Library / Solution chemistry, equilibria and kinetics / SCH-11
Solution chemistry, equilibria and kinetics
Very dilute strong acid including water autoionisation
Hydronium concentration of a fully dissociated monoprotic strong acid at formal concentration C in pure water, including the hydronium from water itself.
- Ref
- SCH-11
- Rev
- v1
- Field
- Solution chemistry, equilibria and kinetics
- Result
- h [mol/L]
For a strong acid at formal concentration C, charge balance gives h = C + Kw/h, so h² − C h − Kw = 0 and h = ½(C + √(C² + 4Kw)). At high concentration h ≈ C, but near 10⁻⁷ M water supplies hydronium of its own: a 10⁻⁸ M acid gives h = 1.05 × 10⁻⁷ M, pH 6.98, not 8. At C = 0 it gives pure water, h = √Kw.
Hydronium concentration of a fully dissociated monoprotic strong acid at formal concentration C in pure water, including the hydronium from water itself. It reduces to h = C for concentrated acid and to √Kw at C = 0.
Calculator
- pH, −log₁₀ h6.978 pH
- Hydroxide concentration, Kw/h9.512×10−8 mol/L
- Share of hydronium from water, (h − C)/h0.9049
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.
- A 1×10−8 mol/L strong acid gives h = 1.051×10−7 mol/L, pH 6.978, instead of the pH 8 the simple formula would suggest.
- 90.5 % of the hydronium comes from water; the correction is negligible above about 10⁻⁶ M.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mol/L) | Local step | Change in result |
|---|---|---|---|---|---|---|
| C Formal acid concentration | 1e-10 to 1e-5 mol/L | 1.001e-7 | 1e-5 | 9.901e-6 | +10% | +0.501 % |
| Kw Water ion product (concentration basis) | 1e-15 to 1e-12 | 3.702e-8 | 1.005e-6 | 9.68e-7 | +10% | +4.64 % |
Sweep of Formal acid concentration
| Formal acid concentration (mol/L) | Hydronium concentration (mol/L) |
|---|---|
| 1e-10 | 1.001e-7 |
| 4.217e-10 | 1.002e-7 |
| 1.778e-9 | 1.009e-7 |
| 7.499e-9 | 1.038e-7 |
| 3.162e-8 | 1.171e-7 |
| 1.334e-7 | 1.869e-7 |
| 5.623e-7 | 5.796e-7 |
| 2.371e-6 | 2.376e-6 |
| 1e-5 | 1e-5 |
Sweep one variable
Variables and how each one contributes
Hydronium concentration
What the equation gives you. Everything else feeds it.
Formal acid concentration
The formal concentration of the strong acid, in mol/L. Raises h; the effect is nearly proportional above 10⁻⁶ M and weak below 10⁻⁸ M. Raising it by 10 % (to 1.1×10−8) moves the result by +0.501 %, from 1.051×10−7 to 1.057×10−7 mol/L.
Water ion product (concentration basis)
The water ion product at the working temperature. A larger Kw adds more hydronium from water, which matters at low acid concentration. Raising it by 10 % (to 1.1×10−14) moves the result by +4.64 %, from 1.051×10−7 to 1.1×10−7 mol/L.
Worked example with the values above
- Discriminant: C² + 4Kw = 1×10−16 + 4×10−14 = 4.01×10−14; root = 2.0025×10−7.
- Add C: 1×10−8 + 2.0025×10−7 = 2.1025×10−7.
- h = ½ × that = 1.0512×10−7 mol/L.
Assumptions and limits
- Dilute solutions in which concentrations stand for activities, with constants on the concentration scale.
- No other acids, bases or dissolved gases such as CO₂ unless included.
Common mistakes
- Using pH = −log C for acids more dilute than about 10⁻⁶ M.
- Using Kw at 25 °C at other temperatures.
- Applying it to a weak acid, which does not dissociate fully.
Statistics and uncertainty
Acid constants change with ionic strength and temperature, so results are best to about 0.05 pH units even with accurate constants. Calibration of a pH electrode contributes about 0.02 pH.
Where it is used
- Calculating pH of dilute acids, salts and intermediate ions.
- Finding which species dominates at a given pH.
Maths you need
- pH: pH = −log₁₀ [H⁺]
- Acid dissociation constant: Ka = [H⁺][A⁻]/[HA]
- Water ion product: Kw = [H⁺][OH⁻]