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 solve the question regarding the temperature-dependent equation, we will derive the Arrhenius equation step by step.

### Step-by-Step Solution:

1. **Understand the Arrhenius Equation**:
The Arrhenius equation describes how the rate constant (k) of a chemical reaction depends on temperature (T). It is given by the formula:
\[ k = A e^{-\frac{E_a}{RT}} \]
where:
- \( k \) = rate constant
- \( A \) = pre-exponential factor (frequency factor)
- \( E_a \) = activation energy
- \( R \) = universal 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:
\[ \ln k = \ln A + \ln \left(e^{-\frac{E_a}{RT}}\right) \]
Using the property of logarithms that states \(\ln(e^x) = x\), we can simplify this further.

3. **Simplify the Equation**:
The equation becomes:
\[ \ln k = \ln A - \frac{E_a}{RT} \]

4. **Final Form of the Temperature-Dependent Equation**:
Thus, the temperature-dependent equation can be expressed as:
\[ \ln k = \ln A - \frac{E_a}{RT} \]
This is the required form of the temperature-dependent equation.

5. **Identify the Correct Option**:
From the options provided, the correct expression is:
\[ \ln k = \ln A - \frac{E_a}{RT} \]
This corresponds to the first option given in the question.

### Conclusion:
The temperature-dependent equation can be written as:
\[ \ln k = \ln A - \frac{E_a}{RT} \]