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

Options

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

Explanation

To solve the question regarding the Arrhenius equation, we will analyze the equation and its implications step by step.

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

### Step 2: Analyze the Effect of Activation Energy
From the equation, we observe that the rate constant \( k \) is affected by the activation energy \( E_a \). Specifically, the term \( e^{-\frac{E_a}{RT}} \) indicates that:
- As \( E_a \) increases, the exponent becomes more negative, leading to a smaller value of \( k \).
- Conversely, as \( E_a \) decreases, the exponent becomes less negative, leading to a larger value of \( k \).

### Step 3: Analyze the Effect of Temperature
The Arrhenius equation also shows that:
- \( k \) is directly proportional to \( e^{-\frac{1}{T}} \).
- As the temperature \( T \) increases, the value of \( k \) increases because the negative exponent decreases in magnitude.

### Step 4: Evaluate the Options
Now, let's evaluate the options provided in the question:
1. **Rate constant increases exponentially with increasing activation energy.**
- This is incorrect. \( k \) decreases with increasing \( E_a \).

2. **Rate constant decreases exponentially with increasing activation energy.**
- This is correct. \( k \) decreases as \( E_a \) increases.

3. **Rate constant increases exponentially with decreasing activation energy and decreasing temperature.**
- This is incorrect. While \( k \) does increase with decreasing \( E_a \), it does not increase with decreasing temperature.

4. **Rate constant increases exponentially with decreasing activation energy and increasing temperature.**
- This is correct. \( k \) increases with both decreasing \( E_a \) and increasing \( T \).

### Conclusion
The correct option is:
- **Rate constant increases exponentially with decreasing activation energy and increasing temperature.**