pH Calculation - Weak Acid

Weak acids are those acids which in solution does not dissociate completely. As acids (Bronsted-Lowry / Arrhenius theories), in solution they donate a proton to H2O (A is a generic counterion, that does not give hydrolysis):

The acid dissociation constant is given by:

Take for example acetic acid, a very common weak acid:

In presence of water the acetic acid will dissociate into his constituents ions   and  , while in part will remain undissociated as:

Then we will have to find out the balance achieved in solution between the species  ,   and   . To calculate the pH we just need the equation of Ka:

We must therefore find  as pH is equal to -log [H3O+].

We call Ca is the initial molar concentration of acid, before the dissociation in water. We denote instead with "X" the amount of acid that dissociates.

If X mol/L of acetic acid dissociate, we will get X mol/L of H3O+ and X mol/L CH3COO-.

So we will have X = [H3O+] = [CH3COO-]

The concentration of CH3COOH that remains undissociated will be (Ca - X) mol/L

Substituting in the expression of the Ka:

This is a quadratic equation that we are perfectly able to solve. It can be however, further simplificated. Since we are dealing with a weak acid, we can suppose that for the most part it remains undissociated.

This means that we can say with good approximation that:

Our expression therefore becomes:

and given that X = [H3O+]

that, rearranged as a function of becomes:

It can be derived that the final formula to calculated calculate the pH is:

  • pH calculation of a weak base

The pH of a weak base can be calculated similarly, by the same steps and the same approximations. Firstly we calculate [OH- ]:

And consequently the pOH:

The pH will be simply found as: