An Arrhenius plot displays ln(k) versus 1/T to determine the activation energy of a chemical reaction from experimental rate constant data. The Arrhenius equation predicts a linear relationship on this transformed scale, where the slope equals -Ea/R (activation energy divided by the gas constant). This visualization is fundamental in physical chemistry and chemical engineering for characterizing reaction kinetics and comparing catalytic performance.
# anyplot.ai
# line-arrhenius: Arrhenius Plot for Reaction Kinetics
# Library: makie 0.22.10 | Julia 1.11.9
# Quality: 84/100 | Created: 2026-06-24
using CairoMakie
using Colors
using Random
using Statistics
using Printf
Random.seed!(42)
# Theme tokens
const THEME = get(ENV, "ANYPLOT_THEME", "light")
const PAGE_BG = THEME == "light" ? colorant"#FAF8F1" : colorant"#1A1A17"
const ELEVATED_BG = THEME == "light" ? colorant"#FFFDF6" : colorant"#242420"
const INK = THEME == "light" ? colorant"#1A1A17" : colorant"#F0EFE8"
const INK_SOFT = THEME == "light" ? colorant"#4A4A44" : colorant"#B8B7B0"
const IMPRINT_PALETTE = [
colorant"#009E73", colorant"#C475FD", colorant"#4467A3", colorant"#BD8233",
colorant"#AE3030", colorant"#2ABCCD", colorant"#954477", colorant"#99B314",
]
# Arrhenius data: first-order thermal decomposition
# k = A * exp(-Ea / R / T), Ea = 80 kJ/mol, A = 1e10 s⁻¹
const R_GAS = 8.314 # J mol⁻¹ K⁻¹
const EA = 80_000.0 # J mol⁻¹
const A_PRE = 1.0e10 # s⁻¹
temperatures_K = [300.0, 330.0, 360.0, 400.0, 440.0, 480.0, 520.0, 560.0, 600.0]
k_ideal = A_PRE .* exp.(-EA ./ (R_GAS .* temperatures_K))
rate_constants = k_ideal .* (1.0 .+ 0.04 .* randn(length(temperatures_K)))
# Arrhenius linearisation using 1000/T on x-axis (conventional; avoids tiny tick numbers)
inv_T_scaled = 1000.0 ./ temperatures_K # values ≈ 1.67 – 3.33
ln_k = log.(rate_constants)
# Linear regression: ln(k) = slope * (1000/T) + intercept
x_bar = mean(inv_T_scaled)
y_bar = mean(ln_k)
slope = sum((inv_T_scaled .- x_bar) .* (ln_k .- y_bar)) /
sum((inv_T_scaled .- x_bar).^2)
intercept = y_bar - slope * x_bar
ln_k_fit = slope .* inv_T_scaled .+ intercept
r_sq = 1.0 - sum((ln_k .- ln_k_fit).^2) / sum((ln_k .- y_bar).^2)
# Ea from slope: since x = 1000/T, slope × 1000 = -Ea/R
ea_kJmol = -slope * R_GAS # ≈ 79.9 kJ/mol
# Extended regression line
margin = 0.08 * (maximum(inv_T_scaled) - minimum(inv_T_scaled))
x_fit = collect(range(minimum(inv_T_scaled) - margin,
maximum(inv_T_scaled) + margin; length = 200))
y_fit = slope .* x_fit .+ intercept
# 95% confidence band (t_{0.025, df=7} = 2.365)
n_pts = length(inv_T_scaled)
sse = sum((ln_k .- ln_k_fit).^2)
s2 = sse / (n_pts - 2)
Sxx = sum((inv_T_scaled .- x_bar).^2)
t_crit = 2.365
se_band = sqrt.(s2 .* (1.0 / n_pts .+ (x_fit .- x_bar).^2 ./ Sxx))
y_upper = y_fit .+ t_crit .* se_band
