Why does the rate of a reaction increase with rise in temperature?

Explanation

### Step-by-Step Solution

1. **Understanding Reaction Rate**: The rate of a chemical reaction is defined as the speed at which reactants are converted into products. It can be influenced by various factors, including temperature.

2. **Effect of Temperature on Molecular Motion**: As temperature increases, the kinetic energy of the molecules also increases. This means that the molecules move faster and collide more frequently with one another.

3. **Activation Energy**: For a reaction to occur, reactant molecules must collide with sufficient energy to overcome the activation energy barrier. Activation energy is the minimum energy required for a reaction to take place.

4. **Increased Fraction of Molecules with Sufficient Energy**: At higher temperatures, a larger fraction of the molecules will have enough energy to overcome the activation energy barrier. This is because more molecules are moving with higher kinetic energy, allowing them to achieve the necessary energy for the reaction.

5. **Arrhenius Equation**: The relationship between the rate constant (k) and temperature (T) is described by the Arrhenius equation:
\[ k = A e^{-\frac{E_a}{RT}} \]
where:
- \( k \) is the rate constant,
- \( A \) is the pre-exponential factor,
- \( E_a \) is the activation energy,
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin.

6. **Impact of Temperature on the Rate Constant**: When temperature increases, the term \(-\frac{E_a}{RT}\) becomes less negative, leading to an increase in the value of \( e^{-\frac{E_a}{RT}} \). This results in an increase in the rate constant \( k \).

7. **Direct Relationship Between Rate and Rate Constant**: The rate of a reaction is directly proportional to the rate constant \( k \). Therefore, as \( k \) increases with temperature, the rate of the reaction also increases.

8. **Conclusion**: In summary, the rate of a reaction increases with a rise in temperature because more molecules have sufficient energy to overcome the activation energy barrier, leading to a higher rate constant and thus a higher reaction rate.