# Arrhenius Equation Calculator

> Uses the two-point Arrhenius equation ln(k2/k1) = (Ea/R)(1/T1 − 1/T2) to find the rate constant at a second temperature and the rate ratio from the activation energy, or the activation energy from rate constants measured at two temperatures, plus the pre-exponential factor A.

- Calculator id: `arrhenius-equation` · Category: Chemistry (`chemistry`) · Tool name: `solve_arrhenius_equation`
- Canonical page: https://tttkmbb.com/chemistry/arrhenius-equation · This document: https://tttkmbb.com/chemistry/arrhenius-equation.md · JSON definition: https://tttkmbb.com/chemistry/arrhenius-equation.json

## Purpose

Uses the two-point Arrhenius equation ln(k2/k1) = (Ea/R)(1/T1 − 1/T2) to find the rate constant at a second temperature and the rate ratio from the activation energy, or the activation energy from rate constants measured at two temperatures, plus the pre-exponential factor A.

**Use when:** You need how much faster a reaction runs at a different temperature, the rate constant at a new temperature, or the activation energy from two measured rate constants.

**Do not use when:** The rate constant is unknown at both temperatures and only Ea and A are known (compute k = A·exp(−Ea/RT) directly), or the reaction changes mechanism between the temperatures.

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `activation_energy_kj_per_mol` | number | kJ/mol | optional | Activation energy in kJ/mol. Leave empty to solve it from two rate constants. (> 0, max 5000) |
| `temperature_1` | number |  | required | First temperature in temperature_unit. |
| `temperature_2` | number |  | required | Second temperature in temperature_unit. |
| `temperature_unit` | enum: celsius \| kelvin |  | optional, default "celsius" | Unit of the temperature inputs; the calculation uses kelvin (K = °C + 273.15). |
| `rate_constant_1` | number |  | optional | Rate constant at temperature_1 in any unit (s⁻¹, L/(mol·s)); k2 is returned in the same unit. (> 0) |
| `rate_constant_2` | number |  | optional | Rate constant at temperature_2 (same unit as k1); needed to solve Ea. (> 0) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `activation_energy_kj_per_mol` | number | kJ/mol | Ea, given or solved. |
| `activation_energy_j_per_mol` | number | J/mol | Ea in joules per mole. |
| `rate_ratio_k2_over_k1` | number |  | exp((Ea/R)(1/T1 − 1/T2)): factor by which the rate changes from T1 to T2. |
| `rate_constant_1` | number |  | Rate constant at T1 (unit of the input). |
| `rate_constant_2` | number |  | Rate constant at T2 (unit of the input). |
| `pre_exponential_factor` | number |  | A = k1 × exp(Ea / (R·T1)), same unit as k (only when k1 is known). |
| `temperature_1_k` | number | K | First temperature in kelvin. |
| `temperature_2_k` | number | K | Second temperature in kelvin. |
| `solved_for` | string |  | rate_constant_2, rate_constant_1, rate_ratio_k2_over_k1 or activation_energy_kj_per_mol. |

## Formula

`k = A × exp(−Ea / (R × T)); ln(k2 / k1) = (Ea / R) × (1/T1 − 1/T2) with R = 8.314462618 J/(mol·K) and T in kelvin; Ea = R × ln(k2 / k1) / (1/T1 − 1/T2); A = k1 × exp(Ea / (R × T1))`

Assumes Ea and A are independent of temperature over the range considered, which holds well for intervals of a few tens of kelvin.

## Data Sources

- OpenStax Chemistry 2e – 12.5 Collision Theory — https://openstax.org/books/chemistry-2e/pages/12-5-collision-theory (textbook, retrieved 2026-09-24)
- Wikipedia – Arrhenius equation — https://en.wikipedia.org/wiki/Arrhenius_equation (reference, retrieved 2026-09-24)
- NIST – CODATA internationally recommended values of the fundamental physical constants (molar gas constant R = 8.314462618 J/(mol·K), exact) — https://physics.nist.gov/cuu/Constants/index.html (standard, retrieved 2026-09-24)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/arrhenius-equation?temperature_1=…&temperature_2=…`
- `POST https://tttkmbb.com/api/v1/calculate/arrhenius-equation` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/arrhenius-equation · OpenAPI operationId `solve_arrhenius_equation` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "arrhenius-equation", "inputs": {…}}`

## Example

- Ea 50 kJ/mol, k1 = 0.001 at 25 °C, k2 at 35 °C: inputs `{"activation_energy_kj_per_mol":50,"temperature_1":25,"temperature_2":35,"rate_constant_1":0.001}` → `{"rate_ratio_k2_over_k1":1.92426,"rate_constant_2":0.00192426,"pre_exponential_factor":574938,"temperature_1_k":298.15,"temperature_2_k":308.15,"solved_for":"rate_constant_2"}`
- Ea from k = 2.5e-3 at 20 °C and 1.0e-2 at 40 °C: inputs `{"temperature_1":20,"temperature_2":40,"rate_constant_1":0.0025,"rate_constant_2":0.01}` → `{"activation_energy_kj_per_mol":52.9056,"activation_energy_j_per_mol":52905.6,"rate_ratio_k2_over_k1":4,"solved_for":"activation_energy_kj_per_mol"}`

```
GET https://tttkmbb.com/api/v1/calculate/arrhenius-equation?activation_energy_kj_per_mol=50&temperature_1=25&temperature_2=35&rate_constant_1=0.001
```

## Limitations

The rate constant is unknown at both temperatures and only Ea and A are known (compute k = A·exp(−Ea/RT) directly), or the reaction changes mechanism between the temperatures. Assumes Ea and A are independent of temperature over the range considered, which holds well for intervals of a few tens of kelvin. All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**Does a reaction really double every 10 °C?**

Only for Ea around 50 kJ/mol near room temperature (this example gives ×1.92). Reactions with larger Ea are more temperature-sensitive, smaller Ea less so.

**Which unit does k need?**

Any: only the ratio k2/k1 enters the equation, so the solved rate constant and A come out in the unit you used for k1.

## Related

- [Half-Life Calculator](https://tttkmbb.com/chemistry/half-life.md) — Convert a first-order rate constant to a half-life (t½ = ln 2 / k).
- [Temperature Converter](https://tttkmbb.com/conversion/temperature.md) — Convert temperatures between °C, °F and K.
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