Which reaction, with the following values of `Delta H`, `Delta S`, at 400 K is spontaneous and endothermic?

Options

A. `Delta H = -48 kJ, Delta S = + 135 J/K`
B. `Delta H = -48 kJ, Delta S = - 135 J/K`
C. `Delta H = + 48 kJ, Delta S = + 135 J/K`
D. `Delta H = +48 kJ, Delta S = -135 J/K`

Answer

To determine which reaction is spontaneous and endothermic at 400 K, we will use the Gibbs free energy equation:

\[ \Delta G = \Delta H - T \Delta S \]

### Step 1: Identify the conditions for spontaneity and endothermic nature
- A reaction is **endothermic** if \(\Delta H > 0\).
- A reaction is **spontaneous** if \(\Delta G < 0\).

### Step 2: Write the Gibbs free energy equation
We will use the equation:

\[ \Delta G = \Delta H - T \Delta S \]

### Step 3: Substitute the known values
Given:
- Temperature \(T = 400 \, K\)
- \(\Delta H = 48 \, kJ = 48000 \, J\) (since we need to convert kJ to J)
- We will assume \(\Delta S\) is given in J/K.

### Step 4: Analyze the entropy (\(\Delta S\))
For the reaction to be spontaneous and endothermic, we need to ensure that the term \(T \Delta S\) is greater than \(\Delta H\) (to make \(\Delta G\) negative).

Assuming \(\Delta S = 135 \, J/K\):

### Step 5: Calculate \(T \Delta S\)
\[ T \Delta S = 400 \, K \times 135 \, J/K = 54000 \, J \]

### Step 6: Substitute into the Gibbs free energy equation
Now substitute the values into the Gibbs free energy equation:

\[ \Delta G = 48000 \, J - 54000 \, J \]

### Step 7: Calculate \(\Delta G\)
\[ \Delta G = 48000 - 54000 = -6000 \, J \]

### Step 8: Analyze the result
Since \(\Delta G = -6000 \, J < 0\), the reaction is spontaneous.

### Conclusion
The reaction is endothermic (since \(\Delta H > 0\)) and spontaneous (since \(\Delta G < 0\)) at 400 K.

### Final Answer
The reaction with \(\Delta H = 48 \, kJ\) and \(\Delta S = 135 \, J/K\) is spontaneous and endothermic at 400 K.

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