In 1889 Svante Arrhenius proposed that the rate constant of a reaction grows exponentially with temperature: k = A·e^(−Ea/RT). The exponential term is the fraction of collisions that carry at least the activation energy Ea; the pre-exponential factor A collects everything else – how often molecules meet and how often they meet in the right orientation. R is the gas constant, 8.314 J/(mol·K), and T is the absolute temperature.
When Ea is quoted per molecule (in joules or electronvolts) rather than per mole, the same equation uses the Boltzmann constant kʙ = 1.380649 × 10⁻²³ J/K instead of R. The two forms give identical rate constants because R = kʙ·N_A.
Taking logarithms turns the equation into a straight line, ln k = ln A − (Ea/R)(1/T). Plotting ln k against 1/T – the Arrhenius plot – therefore gives a line of slope −Ea/R and intercept ln A, which is how activation energies are measured. This calculator solves for whichever quantity you leave out and draws that plot (or k against T) across the temperature range you choose, marking your point on it.
How to use the Arrhenius Equation Calculator
- Solve for
- Form of the equation
- Pre-exponential factor A
- Activation energy Ea (per mole)
- Activation energy Ea (per molecule)
- Temperature T
- Rate constant k
- Chart
- Chart from temperature
- Chart up to temperature
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Choose what to solve for and the form of the equation, then enter the other three quantities. k and A share a unit (s⁻¹ for first-order reactions; for other orders the concentration part is the same for both). Temperatures are converted to kelvin before use.
Worked example
A first-order reaction has A = 1.0 × 10¹³ s⁻¹ and Ea = 100 kJ/mol. At 25 °C (298.15 K), Ea/RT = 100 000 ÷ (8.3145 × 298.15) = 40.34, so k = 10¹³ × e^(−40.34) ≈ 3.03 × 10⁻⁵ s⁻¹ (a half-life of about 6.4 hours). At 35 °C the same reaction has k ≈ 1.12 × 10⁻⁴ s⁻¹, 3.7 times faster.
Working backwards, if you measured k = 3.0 × 10⁻⁵ s⁻¹ at 25 °C for a reaction with A = 10¹³ s⁻¹, solving for Ea gives −RT·ln(k/A) ≈ 100.0 kJ/mol.
Reading the Arrhenius plot
On the ln k vs 1000/T chart, hot temperatures are on the left (small 1/T) and cold on the right. A steep line means a large activation energy and a reaction that is very sensitive to temperature; a flat line means a small barrier. The y-intercept at 1/T = 0 is ln A. If your own data points bend away from a straight line, a single Ea does not describe the reaction over that whole range.
Tips
- Check the equation is balanced before using coefficients.
- Find the limiting reagent before calculating a yield.
- Keep temperatures in kelvin for rate and equilibrium equations.
- Everything is calculated in your browser.
Common mistakes to avoid
- Using the excess reagent to calculate theoretical yield.
- Including pure solids and liquids in equilibrium expressions.
- Entering temperatures in °C into the Arrhenius equation without converting.
Formula
k = A·e^(−Ea/(R·T)) · per molecule: k = A·e^(−Ea/(kʙ·T)) · linear form: ln k = ln A − (Ea/R)·(1/T)Worked example
Inputs — Solve for: Rate constant k; Form of the equation: Per mole – k = A·e^(−Ea/RT); Pre-exponential factor A: 10,000,000,000,000 s⁻¹; Activation energy Ea (per mole): 100 kJ/mol; Temperature T: 298.15 K; Chart: Arrhenius plot – ln k against 1/T; Chart from temperature: 273.15 K; Chart up to temperature: 373.15 K.
Results — Rate constant k = 0.00003; Pre-exponential factor A = 10,000,000,000,000; Activation energy Ea = 100; Temperature T = 298.15; Activation energy per molecule = 1.6605e-19; ln k (k in s⁻¹) = -10.4059.
Key terms
- Rate constant (k)
- Proportionality constant in the rate law; its value depends on temperature.
- Pre-exponential factor (A)
- Collision-frequency and orientation term in the Arrhenius equation.
- Activation energy (Ea)
- The energy barrier between reactants and the transition state.
- Arrhenius plot
- Graph of ln k against 1/T; a straight line with slope −Ea/R.
- Boltzmann constant (kʙ)
- 1.380649 × 10⁻²³ J/K, the gas constant per molecule.
Frequently asked questions
What does the Arrhenius equation tell you?
How the rate constant depends on temperature: k grows exponentially as T rises, and the higher the activation energy, the steeper the growth.
What are the units of A?
The same as k: s⁻¹ for first-order, M⁻¹·s⁻¹ for second-order and M·s⁻¹ for zero-order reactions.
Why must temperature be in kelvin?
The equation uses absolute temperature; 25 °C must be entered as 298.15 K. The unit menu converts °C and °F for you.
What is the difference between the R and kʙ forms?
R = 8.314 J/(mol·K) is used when Ea is per mole; kʙ = 1.380649 × 10⁻²³ J/K when Ea is per molecule. Both give the same k.
How much faster is a reaction at 35 °C than at 25 °C?
For Ea = 100 kJ/mol and A = 10¹³ s⁻¹, k rises from 3.03 × 10⁻⁵ to 1.12 × 10⁻⁴ s⁻¹ – about 3.7 times faster.
How do I find Ea and A from experimental data?
Measure k at several temperatures, plot ln k against 1/T and fit a straight line: Ea = −slope × R and A = e^intercept.
Does the Arrhenius equation always hold?
It works well over modest temperature ranges for most elementary reactions. Curved Arrhenius plots signal a change of mechanism, tunnelling or a temperature-dependent A.
What formula does the Arrhenius Equation Calculator use?
k = A·e^(−Ea/(R·T)) · per molecule: k = A·e^(−Ea/(kʙ·T)) · linear form: ln k = ln A − (Ea/R)·(1/T).