Arrhenius’s electrolytic dissociation: how much of an electrolyte is ions?

Arrhenius proposed that an acid dissolved in water is split, at least in part, into free ions even with no current flowing, and that the degree of dissociation \(\alpha\) distinguishes a strong electrolyte (\(\alpha \approx 1\)) from a weak one (\(\alpha \ll 1\)). The number of dissolved particles per formula unit is then \(i = 1 + \alpha\) for a 1:1 electrolyte – the van ‘t Hoff factor that explained anomalous colligative data.

Here \(\alpha\) is computed with percent_dissociation() for 0.10 M solutions of acids spanning twelve orders of magnitude in \(K_a\).

import matplotlib.pyplot as plt
import numpy as np

from chemistrykit.solutions.systems.acid_base import WeakAcid

Ca = 0.10
Ka_values = np.logspace(-10, 2, 200)
alpha = np.array([WeakAcid(Ca=Ca, Ka=Ka).percent_dissociation() / 100.0 for Ka in Ka_values])

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4.5))
ax1.semilogx(Ka_values, alpha, color="steelblue")
for name, Ka in [("HCN", 6.2e-10), ("acetic", 1.8e-5), ("HF", 6.8e-4), ("HSO$_4^-$", 1.2e-2)]:
    a = WeakAcid(Ca=Ca, Ka=Ka).percent_dissociation() / 100.0
    ax1.plot(Ka, a, "o", color="darkorange")
    ax1.annotate(name, (Ka, a), textcoords="offset points", xytext=(5, -12), fontsize=8)
ax1.set_xlabel("acid dissociation constant $K_a$")
ax1.set_ylabel(r"degree of dissociation $\alpha$")
ax1.set_title("0.10 M acid: from weak to strong electrolyte")

ax2.semilogx(Ka_values, 1.0 + alpha, color="seagreen")
ax2.set_xlabel("acid dissociation constant $K_a$")
ax2.set_ylabel(r"van 't Hoff factor $i = 1 + \alpha$")
ax2.set_title("Dissolved particles per formula unit")
fig.tight_layout()

Printed values for a few familiar acids at 0.10 M:

for name, Ka in [("hydrocyanic acid", 6.2e-10), ("acetic acid", 1.8e-5), ("hydrofluoric acid", 6.8e-4)]:
    acid = WeakAcid(Ca=Ca, Ka=Ka)
    print(f"{name:18s}: {acid.percent_dissociation():6.3f}% dissociated, i = {1 + acid.percent_dissociation() / 100:.4f}")

plt.show()

hydrocyanic acid  :  0.008% dissociated, i = 1.0001
acetic acid       :  1.333% dissociated, i = 1.0133
hydrofluoric acid :  7.913% dissociated, i = 1.0791