Consider the Arrhenius equation given below and mark the correct option.
`k=A e^(-(Ea)/(RT)`

Options

A. Rate constant increase exponentially with increasing activation and decreasing temperature
B. Rate constant decreases exponentially with increasing activation energy and decreasing temperature.
C. Rate constant increase exponentailly with decreasing activation energy and decreasing temperature.
D. Rate constant increases exponentially with decreasing activation energy increasing temperature

Explanation

To solve the problem regarding the Arrhenius equation and determine the correct option, we will analyze the equation step by step.

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

### Step 2: Analyze the Effect of Activation Energy (\( E_a \))
- The term \( e^{-\frac{E_a}{RT}} \) indicates that as the activation energy \( E_a \) increases, the exponent becomes more negative.
- This results in a smaller value for \( e^{-\frac{E_a}{RT}} \), which means that \( k \) (the rate constant) decreases.

### Step 3: Analyze the Effect of Temperature (\( T \))
- Conversely, if the temperature \( T \) increases, the value of \( \frac{E_a}{RT} \) decreases, making the exponent less negative.
- This results in a larger value for \( e^{-\frac{E_a}{RT}} \), which means that \( k \) increases.

### Step 4: Summarize Relationships
From the analysis:
- **Increasing Activation Energy (\( E_a \))**: Decreases \( k \) (rate constant).
- **Increasing Temperature (\( T \))**: Increases \( k \) (rate constant).

### Step 5: Evaluate the Options
Now, let's evaluate the options provided in the question:
1. Rate constant increases exponentially with increasing activation energy.
- **Incorrect**: \( k \) decreases with increasing \( E_a \).

2. Rate constant decreases exponentially with increasing activation energy and decreasing temperature.
- **Correct**: \( k \) decreases with increasing \( E_a \) and decreases with decreasing \( T \).

3. Rate constant increases exponentially with decreasing activation energy and decreasing temperature.
- **Incorrect**: \( k \) increases with decreasing \( E_a \) but decreases with decreasing \( T \).

4. Rate constant increases exponentially with decreasing activation energy and increasing temperature.
- **Correct**: \( k \) increases with decreasing \( E_a \) and increases with increasing \( T \).

### Conclusion
The correct options are **2 and 4**.