Standard entropy of A, B and C are 30, 60 and 100 `JK^-1` `mol^-1` respectively. For the reaction `(2A + 5Brarr 6C)`, If the AH of the reaction is 300 kJ then the temperature at which the reaction will become spontaneous is

Options

A. 1150 K
B. 1240 K
C. 1000 K
D. 1260 K

Answer

To determine the temperature at which the reaction \(2A + 5B \rightarrow 6C\) becomes spontaneous, we need to calculate the change in Gibbs free energy (\(\Delta G\)) and find the temperature at which \(\Delta G\) becomes negative.

### Step 1: Calculate the standard entropy change (\(\Delta S\)) for the reaction

The standard entropy change can be calculated using the formula:

\[ \Delta S = S_{\text{products}} - S_{\text{reactants}} \]

Given the standard entropies:
- \(S_A = 30 \, \text{JK}^{-1}\text{mol}^{-1}\)
- \(S_B = 60 \, \text{JK}^{-1}\text{mol}^{-1}\)
- \(S_C = 100 \, \text{JK}^{-1}\text{mol}^{-1}\)

For the reaction:
- Products: \(6C\)
- Reactants: \(2A + 5B\)

Calculating the entropy for products and reactants:

\[ S_{\text{products}} = 6 \times S_C = 6 \times 100 = 600 \, \text{JK}^{-1} \]
\[ S_{\text{reactants}} = 2 \times S_A + 5 \times S_B = 2 \times 30 + 5 \times 60 = 60 + 300 = 360 \, \text{JK}^{-1} \]

Now, substituting into the \(\Delta S\) formula:

\[ \Delta S = 600 - 360 = 240 \, \text{JK}^{-1} \]

### Step 2: Convert \(\Delta S\) to kJ

Since \(\Delta H\) is given in kJ, we need to convert \(\Delta S\) from J to kJ:

\[ \Delta S = 240 \, \text{JK}^{-1} = \frac{240}{1000} = 0.24 \, \text{kJK}^{-1} \]

### Step 3: Use the Gibbs free energy equation

The Gibbs free energy change is given by:

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

Where:
- \(\Delta H = 300 \, \text{kJ}\)
- \(\Delta S = 0.24 \, \text{kJK}^{-1}\)

### Step 4: Set \(\Delta G\) to zero for spontaneity

To find the temperature at which the reaction becomes spontaneous, we set \(\Delta G = 0\):

\[ 0 = 300 - T \times 0.24 \]

Rearranging gives:

\[ T \times 0.24 = 300 \]

\[ T = \frac{300}{0.24} = 1250 \, \text{K} \]

### Step 5: Conclusion

The temperature at which the reaction becomes spontaneous is:

\[ T \approx 1250 \, \text{K} \]