Arrhenius Equation Calculator
Solve k = A e^(-Ea/RT) for the rate constant at a given temperature, or find activation energy Ea from two rate constant / temperature pairs.
🌡️ What is the Arrhenius Equation?
The Arrhenius equation, k = A e^(-Ea/RT), is the fundamental relationship between a chemical reaction's rate constant k and its temperature T. Proposed by Svante Arrhenius in 1889, it explains why reaction rates almost always increase with temperature, and quantifies exactly how much, using two physically meaningful parameters: the activation energy Ea (the energy barrier reactants must overcome) and the pre-exponential factor A (the maximum possible rate constant if every collision succeeded).
This equation is used constantly across chemistry and chemical engineering: predicting how much faster a reaction runs at a higher process temperature, estimating shelf life and spoilage rates of food and pharmaceuticals through accelerated aging tests at elevated temperature, determining safe operating temperatures for exothermic industrial reactors, and modeling corrosion and material degradation rates, since many non-reaction physical processes also follow Arrhenius-like temperature dependence.
A common misconception is that the Arrhenius equation implies reaction rate scales linearly with temperature. It does not; the relationship is exponential, which is why chemists often use the rough rule that reaction rate roughly doubles for every 10°C temperature rise, and why the k vs T curve on this calculator bends sharply upward rather than following a straight line.
This calculator handles both directions of the problem: given A, Ea, and T, it finds k directly (Solve k mode); given two experimentally measured (k, T) pairs, it finds Ea and A using the linearized two-point form of the equation (Solve Ea mode), the same method used to analyze real kinetics data from a series of rate experiments at different temperatures.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Rate constant at room temperature (Solve k)
A = 1 × 1013 s−1, Ea = 75 kJ/mol, T = 298 K - a typical moderate-barrier reaction near room temperature.
Example 2 — Lower activation energy at 300 K (Solve k)
A = 3.2 × 109 s−1, Ea = 45 kJ/mol, T = 300 K - a faster reaction with a smaller energy barrier.
Example 3 — Finding Ea from two rate measurements (Solve Ea)
k1 = 2.5 × 10−4 at T1 = 280 K; k2 = 3.0 × 10−3 at T2 = 320 K.
Example 4 — A slow reaction over a wide temperature range (Solve Ea)
k1 = 4.0 × 10−5 at T1 = 275 K; k2 = 6.0 × 10−4 at T2 = 315 K.
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Arrhenius equation?
The Arrhenius equation is k = A e^(-Ea/RT), where k is the rate constant, A is the pre-exponential (frequency) factor, Ea is the activation energy in joules per mole, R is the gas constant (8.314 J/(mol·K)), and T is absolute temperature in kelvin. It describes how strongly reaction rate depends on temperature: even a modest activation energy makes k rise sharply as T increases.
What is activation energy?
Activation energy (Ea) is the minimum energy that colliding molecules must have for a reaction to proceed, essentially the height of the energy barrier between reactants and products. A high Ea means only a small fraction of molecular collisions have enough energy to react at a given temperature, making the reaction slow and highly temperature-sensitive; a low Ea means the reaction proceeds readily even at modest temperatures.
What is the pre-exponential factor A?
The pre-exponential factor A, also called the frequency factor, represents the rate constant a reaction would have if every collision had enough energy to react (Ea = 0), accounting for collision frequency and molecular orientation. It shares the same units as the rate constant k. In the two-point (Solve Ea) mode, A is back-calculated from one of the two (k, T) data pairs once Ea is known.
How do you find activation energy from two temperatures?
Using the two-point form of the Arrhenius equation: ln(k2/k1) = -(Ea/R) x (1/T2 - 1/T1). Rearranged for Ea: Ea = -R x ln(k2/k1) / (1/T2 - 1/T1). This only needs two rate constants measured at two different temperatures, no separate measurement of A is required.
Why does temperature affect reaction rate so strongly?
Reaction rate depends exponentially on temperature through the e^(-Ea/RT) term, not linearly. A relatively small increase in T can produce a large increase in the fraction of molecules with enough energy to clear the activation barrier, which is why a common chemistry rule of thumb says reaction rate roughly doubles for every 10°C rise, though the exact factor depends on Ea.
What units should Ea and T be in for this calculator?
Enter Ea in kilojoules per mole (kJ/mol), the calculator converts internally to joules per mole for the R = 8.314 J/(mol·K) constant. Temperature T must always be in kelvin (K), never Celsius or Fahrenheit, since the exponential term requires an absolute temperature scale.
Can the Arrhenius equation give a rate constant greater than the pre-exponential factor A?
No. Since the exponential term e^(-Ea/RT) is always between 0 and 1 for any positive Ea and T, the calculated rate constant k can never exceed A. As T approaches infinity, k approaches A as its theoretical upper limit.
What does the ln(k) vs 1/T Arrhenius plot show?
Taking the natural log of both sides of k = A e^(-Ea/RT) gives ln(k) = ln(A) - (Ea/R)(1/T), a straight line when ln(k) is plotted against 1/T. The slope of that line is -Ea/R, so a steeper (more negative) slope means a higher activation energy, and the y-intercept gives ln(A). This linearization is the standard way chemists extract Ea and A from multiple experimental (k, T) measurements.
Is the Arrhenius equation accurate for all reactions?
It is a very good approximation for the vast majority of elementary reactions over a moderate temperature range, but it can break down for reactions with a strongly temperature-dependent A (some radical and enzyme-catalyzed reactions), or at very wide temperature ranges where a modified form (the Arrhenius-like equation with a T^n prefactor) fits better.
How is the Arrhenius equation used in real chemistry and engineering?
It is used to predict how fast a reaction runs at temperatures not directly tested, to estimate shelf life and degradation rates of pharmaceuticals and food (accelerated aging studies), to design industrial reactor operating temperatures, and to model corrosion rates and material fatigue, since many degradation processes also follow Arrhenius-like temperature dependence.
What happens to the rate constant k if activation energy Ea is zero?
If Ea = 0, the exponential term e^(-Ea/RT) equals 1 regardless of temperature, so k = A at every temperature, meaning the reaction rate becomes independent of temperature. Real reactions with a true activation barrier always have Ea greater than 0, so this is mostly a useful limiting case for understanding the equation rather than a physically common scenario.
Why do I need two (k, T) pairs instead of one to find Ea?
A single (k, T) measurement has two unknowns, A and Ea, and one equation cannot solve for both simultaneously. A second (k, T) pair measured at a different temperature provides the second independent equation needed to solve for both Ea and A uniquely, which is exactly what the Solve Ea mode does.