Infer a weak-acid state from mass, charge and water together
Weak acid strength, total concentration and fractional ionization are distinct quantities.
Learning outcome: Compare total concentrations at fixed Ka. Recover HA+A−=C, h=OH+A− and the quotient; quantify the shortcut error rather than assuming negligible depletion.
Starting recap
Refresh proton equivalents, conjugate pairs, logarithmic dimensionless activity, analytical amount/volume, mass and charge balances, and bounded observational criteria.
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Try
At Ka=10⁻⁵ and C=.010 M, h is close to 3.11267×10⁻⁴ M; the square-root estimate3.16228×10⁻⁴ is slightly high.
Open the experimentPredict
Can dilution increase fractional ionization while decreasing hydronium concentration?
Experiment
Compare total concentrations at fixed Ka. Recover HA+A−=C, h=OH+A− and the quotient; quantify the shortcut error rather than assuming negligible depletion.
Solve weak-acid mass and charge balances together
Worked teaching model · synthetic values · Fictional neutral HA with ideal concentration activities, total analytical C, no added salt, Kw=10⁻¹⁴ at stipulated 25°C. A−=C Ka/(Ka+h), h=OH+A−. All equilibrium roots include water; the square-root shortcut is shown only for comparison.
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- Exact pH at changing analytical concentration: solid
Axes rescale. A concentration scan, imposed-pH distribution and titration curve have different horizontal meanings. Read each axis and preserve the difference between an imposed condition and an inferred equilibrium.
All calculated plot coordinates
| −log10 total acid concentration / 1 M | pH |
|---|---|
| 2 | 3.506866 |
| 2.04 | 3.52719 |
| 2.08 | 3.547529 |
| 2.12 | 3.567884 |
| 2.16 | 3.588255 |
| 2.2 | 3.608644 |
| 2.24 | 3.629052 |
| 2.28 | 3.649478 |
| 2.32 | 3.669925 |
| 2.36 | 3.690392 |
| 2.4 | 3.710882 |
| 2.44 | 3.731395 |
| 2.48 | 3.751932 |
| 2.52 | 3.772494 |
| 2.56 | 3.793082 |
| 2.6 | 3.813699 |
| 2.64 | 3.834344 |
| 2.68 | 3.85502 |
| 2.72 | 3.875727 |
| 2.76 | 3.896468 |
| 2.8 | 3.917244 |
| 2.84 | 3.938056 |
| 2.88 | 3.958907 |
| 2.92 | 3.979797 |
| 2.96 | 4.000729 |
| 3 | 4.021705 |
| 3.04 | 4.042727 |
| 3.08 | 4.063798 |
| 3.12 | 4.084918 |
| 3.16 | 4.106091 |
| 3.2 | 4.127319 |
| 3.24 | 4.148604 |
| 3.28 | 4.16995 |
| 3.32 | 4.19136 |
| 3.36 | 4.212835 |
| 3.4 | 4.234379 |
| 3.44 | 4.255995 |
| 3.48 | 4.277688 |
| 3.52 | 4.299459 |
| 3.56 | 4.321313 |
| 3.6 | 4.343254 |
| 3.64 | 4.365285 |
| 3.68 | 4.387411 |
| 3.72 | 4.409636 |
| 3.76 | 4.431964 |
| 3.8 | 4.454401 |
| 3.84 | 4.47695 |
| 3.88 | 4.499617 |
| 3.92 | 4.522408 |
| 3.96 | 4.545328 |
| 4 | 4.568382 |
| 4.04 | 4.591576 |
| 4.08 | 4.614916 |
| 4.12 | 4.63841 |
| 4.16 | 4.662062 |
| 4.2 | 4.685881 |
| 4.24 | 4.709872 |
| 4.28 | 4.734044 |
| 4.32 | 4.758402 |
| 4.36 | 4.782956 |
| 4.4 | 4.807712 |
| 4.44 | 4.832679 |
| 4.48 | 4.857864 |
| 4.52 | 4.883276 |
| 4.56 | 4.908923 |
| 4.6 | 4.934813 |
| 4.64 | 4.960955 |
| 4.68 | 4.987357 |
| 4.72 | 5.014027 |
| 4.76 | 5.040974 |
| 4.8 | 5.068205 |
| 4.84 | 5.09573 |
| 4.88 | 5.123555 |
| 4.92 | 5.151688 |
| 4.96 | 5.180136 |
| 5 | 5.208905 |
| 5.04 | 5.238003 |
| 5.08 | 5.267433 |
| 5.12 | 5.297201 |
| 5.16 | 5.32731 |
| 5.2 | 5.357765 |
| 5.24 | 5.388567 |
| 5.28 | 5.419717 |
| 5.32 | 5.451215 |
| 5.36 | 5.48306 |
| 5.4 | 5.51525 |
| 5.44 | 5.547781 |
| 5.48 | 5.580648 |
| 5.52 | 5.613844 |
| 5.56 | 5.647363 |
| 5.6 | 5.681193 |
| 5.64 | 5.715325 |
| 5.68 | 5.749746 |
| 5.72 | 5.784441 |
| 5.76 | 5.819395 |
| 5.8 | 5.854588 |
| 5.84 | 5.890002 |
| 5.88 | 5.925613 |
| 5.92 | 5.961396 |
| 5.96 | 5.997325 |
| 6 | 6.033369 |
| Quantity | Value | Unit |
|---|---|---|
| Hydronium concentration | 0.0003112673 | mol/L |
| Conjugate base concentration | 0.0003112673 | mol/L |
| Undissociated acid concentration | 0.009688733 | mol/L |
| Exact ideal pH | 3.506866 | pH units |
| Fraction of acid dissociated | 0.03112673 | fraction |
| Square-root approximate pH | 3.5 | pH units |
| Charge-balance residual | -3.794708e-19 | mol/L |
Water, charge and acid-family mass balance are satisfied by one positive hydrogen concentration.
Mass balance gives A−=C Ka/(Ka+h); insert it into h=Kw/h+A−. A dilute weak acid can have a larger dissociated fraction even while hydronium concentration falls. Acid strength is the equilibrium constant, not the analytical concentration, and an apparent square-root agreement does not independently verify activity coefficients.
These calculated packets are not performed chemical measurements. Keep feasible inventories, activity conventions, fitting observations and independent checks distinct.
Water, charge and acid-family mass balance are satisfied by one positive hydrogen concentration.
Mass balance gives A−=C Ka/(Ka+h); insert it into h=Kw/h+A−. A dilute weak acid can have a larger dissociated fraction even while hydronium concentration falls. Acid strength is the equilibrium constant, not the analytical concentration, and an apparent square-root agreement does not independently verify activity coefficients.
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What omitted response assumption could make a precise concentration-based pH calculation disagree with a real observation?
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