A quantitative relationship between the temperature and rate constant is given by :

Options

A. Nernst equation
B. Arrhenius equation
C. van't Hoff equation
D. Henderson equation

Explanation

To derive the quantitative relationship between temperature and the rate constant, we can follow these steps:

### Step 1: Understand the Arrhenius Equation
The Arrhenius equation describes how the rate constant (k) of a chemical reaction depends on temperature (T) and activation energy (Ea). The equation is given by:

\[ k = A e^{-\frac{E_a}{RT}} \]

where:
- \( k \) = rate constant
- \( A \) = Arrhenius factor (frequency factor)
- \( E_a \) = activation energy
- \( R \) = universal gas constant (8.314 J/(mol·K))
- \( T \) = temperature in Kelvin

### Step 2: Analyze the Components
In the Arrhenius equation:
- The term \( e^{-\frac{E_a}{RT}} \) represents the fraction of molecules that have enough energy to overcome the activation energy barrier.
- As temperature (T) increases, the value of \( \frac{E_a}{RT} \) decreases, leading to an increase in the exponential term, and thus an increase in the rate constant \( k \).

### Step 3: Implications of the Equation
From the Arrhenius equation, we can conclude that:
- The rate constant \( k \) increases with an increase in temperature.
- The relationship is exponential, meaning small increases in temperature can lead to significant increases in the rate constant.

### Step 4: Conclusion
The quantitative relationship between temperature and the rate constant is encapsulated in the Arrhenius equation. This equation allows chemists to predict how changes in temperature will affect the speed of a reaction.

### Summary
The Arrhenius equation provides a quantitative relationship between temperature and the rate constant of a reaction, showing that as temperature increases, the rate constant also increases due to the exponential factor related to activation energy.

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