Which reaction, with the following values of `triangleH, triangleS` at 400 K is spontaneous and endot hermic?

Options

A. `triangleH=-48kJ, triangleS=+135J//K`
B. `triangleH=-48kJ, triangleS=-135J//K`
C. `triangleH=+48kJ, triangleS=+135J//K`
D. `triangleH=-48kJ, triangleS=-135J//K`

Answer

To determine which reaction is spontaneous and endothermic at 400 K, we can follow these steps:

### Step 1: Understand the Definitions
- **Endothermic Reaction**: A reaction that absorbs heat, characterized by a positive change in enthalpy (ΔH > 0).
- **Spontaneous Reaction**: A reaction that occurs without external intervention, characterized by a negative change in Gibbs free energy (ΔG < 0).

### Step 2: Use the Gibbs Free Energy Equation
The Gibbs free energy change (ΔG) can be calculated using the formula:
\[ \Delta G = \Delta H - T\Delta S \]
Where:
- ΔG = change in Gibbs free energy
- ΔH = change in enthalpy
- T = temperature in Kelvin
- ΔS = change in entropy

### Step 3: Identify the Conditions
We need to find a reaction where:
- ΔH is positive (endothermic)
- ΔG is negative (spontaneous)

### Step 4: Analyze the Given Values
Assume we have the following values for four different reactions at 400 K:
1. Reaction A: ΔH = -50 kJ, ΔS = 100 J/K
2. Reaction B: ΔH = -40 kJ, ΔS = 150 J/K
3. Reaction C: ΔH = +48 kJ, ΔS = 135 J/K
4. Reaction D: ΔH = +48 kJ, ΔS = -135 J/K

### Step 5: Convert Units
Since ΔH is given in kJ and ΔS in J/K, we need to convert ΔS to kJ/K by dividing by 1000:
- For Reaction C: ΔS = 135 J/K = 0.135 kJ/K
- For Reaction D: ΔS = -135 J/K = -0.135 kJ/K

### Step 6: Calculate ΔG for Each Reaction
1. **Reaction A**:
\[ \Delta G_A = -50 - 400 \times \frac{100}{1000} = -50 - 40 = -90 \text{ kJ (spontaneous)} \]
2. **Reaction B**:
\[ \Delta G_B = -40 - 400 \times \frac{150}{1000} = -40 - 60 = -100 \text{ kJ (spontaneous)} \]
3. **Reaction C**:
\[ \Delta G_C = 48 - 400 \times 0.135 = 48 - 54 = -6 \text{ kJ (spontaneous)} \]
4. **Reaction D**:
\[ \Delta G_D = 48 - 400 \times (-0.135) = 48 + 54 = 102 \text{ kJ (non-spontaneous)} \]

### Step 7: Conclusion
From the calculations, we find that:
- Reaction C is the only reaction that is both endothermic (ΔH > 0) and spontaneous (ΔG < 0).

Thus, the answer is **Option C**.