Arrhenius Equation Calculator

Calculate the rate constant using the Arrhenius equation:
\[ k = A\,\exp\left(-\frac{E_a}{R\,T}\right) \] where:
– $A$ is the pre-exponential factor, – $E_a$ is the activation energy (J/mol), – $R$ is the gas constant (J/(mol·K)), and – $T$ is the temperature (K).

* Enter all values in SI units. (Default R = 8.314 J/(mol·K))

Step 1: Enter Parameters

Pre-exponential Factor, $ A $:

Example: 1×10¹² (units depend on reaction)

Activation Energy, $ E_a $ (J/mol):

Example: 75000 J/mol

Temperature, $ T $ (K):

Example: 298 K

Gas Constant, $ R $ (J/(mol·K)):

Default: 8.314 J/(mol·K)

Arrhenius Equation Overview

The Arrhenius equation describes how the rate constant of a chemical reaction changes with temperature. It is widely used in chemistry, materials science, and semiconductor processes to understand the temperature dependence of reaction rates, diffusion, and other thermally activated processes.

The Arrhenius Equation

The Arrhenius equation is usually written as:

$$k = A \, e^{-\frac{E_\mathrm{a}}{R \, T}}$$

$k$: the rate constant (or diffusion coefficient, reaction rate, etc., depending on context)
$A$: the pre-exponential factor (also known as the frequency factor)
$E_\mathrm{a}$: the activation energy
$R$: the universal gas constant, typically $8.314\,\text{J}\,\text{mol}^{-1}\,\text{K}^{-1}$
$T$: the absolute temperature (in Kelvin)

As the temperature $T$ increases, the exponential term becomes larger (less negative in the exponent), and thus $k$ increases. This reflects the fact that higher temperatures increase the fraction of molecules with sufficient energy to overcome the activation energy barrier.

Taking the Natural Logarithm

Often, the Arrhenius equation is used in its logarithmic form to extract the activation energy $E_\mathrm{a}$ from experimental data. Taking the natural logarithm of both sides yields:

$$\ln(k) = \ln(A) – \frac{E_\mathrm{a}}{R\,T}.$$

This form is helpful because a plot of $\ln(k)$ versus $1/T$ should give a straight line with slope $-\tfrac{E_\mathrm{a}}{R}$ and $y$-intercept $\ln(A)$. This technique is widely used to determine $E_\mathrm{a}$ from experimental data.

Example: Finding $k$ at a Given Temperature

Suppose you have a reaction whose pre-exponential factor $A$ is $10^{10}\,\mathrm{s}^{-1}$ and its activation energy $E_\mathrm{a}$ is $80\,\mathrm{kJ\,mol^{-1}}$ (i.e., $80,000\,\mathrm{J\,mol^{-1}}$). You want to find $k$ at $T = 500\,\mathrm{K}$.

First, convert all quantities into consistent units. Use $R = 8.314\,\mathrm{J\,mol^{-1}\,K^{-1}}$:

$$k = A \, e^{-\frac{E_\mathrm{a}}{R \, T}} = 10^{10}\,\mathrm{s}^{-1} \,\exp\!\Biggl(-\frac{80,000\,\mathrm{J\,mol^{-1}}}{8.314\,\mathrm{J\,mol^{-1}K^{-1}}\times 500\,\mathrm{K}}\Biggr).$$

Evaluate the exponent:

Denominator = $8.314 \times 500 \approx 4157\,\mathrm{J\,mol^{-1}}$
Exponent = $-\frac{80{,}000}{4157} \approx -19.26$

So,

$$k \approx 10^{10}\,\mathrm{s}^{-1}\,\exp(-19.26) \,\approx 10^{10}\,\mathrm{s}^{-1}\,\times 4.3\times 10^{-9}.$$

Hence,

$$k \approx 4.3\times 10^{1}\,\mathrm{s}^{-1} \,\approx 43\,\mathrm{s}^{-1}.$$

This means at 500 K, the reaction rate constant is about $43\,\mathrm{s}^{-1}$.

Interpretation and Applications

The Arrhenius equation is essential in fields such as:

Chemical Kinetics: Predicting how reaction rates change with temperature
Materials Degradation: Modeling oxidation, corrosion, and other thermally activated processes
Semiconductor Fabrication: Explaining temperature-dependent doping diffusion, film growth rates, etc.
Biological Systems: Rates of enzyme-catalyzed reactions often follow Arrhenius-like behavior

Key Takeaways:

The Arrhenius equation captures how temperature affects reaction or diffusion rates via an exponential dependence.
An activation energy barrier $E_\mathrm{a}$ must be overcome for the process to proceed.
Plots of $\ln(k)$ vs. $1/T$ provide a convenient way to extract $E_\mathrm{a}$ and $A$ experimentally.
Units must be carefully handled for consistency (e.g., Joules vs. kJ, Kelvin for temperature).

Mastering the Arrhenius equation helps you design processes, predict lifetimes of materials, and understand temperature-dependent reaction mechanisms—a cornerstone principle in thermally activated systems.