What is the Arrhenius Equation?
The Arrhenius equation is a fundamental formula in chemical kinetics that describes how the rate constant k of a reaction depends on temperature. Proposed by Swedish chemist Svante Arrhenius in 1889, it remains one of the most widely used equations in both chemistry and chemical engineering:
where:
- k — the rate constant, with units depending on the reaction order (e.g., s⁻¹ for first-order, M⁻¹s⁻¹ for second-order)
- A — the pre-exponential factor (also called the frequency factor or Arrhenius constant), in the same units as k
- e ≈ 2.71828 — Euler's number (base of the natural logarithm)
- Eₐ — the activation energy of the reaction, in J/mol
- R = 8.3145 J/(K·mol) — the universal gas constant
- T — the absolute temperature in Kelvin (K)
What is the Meaning of Activation Energy Eₐ?
The activation energy Eₐ is the minimum energy that colliding molecules must possess for a reaction to occur. Think of it as the energy "barrier" that must be overcome:
- The product R·T gives the average thermal (kinetic) energy of the molecules at temperature T in J/mol.
- The ratio Eₐ/(R·T) is a dimensionless number representing how large the barrier is relative to the available thermal energy.
- The Boltzmann factor e−Eₐ/(RT) equals the fraction of molecular collisions that have enough energy to overcome the activation barrier.
- A smaller Eₐ/(R·T) means more successful collisions → faster reaction. If Eₐ/(R·T) = 0.5, twice as many collisions succeed compared to Eₐ/(R·T) = 1.
Typical activation energies range from ~10 kJ/mol (very fast reactions) to ~200 kJ/mol (very slow reactions) at room temperature.
The Pre-exponential Factor A
The pre-exponential factor A (or frequency factor) accounts for the frequency of molecular collisions and the fraction of collisions with the correct orientation for reaction. It represents the maximum possible rate constant — the value k would reach if the activation energy were zero (T → ∞). In practice, A is determined experimentally from kinetics data or estimated using collision theory.
The Arrhenius Equation with the Boltzmann Constant kB
An alternative form of the equation uses the Boltzmann constant kB = 1.381 × 10⁻²³ J/K (energy per molecule) instead of the gas constant R (energy per mole):
In this form, Eₐ is expressed in joules per molecule (not per mole). The two forms are equivalent since R = NA · kB, where NA = 6.022 × 10²³ mol⁻¹ (Avogadro's number).
The Arrhenius Equation in Logarithmic Form
Taking the natural logarithm of both sides gives the linear (Arrhenius plot) form:
This is the equation of a straight line when ln(k) is plotted against 1/T (the Arrhenius graph). The slope equals −Eₐ/R and the y-intercept equals ln(A). This graphical method is the classic way to determine Eₐ and A experimentally from rate constant measurements at different temperatures.
Comparing Rate Constants at Two Temperatures
To find how much faster a reaction is at temperature T₂ compared to T₁, divide the two Arrhenius expressions:
A common rule of thumb: for biological systems with Eₐ ≈ 50 kJ/mol, a 10 °C increase approximately doubles the reaction rate (Q₁₀ ≈ 2).
Arrhenius Equation Example
Suppose a first-order reaction has:
- A = 1.0 × 10¹³ s⁻¹
- Eₐ = 75,000 J/mol = 75 kJ/mol
- T = 298 K (25 °C, room temperature)
Then: Eₐ/(R·T) = 75,000 / (8.3145 × 298) = 30.27
k = 1.0 × 10¹³ × e−30.27 = 1.0 × 10¹³ × 7.6 × 10⁻¹⁴ ≈ 0.76 s⁻¹
At 50 °C (323 K): Eₐ/(R·T) = 75,000 / (8.3145 × 323) = 27.95
k ≈ 1.0 × 10¹³ × e−27.95 ≈ 7.2 s⁻¹ — about 9.5× faster!
Unit Systems Supported
This Arrhenius equation calculator supports both metric (SI) and American/imperial unit systems:
- Energy: J/mol (SI), kJ/mol (SI), cal/mol (American/historical), kcal/mol (common in biochemistry)
- Temperature: K (Kelvin, SI), °C (Celsius, metric), °F (Fahrenheit, American)
- Rate constant k and factor A: dimensionless input (units depend on reaction order)
The calculator works with any combination: enter your values in the most convenient units and get all results including unit conversions.