How to Calculate Rate Constant Chemistry

Arrhenius Rate Constant Calculator

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Calculation Details:

The rate constant, denoted as $k$, is a crucial coefficient in chemical kinetics that quantifies the speed of a chemical reaction. Unlike a simple rate measurement, the rate constant is specific to a reaction at a specific temperature. Calculating the rate constant allows chemists to predict how fast a reaction will proceed under given conditions.

While the rate constant can be derived experimentally using rate laws, it is theoretically calculated using the Arrhenius Equation. This calculator focuses on the Arrhenius approach, which relates the rate constant to temperature and activation energy.

The Formula: Arrhenius Equation

To calculate the rate constant ($k$) when you know the temperature and the energy barrier of the reaction, you use the Arrhenius equation:

$$k = A \cdot e^{\frac{-E_a}{R \cdot T}}$$

Where:

• $k$ = Rate constant
• $A$ = Pre-exponential factor (or frequency factor). It represents the frequency of collisions with the correct orientation.
• $E_a$ = Activation Energy (usually in J/mol or kJ/mol). This is the minimum energy required for the reaction to occur.
• $R$ = Universal Gas Constant ($8.314 \text{ J}/(\text{mol}\cdot\text{K})$).
• $T$ = Absolute Temperature in Kelvin (K).

Step-by-Step Calculation Guide

Calculating the rate constant manually involves careful handling of units, especially exponential functions. Here is the process used by the calculator above:

1. Convert Temperature to Kelvin

The Arrhenius equation requires absolute temperature. If your data is in Celsius ($^\circ\text{C}$), add 273.15.

$$T_{(K)} = T_{(^\circ\text{C})} + 273.15$$

2. Standardize Energy Units

The gas constant ($R$) is typically given in Joules ($8.314 \text{ J}/\text{mol}\cdot\text{K}$). However, Activation Energy ($E_a$) is often reported in kilojoules (kJ). You must convert $E_a$ to Joules by multiplying by 1000.

$$50 \text{ kJ/mol} = 50,000 \text{ J/mol}$$

3. Calculate the Exponent Term

Divide the activation energy by the product of the gas constant and temperature. Note the negative sign in the formula.

$$\text{Exponent} = \frac{-E_a}{R \cdot T}$$

4. Solve for $k$

Raise the mathematical constant $e$ (approx. 2.718) to the power calculated in step 3, then multiply by the pre-exponential factor ($A$).

Example Calculation

Let's assume a reaction has the following properties:

• Pre-exponential factor ($A$): $1.5 \times 10^{11} \text{ s}^{-1}$
• Activation Energy ($E_a$): $100 \text{ kJ/mol}$
• Temperature ($T$): $300 \text{ K}$

Step 1: Convert $E_a$ to Joules.
$100 \text{ kJ/mol} = 100,000 \text{ J/mol}$

Step 2: Calculate the exponent denominator ($RT$).
$8.314 \times 300 = 2494.2 \text{ J/mol}$

Step 3: Divide $-E_a$ by $RT$.
$-100,000 / 2494.2 \approx -40.09$

Step 4: Calculate $e^{-40.09}$.
$e^{-40.09} \approx 3.89 \times 10^{-18}$

Step 5: Multiply by $A$.
$k = (1.5 \times 10^{11}) \times (3.89 \times 10^{-18}) \approx 5.8 \times 10^{-7} \text{ s}^{-1}$

Understanding the Units of $k$

The numerical value of $k$ changes with temperature, but its units depend on the Reaction Order. The pre-exponential factor ($A$) will share these units.

Reaction Order	Rate Law	Units of $k$
Zero Order	Rate = $k$	$M \cdot s^{-1}$ (Molarity per second)
First Order	Rate = $k[A]$	$s^{-1}$ (per second)
Second Order	Rate = $k[A]^2$	$M^{-1} \cdot s^{-1}$ (per Molarity per second)

Why is $k$ important?

The rate constant connects the concentration of reactants to the actual speed of the reaction. A larger $k$ means a faster reaction. As temperature increases, the $e^{-E_a/RT}$ term increases significantly, leading to a higher $k$ and a faster reaction rate.