pH of Strong Acid and Base

When a strong acid (e.g., HCl) is placed in water, the acid ionizes completely as:

$$ HCl+H_2O→H^++Cl^- $$

Three species are present in the aqueous solution of the strong acid: H+, OH–, and Cl–. H+, generated from HCl, suppresses the ionization of H2O. This leads to the lower concentration of H+ in water than the theoretical concentration of H+ coming from both HCl and H2O. To calculate the concentration of H+ and other species in the aqueous solution, three equations are required:

  1. Equilibrium: Since HCl is fully ionized, there is no reverse reaction (only forward reaction). However, water is dissociated into H+ and OH– at equilibrium with the following relationship:
    $$ (1)\;\;\;K_w=[H^+][OH^-]=10^{-14}\;\;at\;25°C $$where Kw is the ionization constant of water.
  2. Material Balance: The concentration of Cl– is produced only from HCl and is equal to the concentration of HCl initially present in the solution, Ca.
    $$ (2)\;\;\;C_a=[Cl^-] $$
  3. Electroneutrality: The solution containing the ionic species must be electrically neutral in order for the ionic species to be separated from each other so that there is no net charge accumulation. The total concentration of the positive charges in the solution should be equal to the total concentration of negative charges in the solution:
    $$ (3)\;\;\;[H^+]=[OH^-]+[Cl^-] $$

There are three unknown concentration terms and three independent equations. Substituting Equation (1) and Equation (2) into Equation (3) yields:

$$ (4)\;\;\;[H^+]=C_a+\frac{K_w}{[H^+]} $$

This equation becomes a quadratic equation by transposing and the concentration of H+ is given by:

$$ (5a)\;\;\;[H^+]=\frac{C_a+\sqrt{C_a^2+4K_w}}{2} $$

One can follow the same procedure described above for calculating the concentration of OH– in a solution of a strong base (e.g., NaOH). The resulting equation is:

$$ (5b)\;\;\;[OH^-]=\frac{C_b+\sqrt{C_b^2+4K_w}}{2} $$

where Cb is the concentration of the strong base initially present.

If the concentration of a strong acid or strong base is equal to or greater than 10-6 M, the second term of the right-hand side of Equation (4) is negligible compared to the initial concentration of the strong acid or strong base:

$$ [H^+]≅C_a\;\;\;or\;\;\;[OH^-]≅C_b $$

When the concentration of the strong acid or strong base is less than 10-6 M, Equation (5a) or Equation (5b) must be used, respectively.

Reference:

  • Kim, C. (2004). Advanced pharmaceutics : physicochemical principles. London: CRC Press LLC.

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