Chemistry

Arrhenius Equation Calculator | Activation Energy, Rate Constant & Half-Life

Compute the Arrhenius rate constant k, activation energy Ea, pre-exponential factor A, and reaction half-life using k = Ae^(−Ea/RT). Given two rate constants at two temperatures, back-calculates Ea. Shows temperature effect on rate with a comparison table.

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Ea UNITS

kJ/molJ/mol

TEMPERATURE UNITS

Kelvin°C

PRESET: N₂O₅ DECOMPOSITION

Pre-exponential factor A (s⁻¹)

Activation energy Ea (kJ/mol)

Temperature T (K)

Rate constant k (s⁻¹)

4.3886e-5

Half-life t½ (s)

1.5794e+4

First-order: t½ = ln2/k

RATE CONSTANT vs TEMPERATURE

T (K)	T (°C)	k (s⁻¹)	k / k₂₉₈
273	0	9.7486e-7	0.022
298	25	4.3886e-5	1.000
310	37	2.1939e-4	4.999
323	50	1.0959e-3	24.971
373	100	1.8736e-1	4269.168
400	127	1.7633e+0	40179.196
500	227	8.6398e+2	19687127.829

What Is the Arrhenius Equation Calculator | Activation Energy, Rate Constant & Half-Life?

The Arrhenius equation, proposed by Svante Arrhenius in 1889, describes how the rate constant k of a chemical reaction depends on temperature. It provides a quantitative basis for the empirical observation that reaction rates roughly double every 10°C.

The activation energy Ea is the minimum energy reactant molecules must possess to form the transition state and complete a reaction. The pre-exponential factor A (also called the frequency factor or collision frequency) represents how often reactant molecules collide with the correct orientation, regardless of energy.

The exponential factor exp(−Ea/RT) gives the fraction of molecular collisions with enough energy to overcome the activation barrier. At higher temperatures, this fraction increases rapidly, explaining the strong temperature dependence of reaction rates. An Arrhenius plot (ln k vs 1/T) gives a straight line with slope −Ea/R, allowing Ea to be determined experimentally.

Formula

Arrhenius Equation:    k = A · exp(−Ea / (R · T))

Two-Point Form:        ln(k₁/k₂) = (Ea/R) · (1/T₂ − 1/T₁)
Solving for Ea:        Ea = R · ln(k₁/k₂) / (1/T₂ − 1/T₁)
First-order half-life: t½ = ln(2) / k

Constants:
  R = 8.314 J/(mol·K)    (universal gas constant)
  A = pre-exponential (frequency) factor (same units as k)
  Ea = activation energy (J/mol or kJ/mol)
  T  = absolute temperature (K)

How to Use

1
Select mode: Compute k to find the rate constant at a given temperature, Two-point to extract activation energy from two measurements, or Temperature Table to see k across a range.
2
Choose activation energy units (kJ/mol or J/mol) and temperature units (K or °C).
3
Click the N₂O₅ decomposition preset to load A=4.98×10¹³ s⁻¹, Ea=103 kJ/mol.
4
For Compute k: enter A, Ea, and T. Read k (s⁻¹) and first-order half-life t½.
5
For Two-point: enter k₁ at T₁ and k₂ at T₂. The calculator returns Ea (kJ/mol and kcal/mol) and A.
6
Check the temperature table for k at standard temperatures from 273 K (0°C) to 500 K.

Choose Compute k (forward), Two-Point (find Ea from two rate constants), or Temperature Table. Enter values and read results. Switch Ea units between kJ/mol and J/mol and temperature between K and °C.

Example Calculation

Problem: Decomposition of N₂O₅: A = 4.98×10¹³ s⁻¹, Ea = 103 kJ/mol. Find k at 25°C and 50°C, and the ratio k₅₀/k₂₅.

Solution at 25°C (298 K):

k = 4.98×10¹³ × exp(−103,000 / (8.314 × 298)) = 4.98×10¹³ × exp(−41.58) ≈ 4.87×10⁻⁵ s⁻¹

t½ = ln(2) / 4.87×10⁻⁵ ≈ 14,230 s ≈ 3.95 h

Solution at 50°C (323 K):

k = 4.98×10¹³ × exp(−103,000 / (8.314 × 323)) ≈ 1.70×10⁻³ s⁻¹

Ratio k₅₀/k₂₅ ≈ 34.9 — rate increases ~35× over a 25°C interval.

Understanding Arrhenius Equation | Activation Energy, Rate Constant & Half-Life

Activation Energies of Common Reactions

Reaction	Ea (kJ/mol)	A (s⁻¹)	Notes
N₂O₅ decomposition	103	4.98×10¹³	Gas phase, first-order
H₂ + I₂ → 2HI	165	2.45×10¹⁰	Bimolecular gas phase
Protein denaturation (egg)	200–300	—	Biological irreversible
Diamond → graphite	~350	—	Extremely slow at 25°C
Sucrose hydrolysis	108	—	Acid-catalyzed
O₃ decomposition	113	9.4×10¹¹	Atmospheric chemistry

Limitations and Extensions of the Arrhenius Equation

›The Arrhenius equation assumes Ea and A are temperature-independent; this holds over moderate temperature ranges but fails at extremes.
›Transition State Theory (Eyring equation) refines the model: k = (kBT/h)·exp(−ΔG‡/RT), separating enthalpic (ΔH‡) and entropic (ΔS‡) contributions.
›Quantum tunnelling allows proton-transfer reactions to proceed below the classical barrier, causing Arrhenius plots to curve at low temperatures.
›Enzyme kinetics show an apparent Arrhenius behaviour up to the denaturation temperature, then activity drops precipitously.
›Marcus theory handles outer-sphere electron transfer, where Ea depends on reorganisation energy λ and driving force ΔG°.

Frequently Asked Questions

What does activation energy Ea physically represent?

Ea is the minimum energy required for reactant molecules to collide and convert into products via the transition state. It is the height of the energy barrier on the reaction coordinate diagram. A catalyst lowers Ea by providing an alternative lower-energy pathway, dramatically increasing k without changing the thermodynamics (ΔG) of the reaction.

How do I use the two-point Arrhenius equation to find Ea?

Measure k at two temperatures T₁ and T₂. Then: Ea = R × ln(k₁/k₂) / (1/T₂ − 1/T₁). Make sure temperatures are in Kelvin. Note that the sign and magnitude depend on the ratio k₁/k₂ and which temperature is larger. The two-point form is an approximation that improves with a larger temperature difference.

What is the pre-exponential factor A?

A (also called the frequency factor or Arrhenius factor) is the rate constant extrapolated to infinite temperature where the Boltzmann factor equals 1. It encodes collision frequency and steric probability. For unimolecular reactions A is typically 10¹³–10¹⁵ s⁻¹; for bimolecular reactions the units are L·mol⁻¹·s⁻¹ with similar magnitudes.

Why does the rule of thumb say rate doubles every 10°C?

For a typical activation energy of ~50 kJ/mol, computing k at T+10 versus T near 300 K gives a ratio of about 1.8–2.4. This arises from the exponential sensitivity of the Boltzmann factor to temperature. The doubling rule is a rough guide; actual ratios depend strongly on Ea — reactions with high Ea show much greater temperature sensitivity.

How does catalysis affect the Arrhenius parameters?

A catalyst provides an alternative reaction pathway with lower activation energy. In Arrhenius terms, Ea decreases significantly. A may also change if the catalyst alters the steric factor or collision geometry. The equilibrium constant K is unchanged because the catalyst lowers both forward and reverse barriers equally, but k_forward and k_reverse both increase.

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