The logarithmic from of Arrhenius equation is represented by

Options

A. `In k = (E_(a))/(RT)- In A`
B. `In k = In A - (E_(a))/(2.303RT)`
C. `In k = In A - (E_(a))/(RT)`
D. `In k = In A + (E_(a))/(RT)`

Explanation

To derive the logarithmic form of the Arrhenius equation step by step, follow these instructions:

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

### Step 2: Take the Natural Logarithm of Both Sides
To convert the equation into its logarithmic form, take the natural logarithm (ln) of both sides:
\[ \ln K = \ln(A e^{-\frac{E_a}{RT}}) \]

### Step 3: Apply Logarithm Properties
Using the property of logarithms that states \( \ln(ab) = \ln a + \ln b \), we can separate the terms:
\[ \ln K = \ln A + \ln\left(e^{-\frac{E_a}{RT}}\right) \]

### Step 4: Simplify the Logarithm of the Exponential
Using the property \( \ln(e^x) = x \), we can simplify the second term:
\[ \ln K = \ln A - \frac{E_a}{RT} \]

### Step 5: Final Form of the Logarithmic Arrhenius Equation
Thus, the logarithmic form of the Arrhenius equation is:
\[ \ln K = \ln A - \frac{E_a}{RT} \]

### Summary of the Steps
1. Start with the Arrhenius equation: \( K = A e^{-\frac{E_a}{RT}} \).
2. Take the natural logarithm of both sides: \( \ln K = \ln(A e^{-\frac{E_a}{RT}}) \).
3. Use the logarithm property to separate: \( \ln K = \ln A + \ln\left(e^{-\frac{E_a}{RT}}\right) \).
4. Simplify the logarithm of the exponential: \( \ln K = \ln A - \frac{E_a}{RT} \).
5. This gives the final form: \( \ln K = \ln A - \frac{E_a}{RT} \).