What is activation energy?

Activation energy (Eₐ) is the minimum energy that reacting molecules must collectively possess at the moment of collision for a reaction to occur. It is the height of the energy barrier separating reactants from products on a potential-energy surface. Reactions with high Eₐ (typically above 100 kJ/mol) are slow at room temperature; those with low Eₐ (below 40 kJ/mol) can be fast even at 0 °C. Catalysts lower Eₐ by providing an alternative reaction pathway.

What is the Arrhenius equation?

The Arrhenius equation is k = A·exp(−Eₐ/(R·T)), where k is the rate constant, A is the pre-exponential (frequency) factor (same units as k), Eₐ is the activation energy in J/mol, R = 8.314 J/(mol·K) is the gas constant, and T is the absolute temperature in Kelvin. Taking the natural logarithm gives the linear form ln(k) = ln(A) − Eₐ/(R·T), which is the basis of the two-point and graphical methods.

How does the two-point method work?

If you measure k at two different temperatures T₁ and T₂, you can eliminate the unknown A by writing the Arrhenius equation for each temperature and subtracting: ln(k₂/k₁) = −(Eₐ/R)·(1/T₂ − 1/T₁). Rearranging gives Eₐ = −R·ln(k₂/k₁) / (1/T₂ − 1/T₁). This is the most common experimental method for determining activation energies in undergraduate labs.

What is the pre-exponential factor A?

The pre-exponential factor A (also called the frequency factor or collision frequency) represents the maximum possible rate constant if every collision were energetically successful — in other words, the rate constant at infinite temperature. For simple bimolecular gas reactions A is on the order of 10¹⁰ to 10¹³ L/mol·s. It encodes the collision frequency and a steric factor (the fraction of correctly oriented collisions). In Eyring–Polanyi theory A is replaced by the more fundamental (kB·T/h)·exp(ΔS‡/R).

What is the Eyring–Polanyi equation?

The Eyring–Polanyi (transition-state theory) equation is k = (kB·T/h)·exp(−ΔG‡/(R·T)), where kB = 1.381×10⁻²³ J/K is the Boltzmann constant, h = 6.626×10⁻³⁴ J·s is the Planck constant, and ΔG‡ = ΔH‡ − T·ΔS‡ is the Gibbs energy of activation. Unlike the empirical Arrhenius equation, it derives the rate constant from the thermodynamic properties of the transition state and gives mechanistic insight: a highly negative ΔS‡ indicates a tight, ordered transition state (common in association reactions), while a positive ΔS‡ suggests bond breaking or a loose transition state.

How do I convert Kelvin and Celsius?

T(K) = T(°C) + 273.15. Always enter temperature in Kelvin in this calculator. Room temperature is approximately 298 K (25 °C), body temperature is about 310 K (37 °C), and the boiling point of water is 373 K (100 °C). Never use Celsius directly in the Arrhenius or Eyring equations — the exponential is extremely sensitive to temperature, and using Celsius instead of Kelvin introduces very large errors.

Activation Energy Calculator

Activation energy is the concept at the heart of chemical kinetics. It explains why reactions speed up with temperature, why enzymes make life possible, and why you can store petrol in a tank without it exploding: the reactant molecules need enough energy to climb over the barrier before products can form.

This calculator covers four interconnected tools:

Arrhenius equation — solve for k, Eₐ, A, or T in any direction
Two-point Eₐ extraction — find activation energy from rate constants measured at two temperatures
First-order half-life — convert between k and t₁/₂ for radioactive decay, drug clearance, and unimolecular reactions
Eyring–Polanyi — transition-state theory rate constant from ΔH‡ and ΔS‡

Every mode shows the full substitution so you can follow the arithmetic step by step.

How it works

The Arrhenius equation

The Arrhenius equation (1889) quantifies how the rate constant k depends on temperature:

k = A · exp(−Eₐ / (R·T))

where A is the pre-exponential factor (same units as k, often s⁻¹ for unimolecular reactions), Eₐ is the activation energy in J/mol, R = 8.314 J/(mol·K), and T is in Kelvin. Taking the natural log linearises the equation:

ln k = ln A − Eₐ/(R·T)

A plot of ln k versus 1/T is a straight line with slope −Eₐ/R — the classic Arrhenius plot.

Two-point method

Writing the Arrhenius equation at two temperatures and subtracting eliminates ln A:

ln(k₂/k₁) = −(Eₐ/R) · (1/T₂ − 1/T₁)

This is the standard undergraduate method: measure k at a low and a high temperature, plug both pairs in, and recover Eₐ without needing to know A at all.

First-order half-life

For any first-order reaction (or first-order process such as radioactive decay):

t₁/₂ = ln(2) / k ≈ 0.6931 / k

The half-life is independent of concentration — only k matters. Given t₁/₂ the reverse gives k = ln(2) / t₁/₂.

Eyring–Polanyi (transition-state theory)

Going deeper than the empirical Arrhenius equation, transition-state theory expresses k in terms of the activation enthalpy ΔH‡ and activation entropy ΔS‡:

k = (kB·T / h) · exp(−ΔG‡ / (R·T))

where ΔG‡ = ΔH‡ − T·ΔS‡ is the Gibbs energy of activation, kB = 1.381×10⁻²³ J/K, and h = 6.626×10⁻³⁴ J·s. The prefactor kB·T/h ≈ 6.25×10¹² s⁻¹ at 298 K sets the universal speed limit for any elementary step.

Worked example

Problem: The rate constant for the thermal decomposition of N₂O₅ is k = 3.46×10⁻⁵ s⁻¹ at 25 °C (298 K) and k = 1.35×10⁻³ s⁻¹ at 45 °C (318 K). Find Eₐ.

Using the two-point formula:

ln(k₂/k₁) = ln(1.35×10⁻³ / 3.46×10⁻⁵) = ln(39.02) ≈ 3.664
1/T₂ − 1/T₁ = 1/318 − 1/298 = 3.145×10⁻³ − 3.356×10⁻³ = −2.11×10⁻⁴ K⁻¹
Eₐ = −R · 3.664 / (−2.11×10⁻⁴) = 8.314 × 17 360 ≈ 144.3 kJ/mol

Enter k₁ = 3.46e-5, T₁ = 298, k₂ = 1.35e-3, T₂ = 318 in the Two-Point tab and the calculator gives the same answer with full working shown.

Formula reference

Equation	Formula	Constants used
Arrhenius	k = A·exp(−Eₐ/(R·T))	R = 8.314 J/mol·K
Two-point Eₐ	Eₐ = −R·ln(k₂/k₁)/(1/T₂ − 1/T₁)	R = 8.314 J/mol·K
First-order half-life	t₁/₂ = ln(2)/k	—
Eyring–Polanyi	k = (kB·T/h)·exp(−ΔG‡/(RT))	kB = 1.381×10⁻²³ J/K, h = 6.626×10⁻³⁴ J·s
Gibbs of activation	ΔG‡ = ΔH‡ − T·ΔS‡	—

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