The iterative weak acid approximation

P2.5.2The iterative weak acid approximation

The iterative weak acid approximation determines the hydrogen ion concentration, $[\mathrm{H^+}]$ of an acid solution from the acid dissociation constant, $$K_a$$ , and the acid concentration, $$c$$ , by succesive application of the formula

[\mathrm{H^+}]_{n+1} = \sqrt{K_a\left( c - [\mathrm{H^+}]_n \right)},

starting with $[\mathrm{H^+}]_0 = 0$ . The iterations are continued until $[\mathrm{H^+}]$ changes by less than some pre-determined, small tolerance value.

Use this method to determine the hydrogen ion concentration, and hence the pH ( $=-\log_{10}[\mathrm{H^+}]$ ) of a $c=0.01\;\mathrm{M}$ solution of acetic acid ( $K_a = 1.78 \times 10^{-5}$ ). Use the tolerance TOL = 1.e-10.

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Here is one approach.

import math
TOL = 1.e-10

# Acid dissociation constant, acid concentration (M)
Ka, c = 1.78e-5, 0.01

Hp = 0.
while True:
    lastHp, Hp = Hp, math.sqrt(Ka * (c - Hp))
    if abs(lastHp - Hp) < TOL:
        break
pH = -math.log10(Hp)

print('[H+] = {:g} M, pH = {:.2f}'.format(Hp, pH))

For a 0.01 M solution of acetic acid, the output is:

[H+] = 0.000413094 M, pH = 3.38