# pH and Activity Corrections

*pH Value*

pH (Potential Hydrogen) is defined as the negative decimal logarithm of the hydrogen ion activity:

(1) | pH = – log {H^{+}} |

Curly brackets {..} symbolize the activity, whereas rectangular brackets [..] are reserved for concentrations. Inserting the relationship, {H^{+}} = γ ∙ [H^{+}], one gets

(2) | pH = – log γ – log [H^{+}] |

Formulas for the activation coefficient (log γ) are presented here. In diluted solutions we have γ = 1, and Eq.(2) simplifies to (because log 1 = 0):

(3) | pH = – log [H^{+}] |
(only for diluted solutions) |

*Example*

What is the pH of a 0.1 molar HCl solution without and with activity corrections?

HCl is a *strong acid*. The hydrogen ion concentration is then given by [H^{+}] = 0.1 M. Inserting it into Eq.(3) yields an approximate pH value of

(4) | pH = – log 0.1 = 1.0 | (without activity correction) |

Now with activity correction. For this purpose we adopt the extended Debye-Hückel equation

(5) | \(log \, \gamma_{i} = -\dfrac{Az^{2}_{i}\sqrt{I}}{1+Ba\sqrt{I}}\) |

with A = 0.5085 M^{-1/2}, B = 3.281 M^{-1/2} nm^{-1} and the effective ion-size parameter *a* = 0.9 nm. With z = 1 and the ionic strength I = 0.1 M we obtain log γ = – 0.083. Inserting this result into Eq.(2) gives us the exact pH value:

(6) | pH = – log γ – log [H^{+}] = 0.083 + 1.0 = 1.083 |

The pH of the 0.1 molar HCl solution is 1.083 (rather than 1.0 as often assumed). This exact pH value is also predicted by *aqion* – see the initial pH in the titration example.