y_lower = y_fit .- t_crit .* se_band
title_str = "line-arrhenius · julia · makie · anyplot.ai"
title_size = length(title_str) > 67 ? max(14, round(Int, 20 * 67 / length(title_str))) : 20
fig = Figure(
size = (1600, 900),
fontsize = 14,
backgroundcolor = PAGE_BG,
)
ax = Axis(
fig[1, 1];
title = title_str,
titlesize = title_size,
titlecolor = INK,
xlabel = "10³/T (K⁻¹)",
ylabel = "ln(k)",
xlabelsize = 16,
ylabelsize = 16,
xticklabelsize = 13,
yticklabelsize = 13,
xlabelcolor = INK,
ylabelcolor = INK,
xticklabelcolor = INK_SOFT,
yticklabelcolor = INK_SOFT,
xtickcolor = INK_SOFT,
ytickcolor = INK_SOFT,
backgroundcolor = PAGE_BG,
topspinevisible = false,
rightspinevisible = false,
leftspinecolor = INK_SOFT,
bottomspinecolor = INK_SOFT,
xgridvisible = false,
ygridvisible = true,
xminorgridvisible = false,
yminorgridvisible = false,
ygridcolor = RGBAf(INK.r, INK.g, INK.b, 0.12),
xticks = LinearTicks(4),
)
# True secondary x-axis: temperature reference in K (overlapping Axis at top)
ax_top = Axis(
fig[1, 1];
xaxisposition = :top,
yaxisposition = :right,
backgroundcolor = :transparent,
topspinevisible = true,
topspinecolor = INK_SOFT,
bottomspinevisible = false,
leftspinevisible = false,
rightspinevisible = false,
xgridvisible = false,
ygridvisible = false,
yticksvisible = false,
yticklabelsvisible = false,
xlabel = "Temperature (K)",
xlabelsize = 13,
xlabelcolor = INK_SOFT,
xticklabelsize = 11,
xticklabelcolor = INK_SOFT,
xtickcolor = INK_SOFT,
xticks = (
1000.0 ./ [300.0, 350.0, 400.0, 500.0, 600.0],
["300", "350", "400", "500", "600"],
),
)
linkxaxes!(ax, ax_top)
linkyaxes!(ax, ax_top)
# Confidence band (Makie band! — highlights fit quality)
band!(ax, x_fit, y_lower, y_upper;
color = (IMPRINT_PALETTE[3], 0.15),
)
# Regression line (drawn before markers so they render on top)
lines!(ax, x_fit, y_fit;
color = IMPRINT_PALETTE[3],
linewidth = 2.5,
linestyle = :dash,
label = "Linear fit (Arrhenius)",
)
# Experimental data points
scatter!(ax, inv_T_scaled, ln_k;
color = IMPRINT_PALETTE[1],
markersize = 14,
strokewidth = 1.5,
strokecolor = PAGE_BG,
label = "Measured k(T)",
)
# Annotation: kinetic parameters — lower-left corner
x_range = maximum(inv_T_scaled) - minimum(inv_T_scaled)
y_range = maximum(ln_k) - minimum(ln_k)
x_ann = minimum(inv_T_scaled) + 0.03 * x_range
y_ann = minimum(ln_k) + 0.08 * y_range
text!(ax, x_ann, y_ann;
text = @sprintf("-Ea/R = %d K\nEa = %.1f kJ mol⁻¹\nR² = %.4f",
round(Int, -slope * 1000.0), ea_kJmol, r_sq),
color = INK,
fontsize = 12,
align = (:left, :bottom),
)
axislegend(ax;
position = :rt,
labelsize = 12,
framevisible = true,
framecolor = INK_SOFT,
backgroundcolor = ELEVATED_BG,
labelcolor = INK,
)
resize_to_layout!(fig)
save("plot-$(THEME).png", fig; px_per_unit = 2)
Part of Arrhenius Plot for Reaction Kinetics on anyplot.ai.