Temperature dependent equation can be written as:

Options

A. `Ink =InA - E_a/(RT)`
B. `Ink = InA + e^(E_2//RT)`
C. `InK = InA -e^(RT//E_a)`
D. All of these

Answer

To derive the temperature-dependent equation from the Arrhenius equation, we will follow these steps:

### Step-by-Step Solution:

1. **Start with the Arrhenius Equation**:
The Arrhenius equation is given by:
\[ k = A e^{-\frac{E_a}{RT}} \]
where:
- \( k \) = rate constant
- \( A \) = pre-exponential factor
- \( E_a \) = activation energy
- \( R \) = gas constant
- \( T \) = temperature in Kelvin

2. **Take the Natural Logarithm of Both Sides**:
To linearize the equation, we take the natural logarithm (ln) of both sides:
\[ \ln k = \ln(A e^{-\frac{E_a}{RT}}) \]

3. **Apply the Properties of Logarithms**:
Using the property of logarithms that states \(\ln(xy) = \ln x + \ln y\), we can separate the terms:
\[ \ln k = \ln A + \ln e^{-\frac{E_a}{RT}} \]

4. **Simplify the Logarithmic Expression**:
Since \(\ln e^x = x\), we can simplify the second term:
\[ \ln k = \ln A - \frac{E_a}{RT} \]

5. **Final Form of the Temperature-Dependent Equation**:
Rearranging the equation gives us the final form:
\[ \ln k = \ln A - \frac{E_a}{RT} \]

### Conclusion:
Thus, the temperature-dependent equation can be expressed as:
\[ \ln k = \ln A - \frac{E_a}{RT} \]