MAT-10

Arrhenius rate

k = A exp(−Ea / RT). Thermal activation over a barrier Ea.

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Mechanics of materialsArrhenius

Governing equation

k=A\exp(-E_a/RT)

$T \in \left[250, 900\right]$

where

A: Pre-factor (1/s)
E_a: Activation energy (kJ/mol)
T: Temperature (K)
k: Rate constant (1/s)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

k=A\exp(-E_a/RT)

$T \in \left[250, 900\right]$

$T = 250$ $k = 1.92\times 10^{-5}$

Along the curve

②

Domain of the variable — blue

T_{\min} = 250,\quad T_{\max} = 900

$T \in \left[250, 900\right]$

$T = 250$ $T = 250$

Along the curve

③

Response curve — green

k=A\exp(-E_a/RT)

$T \in \left[250, 900\right]$

$T = 250$ $k = 1.92\times 10^{-5}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}k}{\mathrm{d}T}

$T \in \left[250, 900\right]$

$T = 254.1$ $k^{\prime} = 5.61\times 10^{-6}$

Along the curve

Lecture brief

Historical brief

Hooke (1678), Hall–Petch grain size, Arrhenius activation and Paris fatigue-crack growth are the spine of materials selection. The sheets relate stress, microstructure and life. This sheet (MAT-10 — Arrhenius rate) is the form associated with Arrhenius. Working symbols: $A$ , $E_a$ , $T$ $\rightarrow$ $k$ . A is the attempt frequency. Metals creep, polymers oxidise, concrete hydrates.

Purpose

Purpose: compute

k

from

A

E_a

T

in Materials via

k=A\exp(-E_a/RT)

k = A exp(−Ea / RT). Thermal activation over a barrier Ea. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

A = 1.000e+12\,\mathrm{1/s}

E_a = 80.000\,\mathrm{kJ/mol}

T = 400.000\,\mathrm{K}

, the governing relation

k=A\exp(-E_a/RT)

yields

k = 35.702176\,\mathrm{1/s}

. A barrier, a Boltzmann tail, a rate. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Pre-factor A1000000000000 1/sActivation energy E_a80 kJ/molTemperature T400 K

Outputs

Rate constant k35.702176 1/s

Reading speed

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Narration of this film

A barrier, a Boltzmann tail, a rate.

A is the attempt frequency. Metals creep, polymers oxidise, concrete hydrates.

Reading speed

Watch on YouTube