At `527^(@)`C temperature the activation energy is 54.7 KJ/mole. The value of Arrhenius factor is `4 xx 10^(10)` . The rate constant will be

Options

A. ` 10^(11)`
B. ` 10^(13)`
C. `10^(7)`
D. `10^(-13)`

Answer

To solve the problem, we will use the Arrhenius equation, which relates the rate constant (k) to the activation energy (Ea), the temperature (T), and the Arrhenius factor (A). The equation is given by:

\[ k = A e^{-\frac{E_a}{RT}} \]

Where:
- \( k \) = rate constant
- \( A \) = Arrhenius factor
- \( E_a \) = activation energy (in Joules)
- \( R \) = universal gas constant (8.314 J/(mol·K))
- \( T \) = temperature (in Kelvin)

### Step-by-step Solution:

1. **Convert the temperature from Celsius to Kelvin**:
\[ T = 527^\circ C + 273.15 = 800.15 \approx 800 \, K \]

**Hint**: To convert Celsius to Kelvin, add 273.15 to the Celsius temperature.

2. **Convert the activation energy from kJ/mol to J/mol**:
\[ E_a = 54.7 \, \text{kJ/mol} \times 1000 \, \text{J/kJ} = 54700 \, \text{J/mol} \]

**Hint**: To convert kJ to J, multiply by 1000.

3. **Substitute the values into the Arrhenius equation**:
Using the equation in logarithmic form:
\[ \ln k = \ln A - \frac{E_a}{RT} \]
We can also express it in terms of log base 10:
\[ \log k = \log A - \frac{E_a}{2.303 \cdot R \cdot T} \]

4. **Substituting the known values**:
- \( A = 4 \times 10^{10} \)
- \( R = 8.314 \, \text{J/(mol·K)} \)
- \( T = 800 \, K \)
- \( E_a = 54700 \, \text{J/mol} \)

\[ \log k = \log(4 \times 10^{10}) - \frac{54700}{2.303 \times 8.314 \times 800} \]

5. **Calculate the second term**:
\[ \frac{54700}{2.303 \times 8.314 \times 800} \approx 3.57 \]

6. **Calculate \(\log(4 \times 10^{10})\)**:
\[ \log(4 \times 10^{10}) = \log 4 + \log(10^{10}) = \log 4 + 10 \]
\[ \log 4 \approx 0.602 \quad \Rightarrow \quad \log(4 \times 10^{10}) \approx 0.602 + 10 = 10.602 \]

7. **Combine the results**:
\[ \log k = 10.602 - 3.57 \approx 7.032 \]

8. **Convert back to k**:
\[ k = 10^{7.032} \approx 10^7 \]

Thus, the rate constant \( k \) is approximately \( 10^7 \).

### Final Answer:
The rate constant \( k \) is \( 10^7 \).