Ion Product of Water and the pH Scale: Definition, Calculations, and Applications

PART 3.
TOPIC 5. Ion Product of Water. Hydrogen Ion Concentration and pH Scale.
The hydrogen ion concentration (pH) is a value that characterizes the activity or concentration of hydrogen ions in solutions. The hydrogen ion concentration is denoted as pH.
The pH is numerically equal to the negative base-10 logarithm of the activity or concentration of hydrogen ions, expressed in moles per liter:
pH = -lg[ H+ ]
In water, the concentration of hydrogen ions is determined by the electrolytic dissociation of water according to the equation:
H2O ⇌ H+ + OH-
The dissociation constant at 22°C is:

Neglecting the small fraction of dissociated molecules, the concentration of undissociated water can be taken as the total concentration of water, which is: C[H2O] = 1000/18 = 55.55 mol/L.
Then: C[ H+ ] · C[ OH- ] = K · C[H2O] = 1.8 × 10^-16 · 55.55 = 10^-14
For water and its solutions, the product of the concentrations of H+ and OH- ions is constant at a given temperature. This is called the ion product of water, K_W, and at 25°C it equals 10^-14.
The constancy of the ion product of water allows for the calculation of the concentration of H+ ions if the concentration of OH- ions is known, and vice versa.
The concepts of acidic, neutral, and basic environments acquire a quantitative meaning.
If [ H+ ] = [ OH- ], these concentrations (each of them) are equal to 10^-7 mol/L, i.e., [ H+ ] = [ OH- ] = 10^-7 mol/L, and the environment is neutral. In these solutions:
pH = -lg[ H+ ] = 7 and pOH = -lg[ OH- ] = 7.
If [ H+ ] > 10^-7 mol/L and [ OH- ] < 10^-7 mol/L, the environment is acidic; pH < 7.
If [ H+ ] < 10^-7 mol/L and [ OH- ] > 10^-7 mol/L, the environment is basic; pH > 7.
In any aqueous solution, pH + pOH = 14, where pOH = -lg[ OH- ].
The pH value is of great importance for biochemical processes, various industrial processes, the study of natural water properties, their potential applications, etc.

Calculating pH of Acid and Base Solutions.

To calculate the pH of acid and base solutions, one should first calculate the molar concentration of free hydrogen ions (H+) or free hydroxyl ions (OH-), and then use the following formulas:
pH = -lg[ H+ ]; pOH = -lg[ OH- ]; pH + pOH = 14.
The concentration of any ion in mol/L in an electrolyte solution can be calculated using the equation:

where C_ion is the molar concentration of the ion in mol/L;
C_electrolyte is the molar concentration of the electrolyte in mol/L;
α is the degree of dissociation of the electrolyte;
n is the number of ions of that type produced when one molecule of the electrolyte dissociates.
If the electrolyte is weak, the degree of dissociation can be determined using Ostwald’s dilution law:

then C_ion = C_electrolyte · α · n = v · C_Mdiss.

Example 1. Calculate the pH of a 0.001 M NaOH solution.
Solution: Sodium hydroxide is a strong electrolyte, and dissociation in aqueous solution follows the scheme:
NaOH → Na+ + OH-
The degree of dissociation in a dilute solution can be assumed to be 1. The concentration of OH- ions (mol/L) in the solution is:

Example 2. Calculate the pH of a 1% formic acid solution, assuming the solution density is 1 g/mL and K_diss = 2.1 × 10^-4.
Solution: 1 liter of solution contains 10 g of HCOOH, which is 10/46 = 0.22 mol, where 46 g/mol is the molar mass of formic acid. Therefore, the molar concentration of the solution is 0.22 mol/L. Formic acid is a weak electrolyte, so

since HCOOH ⇌ H+ + HCOO-.

Example 3. The pH of the solution is 4.3. Calculate [ H+ ] and [ OH- ].
Solution:
[ H+ ] = 10^-pH = 10^-4.3 = 5 × 10^-5 mol/L

[ OH- ] = 10^-14 / 5 × 10^-5 = 2 × 10^-10 mol/L.