Library / Foundations: Chemistry / FCH-29
Foundations: Chemistry
Hydrogen-ion concentration of a weak acid without the approximation
Exact hydrogen-ion concentration of a weak acid that dissociates only partly, from solving the quadratic instead of assuming the dissociation is small.
- Ref
- FCH-29
- Rev
- v1
- Field
- Foundations: Chemistry
- Result
- [H⁺] [mol/dm³]
For a weak monoprotic acid HA with dissociation constant Ka and initial concentration c₀, equilibrium needs [H⁺]²/(c₀ − [H⁺]) = Ka. Solving the quadratic exactly gives [H⁺] = (−Ka + √(Ka² + 4Kac₀))/2. The second form on the page is algebraically the same and avoids cancellation errors when Ka is tiny. It improves on the shortcut [H⁺] ≈ √(Kac₀), which fails when the acid is more than a few per cent dissociated.
Exact hydrogen-ion concentration of a weak acid that dissociates only partly, from solving the quadratic instead of assuming the dissociation is small. The second form avoids losing accuracy when Ka is very small.
Calculator
- pH, −log₁₀[H⁺]3.002
- Approximation √(Kac₀)0.001 mol/dm³
- Error of the approximation relative to the exact value0.5012 %
- Fraction dissociated, [H⁺]/c₀0.995 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- For Ka = 1×10−5 and c₀ = 0.1 mol/dm³ the exact [H⁺] is 9.95×10−4 mol/dm³ (pH 3.002). The shortcut √(Kac₀) = 0.001 is about 0.501 % too high.
- Make the acid more dilute or stronger and the approximation worsens: the exact formula is needed once more than about 5 % of the acid has dissociated.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mol/dm³) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Ka Acid dissociation constant | 1e-10 to 0.1 mol/dm³ | 3.162e-6 | 0.0618 | 0.0618 | +10% | +4.86 % |
| c0 Initial concentration of the acid | 1e-4 to 5 mol/dm³ | 2.702e-5 | 0.007066 | 0.007039 | +10% | +4.91 % |
Sweep of Initial concentration of the acid
| Initial concentration of the acid (mol/dm³) | Hydrogen-ion concentration (mol/dm³) |
|---|---|
| 1e-4 | 2.702e-5 |
| 3.867e-4 | 5.739e-5 |
| 0.001495 | 1.174e-4 |
| 0.005782 | 2.355e-4 |
| 0.02236 | 4.679e-4 |
| 0.08647 | 9.249e-4 |
| 0.3344 | 0.001824 |
| 1.293 | 0.003591 |
| 5 | 0.007066 |
Sweep one variable
Variables and how each one contributes
Hydrogen-ion concentration
What the equation gives you. Everything else feeds it.
Acid dissociation constant
The acid dissociation constant, in mol/dm³. For ethanoic acid it is about 1.8 × 10⁻⁵. The slider is logarithmic. Raises [H⁺] roughly with the square root of Ka for a weak acid, so a stronger acid gives more hydrogen ions. Raising it by 1×10−5 (to 2×10−5) moves the result by +41.1 %, from 9.95×10−4 to 0.001404 mol/dm³.
Initial concentration of the acid
The initial concentration of the acid, in mol/dm³. The slider is logarithmic. Raises [H⁺] roughly with the square root of the concentration; a more dilute acid is a larger fraction dissociated but has fewer ions. Raising it by 0.1 (to 0.2) moves the result by +41.6 %, from 9.95×10−4 to 0.001409 mol/dm³.
Worked example with the values above
- Discriminant: Ka² + 4Kac₀ = 1×10−10 + 4×10−6 = 4×10−6; its square root is 0.002.
- Using the stable form: denominator Ka + √ = 0.00201; numerator 2Kac₀ = 2×10−6.
- [H⁺] = 2×10−6 / 0.00201 = 9.95×10−4 mol/dm³, a pH of 3.002.
Assumptions and limits
- A monoprotic weak acid, with the auto-ionisation of water neglected (valid when [H⁺] is well above 10⁻⁶ mol/dm³).
- Concentrations used in place of activities, which is acceptable for dilute solutions.
Common mistakes
- Using the shortcut √(Kac₀) for a dilute or fairly strong acid.
- Using the formula for a polyprotic acid, which needs each step treated separately.
- Entering pKa where Ka is required.
Statistics and uncertainty
The pH depends on the logarithm of [H⁺], so a 10 % error in Ka shifts the pH by about 0.02 for a weak acid. Published Ka values depend on temperature and ionic strength, which dominates the uncertainty.
Where it is used
- pH of a weak acid solution, such as a cleaning agent or a chemical reagent.
- Checking how much error the common shortcut introduces.
Maths you need
- Acid dissociation constant: ratio of products to the undissociated acid at equilibrium
- Quadratic formula: solution of ax² + bx + c = 0