Following are the values of `E_(a)` and `DeltaH` for three reactions carried out at the same temperature `:`
`I :E_(a)=20kJ mol^(-1),DeltaH=-60kJmol^(-1)`
`II :E_(a)=10kJ mol^(-1),DeltaH=-20kJ mol^(-1)`
`III : E_(a)=40 kJ mol^(-1),DeltaH=+15kJ mol^(-1)`
If all the three reaction have same frequency factor then fastest and slowest reaction are `:`

Options

A. `{:("Faster","Slowest"),(I,II):}`
B. `{:("Faster","Slowest"),(II,III):}`
C. `{:("Faster","Slowest"),(I,III):}`
D. can't be predicted

Explanation

To determine the fastest and slowest reactions among the three given reactions based on their activation energies (Ea), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Activation Energies**:
- For Reaction I: \( E_a = 20 \, \text{kJ mol}^{-1} \)
- For Reaction II: \( E_a = 10 \, \text{kJ mol}^{-1} \)
- For Reaction III: \( E_a = 40 \, \text{kJ mol}^{-1} \)

2. **Understand the Relationship Between Activation Energy and Reaction Rate**:
- The rate of a chemical reaction is inversely related to the activation energy. This means that a lower activation energy results in a faster reaction rate, while a higher activation energy results in a slower reaction rate.

3. **Compare the Activation Energies**:
- Reaction I: \( E_a = 20 \, \text{kJ mol}^{-1} \) (moderate)
- Reaction II: \( E_a = 10 \, \text{kJ mol}^{-1} \) (fastest)
- Reaction III: \( E_a = 40 \, \text{kJ mol}^{-1} \) (slowest)

4. **Determine the Fastest and Slowest Reactions**:
- The fastest reaction is the one with the lowest activation energy, which is Reaction II with \( E_a = 10 \, \text{kJ mol}^{-1} \).
- The slowest reaction is the one with the highest activation energy, which is Reaction III with \( E_a = 40 \, \text{kJ mol}^{-1} \).

5. **Conclusion**:
- Fastest Reaction: Reaction II
- Slowest Reaction: Reaction III

### Final Answer:
- Fastest Reaction: **II**
- Slowest Reaction: **III**