_3 (g) \)
- Reactants: 0.5 moles of N2 + 1.5 moles of H2 = 2 gaseous moles.
- Products: 1 mole of NH3 = 1 gaseous mole.
- **ΔS°**: Negative (decrease in gaseous moles).
- **Conclusion**: Not the answer.

- **Option D**: \( \text{C (s)} + \text{H}_2 (g) \rightarrow \text{C}_2\text{H}_2 (g) \)
- Reactants: 1 mole of C (solid) + 1 mole of H2 (gas) = 1 gaseous mole.
- Products: 1 mole of C2H2 (gas) = 1 gaseous mole.
- **ΔS°**: Zero (no change in gaseous moles).
- **Conclusion**: Not the answer.

3. **Identifying the Correct Option**:
- The only option that shows a positive ΔS° and a decrease in ΔG° with increasing temperature is **Option C**.

### Final Answer:
The correct reaction is **Option C**.