Calculating pH, pOH, and Hydronium Concentration
The pH, pOH and hydronium concentration of a solution are connected by pH = −log[H3O+], pOH = −log[OH−], pH + pOH = 14.00 (at 25 °C), [H3O+] = 10−pH and [OH−] = 10−pOH. In brief: for a strong acid, [H+] equals the acid concentration (0.0020 mol/L HCl has pH 2.70); for a strong base, find [OH−] first and use pH = 14 − pOH (0.050 mol/L Ba(OH)2 has [OH−] = 0.100 mol/L and pH 13.00); for weak acids and bases use the equilibrium constant.
This article gives a decision procedure for pH problems, the formulas for strong and weak acids and bases, a table of the conversions, worked examples including mixtures and dilutions, and advice about significant figures and the common pitfalls.
What are the key relations?
- pH = −log[H+] and [H+] = 10−pH
- pOH = −log[OH−] and [OH−] = 10−pOH
- [H+][OH−] = 1.0 × 10−14 and pH + pOH = 14.00 at 25 °C
How do you calculate the pH of strong acids and bases?
A strong monoprotic acid gives [H+] equal to its concentration. A strong base gives [OH−] equal to its concentration times the number of hydroxide ions per formula unit. Then convert with the logarithm and with pH + pOH = 14.
How do you calculate the pH of weak acids and bases?
- Write the ionization equilibrium and the expression for Ka (or Kb).
- Set up the concentrations: initial c, change −x for the acid and +x for each ion, equilibrium c − x, x, x.
- Solve Ka = x2 ÷ (c − x). If x is less than 5% of c, use x ≈ √(Kac). Otherwise solve the quadratic equation.
- For an acid x = [H+]; for a base x = [OH−].
Summary of the methods
| Solution | Method | Example |
|---|---|---|
| Strong acid | [H+] = c | 0.0020 M HCl: pH 2.70 |
| Strong base | [OH−] = c × n, then pOH, pH | 0.0010 M NaOH: pH 11.00 |
| Weak acid | [H+] = √(Kac) | 0.10 M CH3COOH: pH 2.87 |
| Weak base | [OH−] = √(Kbc) | 0.10 M NH3: pH 11.13 |
| Mixture of strong acid and base | Net moles ÷ total volume | See example 5 |
Worked examples
- Strong acid. 0.0020 mol/L HCl: pH = −log(0.0020) = 2.70, pOH = 11.30.
- Strong base with two OH. 0.050 mol/L Ba(OH)2: [OH−] = 0.100 mol/L, pOH = 1.00, pH = 13.00.
- From pH to concentration. pH 3.40: [H+] = 10−3.40 = 4.0 × 10−4 mol/L. For a strong monoprotic acid, that is its concentration.
- Weak acid. 0.250 mol/L formic acid (Ka = 1.8 × 10−4): x = √(1.8 × 10−4 × 0.250) = 6.7 × 10−3 mol/L, which is 2.7% of 0.250 (less than 5%). pH = 2.17.
- Mixing a strong acid and base. 40.0 mL of 0.100 mol/L HCl (4.00 mmol H+) and 30.0 mL of 0.100 mol/L NaOH (3.00 mmol OH−). Excess H+ = 1.00 mmol in 70.0 mL: [H+] = 0.01429 mol/L, pH 1.85.
- Dilution. 10.0 mL of 0.100 mol/L HCl is diluted to 1.00 L. c = 1.00 × 10−3 mol/L, pH 3.00. A 100-fold dilution raised the pH by 2 units.
- Weak base. 0.200 mol/L methylamine (Kb = 4.4 × 10−4): x = √(8.8 × 10−5) = 9.4 × 10−3 mol/L (4.7% of c, acceptable). pOH = 2.03, pH = 11.97.
- Hydronium from pOH. pOH = 5.20, so pH = 8.80 and [H3O+] = 1.6 × 10−9 mol/L.
How many significant figures does pH have?
Only the digits after the decimal point in a pH are significant, and their number equals the number of significant figures in the concentration. A concentration of 2.5 × 10−4 mol/L (two significant figures) gives pH = 3.60 (two decimal places). The digit before the decimal point only locates the power of ten.
Where does the logarithmic scale come from?
Sørensen introduced pH in 1909 so that the very small hydrogen ion concentrations in biochemical work could be written as simple numbers. Logarithmic quantities such as pOH, pKa and pKw follow the same idea.
Which mistakes are common?
Mistake 1: Forgetting the second OH− in Ba(OH)2 or Ca(OH)2
Multiply the concentration by 2.
Mistake 2: Using pH = −log(c) for a weak acid
Only strong acids ionize completely.
Mistake 3: Mixing up pH and pOH for bases
Calculate pOH from [OH−] and then subtract from 14.
Mistake 4: Averaging the pH values when mixing
Calculate with moles and total volume, not with the pH numbers.