Arrhenius Plot for Reaction Kinetics — plotnine

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.

Arrhenius Plot for Reaction Kinetics rendered with plotnine

Python source (plotnine)

""" anyplot.ai
line-arrhenius: Arrhenius Plot for Reaction Kinetics
Library: plotnine 0.15.7 | Python 3.13.14
Quality: 91/100 | Updated: 2026-06-24
"""

import os

import numpy as np
import pandas as pd
from plotnine import (
    aes,
    annotate,
    element_blank,
    element_line,
    element_rect,
    element_text,
    geom_line,
    geom_point,
    geom_ribbon,
    ggplot,
    labs,
    scale_x_continuous,
    scale_y_continuous,
    theme,
    theme_minimal,
)
from scipy.stats import t as t_dist


# Theme tokens
THEME = os.getenv("ANYPLOT_THEME", "light")
PAGE_BG = "#FAF8F1" if THEME == "light" else "#1A1A17"
ELEVATED_BG = "#FFFDF6" if THEME == "light" else "#242420"
INK = "#1A1A17" if THEME == "light" else "#F0EFE8"
INK_SOFT = "#4A4A44" if THEME == "light" else "#B8B7B0"
INK_MUTED = "#6B6A63" if THEME == "light" else "#A8A79F"

# Imprint palette — first series always #009E73, position 3 for fit line
BRAND = "#009E73"
BLUE = "#4467A3"

# Data — first-order decomposition, 300–600 K with realistic experimental scatter
temperature_K = np.array([300, 350, 400, 450, 500, 550, 600])
noise = np.array([0.22, -0.18, 0.25, -0.14, 0.19, -0.21, 0.15])
rate_constant_k = np.exp(-4878.0 / temperature_K + 9.5 + noise)

inv_T = 1.0 / temperature_K
ln_k = np.log(rate_constant_k)

# Regression statistics
coeffs = np.polyfit(inv_T, ln_k, 1)
slope, intercept = coeffs
ln_k_pred = np.polyval(coeffs, inv_T)
ss_res = np.sum((ln_k - ln_k_pred) ** 2)
ss_tot = np.sum((ln_k - ln_k.mean()) ** 2)
r_squared = 1 - ss_res / ss_tot

R_gas = 8.314
Ea_kJ = -slope * R_gas / 1000

df = pd.DataFrame({"inv_T": inv_T, "ln_k": ln_k})

# Regression line with 95% confidence band
inv_T_fine = np.linspace(inv_T.min(), inv_T.max(), 200)
ln_k_fit = np.polyval(coeffs, inv_T_fine)

n = len(inv_T)
se_residual = np.sqrt(ss_res / (n - 2))
inv_T_mean = inv_T.mean()
inv_T_ss = np.sum((inv_T - inv_T_mean) ** 2)
se_fit = se_residual * np.sqrt(1.0 / n + (inv_T_fine - inv_T_mean) ** 2 / inv_T_ss)
t_val = t_dist.ppf(0.975, n - 2)
ci_lower = ln_k_fit - t_val * se_fit
ci_upper = ln_k_fit + t_val * se_fit

df_fit = pd.DataFrame({"inv_T": inv_T_fine, "ln_k_fit": ln_k_fit, "ci_lower": ci_lower, "ci_upper": ci_upper})

# X-axis ticks with dual annotation (1/T + temperature in K)
tick_temps = [300, 400, 500, 600]
tick_positions = [1.0 / t for t in tick_temps]
tick_labels = [f"{1.0 / t:.2e}\n({t} K)" for t in tick_temps]

# Annotation placement — upper-left quadrant
anno_x = inv_T.min() + 0.35 * (inv_T.max() - inv_T.min())
anno_y_top = ln_k.max() - 0.2

anno_r2 = f"R² = {r_squared:.4f}"
anno_ea = f"Eₐ = {Ea_kJ:.1f} kJ/mol"
anno_slope = f"slope = −Eₐ/R = {slope:.0f} K"

title = "line-arrhenius · python · plotnine · anyplot.ai"

# Plot
plot = (
    ggplot(df, aes(x="inv_T", y="ln_k"))
    + geom_ribbon(
        aes(x="inv_T", ymin="ci_lower", ymax="ci_upper"), data=df_fit, fill=BRAND, alpha=0.12, inherit_aes=False
    )
    + geom_line(aes(x="inv_T", y="ln_k_fit"), data=df_fit, color=BLUE, size=1.5, inherit_aes=False)
    + geom_point(color=BRAND, size=4.5, stroke=0.8, shape="o")
    + scale_x_continuous(breaks=tick_positions, labels=tick_labels, expand=(0.05, 0))
    + scale_y_continuous(expand=(0.08, 0))
    + annotate(
        "label",
        x=anno_x,
        y=anno_y_top,
        label=anno_r2,
        size=4.5,
        color=INK,
        fontweight="bold",
        ha="center",
        fill=ELEVATED_BG,
        alpha=0.92,
        label_padding=0.4,
        label_size=0,
    )
    + annotate("text", x=anno_x, y=anno_y_top - 0.9, label=anno_ea, size=4.0, color=INK, fontweight="bold", ha="center")
    + annotate(
        "text",
        x=anno_x,
        y=anno_y_top - 1.65,
        label=anno_slope,
        size=4.0,
        color=INK_MUTED,
        fontstyle="italic",
        ha="center",
    )
    + labs(x="1/T (K⁻¹)", y="ln(k)", title=title)
    + theme_minimal()
    + theme(
        figure_size=(8, 4.5),
        plot_title=element_text(size=12, weight="bold", color=INK, margin={"b": 10}),
        axis_title=element_text(size=10, color=INK),
        axis_text=element_text(size=8, color=INK_SOFT),
        axis_ticks=element_line(color=INK_SOFT, size=0.3),
        plot_background=element_rect(fill=PAGE_BG, color=PAGE_BG),
        panel_background=element_rect(fill=PAGE_BG),
        panel_grid_major_x=element_blank(),
        panel_grid_minor=element_blank(),
        panel_grid_major_y=element_line(color=INK, size=0.3, alpha=0.15),
        axis_ticks_minor=element_blank(),
        plot_margin=0.08,
    )
)

# Save
plot.save(f"plot-{THEME}.png", dpi=400, width=8, height=4.5, units="in", verbose=False)

Part of Arrhenius Plot for Reaction Kinetics on anyplot.ai.

Arrhenius Plot for Reaction Kinetics — plotnine

Python source (plotnine)

Other implementations