SCIENCE · INVESTIGATE AND EXPLAIN

Spend a finite buffer reserve before using a logarithmic ratio

Buffer calculations need a proton inventory before an equilibrium relation.

Learning outcome: Compare additions below, at and above the reserve. Preserve the stoichiometric table, exact equilibrium species and the domain where the HH ratio remains meaningful.

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

Initially 5 mmol each HA and A− in1 L, pKa 5; adding2 mmol strong acid leaves nominal3 mmol A− and 7 mmol HA, giving HH pH≈4.6320 before exact equilibrium correction.

Open the experiment

Predict

What fails if the added strong acid exhausts the nominal conjugate-base reserve?

Your prediction and reason

Experiment

Compare additions below, at and above the reserve. Preserve the stoichiometric table, exact equilibrium species and the domain where the HH ratio remains meaningful.

Apply finite neutralization before an equilibrium ratio

Worked teaching model · synthetic values · Fictional pKa 5 buffer with equal initial HA and A− amounts in1 L. Strong acid adds chloride without appreciable volume change. Solve h+Na=OH+A−+Cl exactly; HH stoichiometric ratio is only reported while both nominal buffer forms remain. Ideal activities, Kw10⁻¹⁴; no mixing protocol.

Strong acid added to fixed1L packet: 2 mmol Initial amount of each HA and sodium A: 5 mmol each

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Exact pH after acid addition: 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

Exact pH after acid addition
strong acid added (mmol to fixed 1 L)	pH
0	5.00173
0.06	4.991349
0.12	4.980966
0.18	4.970578
0.24	4.960184
0.3	4.949779
0.36	4.939361
0.42	4.928926
0.48	4.918473
0.54	4.907997
0.6	4.897496
0.66	4.886966
0.72	4.876405
0.78	4.865809
0.84	4.855174
0.9	4.844499
0.96	4.833778
1.02	4.823009
1.08	4.812188
1.14	4.801312
1.2	4.790376
1.26	4.779377
1.32	4.768311
1.38	4.757174
1.44	4.74596
1.5	4.734667
1.56	4.723289
1.62	4.711822
1.68	4.700261
1.74	4.6886
1.8	4.676834
1.86	4.664958
1.92	4.652966
1.98	4.640852
2.04	4.628609
2.1	4.616231
2.16	4.60371
2.22	4.59104
2.28	4.578213
2.34	4.565219
2.4	4.552051
2.46	4.5387
2.52	4.525156
2.58	4.511408
2.64	4.497445
2.7	4.483257
2.76	4.46883
2.82	4.454151
2.88	4.439207
2.94	4.423982
3	4.408461
3.06	4.392625
3.12	4.376457
3.18	4.359937
3.24	4.343043
3.3	4.325751
3.36	4.308038
3.42	4.289876
3.48	4.271236
3.54	4.252087
3.6	4.232393
3.66	4.21212
3.72	4.191225
3.78	4.169666
3.84	4.147397
3.9	4.124365
3.96	4.100516
4.02	4.075792
4.08	4.05013
4.14	4.023464
4.2	3.995725
4.26	3.966843
4.32	3.936746
4.38	3.905368
4.44	3.872648
4.5	3.838538
4.56	3.803012
4.62	3.766072
4.68	3.727759
4.74	3.688168
4.8	3.647452
4.86	3.60583
4.92	3.563585
4.98	3.521048
5.04	3.478581
5.1	3.436548
5.16	3.395286
5.22	3.35508
5.28	3.316155
5.34	3.278667
5.4	3.242709
5.46	3.208319
5.52	3.175495
5.58	3.1442
5.64	3.114379
5.7	3.085963
5.76	3.058875
5.82	3.033037
5.88	3.008368
5.94	2.984793
6	2.962239

Calculated under the declared model
Quantity	Value	Unit
Total acid-family concentration	0.01	mol/L
Strong acid added	0.002	mol
Exact equilibrated pH	4.636786	pH units
Equilibrium A− concentration	0.003023078	mol/L
Equilibrium HA concentration	0.006976922	mol/L
Net inert cation minus anion concentration	0.003	mol/L
Stoichiometric HH approximation	4.632023	pH units

Both nominal buffer forms survive the stoichiometric step; compare its ratio approximation with exact equilibrium.

First account for transferred protons: nominal A− amount b−a and HA b+a mmol while the base reserve remains. Then equilibrium redistributes those species. The exact charge balance handles the exhaustion boundary and excess strong acid without an invalid HH logarithm. A buffer resists a limited perturbation; it does not hold pH fixed for arbitrary addition.

These calculated packets are not performed chemical measurements. Keep feasible inventories, activity conventions, fitting observations and independent checks distinct.

Both nominal buffer forms survive the stoichiometric step; compare its ratio approximation with exact equilibrium.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Perform the finite proton bookkeeping first

A buffer contains a weak acid and its conjugate base in useful amounts. In the fictional 1 L packet, begin with 5 mmol HA and 5 mmol sodium A. A 2 mmol strong-acid addition converts nominal A− into HA, leaving3 and 7 mmol. The sodium remains; the added acid contributes an inert anion.

The concentration-form Henderson–Hasselbalch relation follows from Ka=h[A−]/[HA]: pH≈pKa+log([A−]/[HA]) in the ideal approximation. Applying it to the nominal postreaction amounts gives 5+log(3/7)≈4.6320. These amounts are a useful stoichiometric starting point, not the exact final equilibrium species.

Keep charge balance across the boundary

The exact model solves h+Na=OH+A−+Cl with the conserved acid-family total and water equilibrium. It therefore handles the exhaustion point and excess acid without taking a logarithm of zero or negative base inventory. When the added acid reaches5 mmol, the nominal base reserve is exhausted; do not continue the buffer shortcut by force.

A buffer resists limited additions rather than fixing pH absolutely. Dilution and finite volume changes would add further constraints; this model explicitly holds volume fixed to isolate the inventory issue.

Transfer and check

Begin instead with 4 mmol of each form and add1 mmol acid. Nominal amounts become3 mmol A− and 5 mmol HA, giving pH≈5+log(.6)=4.77815. If6 mmol acid were added, the buffer-reserve picture would fail: solve the remaining weak-acid and excess strong-acid balance rather than reporting log(−2/10).

Source and evidence boundary

Conceptual reference checked 30 October 2026: OpenStax Chemistry 2e. Prose, numerical packets and diagrams are original. All chemistry is fictional or supplied; no substance handling, mixing, titration apparatus or health interpretation is requested. The ideal concentration approximation, fixed temperature and inert-ion assumptions are explicit, not measured calibration. Retain your charge/mass balance, prediction, changed-case check and limits.

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Why can a correct finite reaction ledger still require a small equilibrium redistribution afterward?

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Previous: Normalize stepwise proton populations without inventing the pH Next: Distinguish a pH ratio from a differential amount reserve

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