If for a certain reaction `triangle_rH` is 30KJ `mol^(-1)` at 450K, the value of `triangle_rS` (in `JK^(-1)mol^(-1))` for which the same reaction will be spontaneous at the same temperature is

Options

A. 70
B. `-33`
C. 33
D. `-70`

Answer

To determine the value of \( \Delta_r S \) (change in entropy) for which the reaction will be spontaneous at a temperature of 450 K, we can use the Gibbs free energy equation:

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

For a reaction to be spontaneous, \( \Delta G \) must be less than zero:

\[ \Delta G < 0 \]

This implies:

\[ \Delta H - T \Delta S < 0 \]

Rearranging this inequality gives:

\[ \Delta H < T \Delta S \]

From this, we can express \( \Delta S \):

\[ \Delta S > \frac{\Delta H}{T} \]

### Step 1: Convert \( \Delta H \) to Joules
Given that \( \Delta H = 30 \text{ kJ/mol} \), we need to convert this to Joules:

\[ \Delta H = 30 \text{ kJ/mol} = 30 \times 10^3 \text{ J/mol} = 30000 \text{ J/mol} \]

### Step 2: Substitute the values into the inequality
Now, we substitute \( \Delta H \) and \( T \) into the inequality:

\[ \Delta S > \frac{30000 \text{ J/mol}}{450 \text{ K}} \]

### Step 3: Calculate \( \Delta S \)
Now we perform the calculation:

\[ \Delta S > \frac{30000}{450} \approx 66.67 \text{ J/K/mol} \]

### Step 4: Conclusion
Thus, for the reaction to be spontaneous at 450 K, \( \Delta S \) must be greater than 66.67 J/K/mol.

To ensure spontaneity, we can choose a value greater than this threshold. For instance, if \( \Delta S = 70 \text{ J/K/mol} \), the reaction will be spontaneous.

### Final Answer:
The value of \( \Delta_r S \) for which the reaction will be spontaneous at 450 K is:

\[ \Delta_r S > 66.67 \text{ J/K/mol} \]