SCIENCE · INVESTIGATE AND EXPLAIN

Normalize stepwise proton populations without inventing the pH

A conditional species distribution is not automatically a self-consistent solution composition.

Learning outcome: Scan imposed pH and verify normalized fractions. Change total family concentration without changing the imposed pH, then identify the omitted reservoir that maintains charge balance.

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

For pKa 1=3,pKa 2=7 at imposed pH5, fractions are approximately .009804,.980392,.009804 for H2A,HA−,A²−.

Open the experiment

Predict

Does the dominant middle form at pH5 mean the other forms are exactly absent?

Your prediction and reason

Experiment

Scan imposed pH and verify normalized fractions. Change total family concentration without changing the imposed pH, then identify the omitted reservoir that maintains charge balance.

Distribute a diprotic family at externally controlled pH

Worked teaching model · synthetic values · Fictional H2A/HA−/A²− with pKa 1=3 and pKa 2=7, ideal activities at fixed temperature. An unspecified external reservoir fixes pH and supplies charge balance; this is a conditional distribution, not the self-consistent pH of pure acid. No real buffer preparation.

Externally buffered pH: 5 pH units Analytical family concentration: 0.02 mol/L

Reset keeps your prediction, captures and practice history.

H2A fraction: solid
HA− fraction: long dashes
A²− fraction: short dashes

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

H2A fraction
externally maintained pH	family fraction
1	0.990099
1.08	0.9881202
1.16	0.9857515
1.24	0.9829188
1.32	0.9795346
1.4	0.9754966
1.48	0.9706857
1.56	0.9649641
1.64	0.958174
1.72	0.9501359
1.8	0.9406487
1.88	0.9294904
1.96	0.9164207
2.04	0.9011859
2.12	0.8835269
2.2	0.8631912
2.28	0.8399481
2.36	0.8136087
2.44	0.784049
2.52	0.7512346
2.6	0.7152446
2.68	0.6762912
2.76	0.63473
2.84	0.5910589
2.92	0.5459013
3	0.499975
3.08	0.4540483
3.16	0.4088895
3.24	0.3652164
3.32	0.3236526
3.4	0.2846961
3.48	0.2487028
3.56	0.2158849
3.64	0.1863216
3.72	0.1599788
3.8	0.1367324
3.88	0.1163937
3.96	0.09873202
4.04	0.08349472
4.12	0.07042281
4.2	0.05926259
4.28	0.04977374
4.36	0.04173421
4.44	0.03494287
4.52	0.02922032
4.6	0.02440858
4.68	0.02036988
4.76	0.0169851
4.84	0.01415193
4.92	0.01178298
5	0.009803922
5.08	0.008151827
5.16	0.006773541
5.24	0.005624296
5.32	0.004666469
5.4	0.003868499
5.48	0.003203944
5.56	0.002650687
5.64	0.002190243
5.72	0.001807175
5.8	0.001488609
5.88	0.001223808
5.96	0.001003827
6.04	0.0008212172
6.12	0.0006697758
6.2	0.0005443416
6.28	0.0004406182
6.36	0.0003550285
6.44	0.0002845917
6.52	0.0002268196
6.6	0.0001796311
6.68	0.0001412795
6.76	0.0001102936
6.84	8.542912e-5
6.92	6.562994e-5
7	4.999750e-5
7.08	3.776714e-5
7.16	2.828986e-5
7.24	2.101789e-5
7.32	1.549294e-5
7.4	1.133586e-5
7.48	8.237124e-6
7.56	5.947620e-6
7.64	4.269879e-6
7.72	3.049667e-6
7.8	2.168238e-6
7.88	1.535394e-6
7.96	1.083464e-6
8.04	7.622455e-7
8.12	5.348659e-7
8.2	3.744790e-7
8.28	2.616892e-7
8.36	1.825763e-7
8.44	1.272070e-7
8.52	8.852759e-8
8.6	6.154967e-8
8.68	4.275824e-8
8.76	2.968367e-8
8.84	2.059527e-8
8.92	1.428268e-8
9	9.900990e-9

HA− fraction
externally maintained pH	family fraction
1	0.00990099
1.08	0.01187982
1.16	0.01424844
1.24	0.01708117
1.32	0.02046538
1.4	0.02450337
1.48	0.02931424
1.56	0.03503573
1.64	0.04182581
1.72	0.04986384
1.8	0.05935092
1.88	0.07050906
1.96	0.08357856
2.04	0.09881306
2.12	0.1164715
2.2	0.1368066
2.28	0.1600488
2.36	0.186387
2.44	0.215945
2.52	0.2487572
2.6	0.284744
2.68	0.3236933
2.76	0.365249
2.84	0.4089128
2.92	0.4540609
3	0.499975
3.08	0.5458861
3.16	0.5910251
3.24	0.6346733
3.32	0.6762061
3.4	0.7151243
3.48	0.7510704
3.56	0.7838305
3.64	0.8133234
3.72	0.8395806
3.8	0.8627232
3.88	0.8829365
3.96	0.9004468
4.04	0.9155014
4.12	0.9283534
4.2	0.9392488
4.28	0.9484191
4.36	0.9560755
4.44	0.9624064
4.52	0.9675757
4.6	0.9717229
4.68	0.9749637
4.76	0.9773906
4.84	0.9790745
4.92	0.9800652
5	0.9803922
5.08	0.9800652
5.16	0.9790745
5.24	0.9773906
5.32	0.9749637
5.4	0.9717229
5.48	0.9675757
5.56	0.9624064
5.64	0.9560755
5.72	0.9484191
5.8	0.9392488
5.88	0.9283534
5.96	0.9155014
6.04	0.9004468
6.12	0.8829365
6.2	0.8627232
6.28	0.8395806
6.36	0.8133234
6.44	0.7838305
6.52	0.7510704
6.6	0.7151243
6.68	0.6762061
6.76	0.6346733
6.84	0.5910251
6.92	0.5458861
7	0.499975
7.08	0.4540609
7.16	0.4089128
7.24	0.365249
7.32	0.3236933
7.4	0.284744
7.48	0.2487572
7.56	0.215945
7.64	0.186387
7.72	0.1600488
7.8	0.1368066
7.88	0.1164715
7.96	0.09881306
8.04	0.08357856
8.12	0.07050906
8.2	0.05935092
8.28	0.04986384
8.36	0.04182581
8.44	0.03503573
8.52	0.02931424
8.6	0.02450337
8.68	0.02046538
8.76	0.01708117
8.84	0.01424844
8.92	0.01187982
9	0.00990099

A²− fraction
externally maintained pH	family fraction
1	9.900990e-9
1.08	1.428268e-8
1.16	2.059527e-8
1.24	2.968367e-8
1.32	4.275824e-8
1.4	6.154967e-8
1.48	8.852759e-8
1.56	1.272070e-7
1.64	1.825763e-7
1.72	2.616892e-7
1.8	3.744790e-7
1.88	5.348659e-7
1.96	7.622455e-7
2.04	1.083464e-6
2.12	1.535394e-6
2.2	2.168238e-6
2.28	3.049667e-6
2.36	4.269879e-6
2.44	5.947620e-6
2.52	8.237124e-6
2.6	1.133586e-5
2.68	1.549294e-5
2.76	2.101789e-5
2.84	2.828986e-5
2.92	3.776714e-5
3	4.999750e-5
3.08	6.562994e-5
3.16	8.542912e-5
3.24	0.0001102936
3.32	0.0001412795
3.4	0.0001796311
3.48	0.0002268196
3.56	0.0002845917
3.64	0.0003550285
3.72	0.0004406182
3.8	0.0005443416
3.88	0.0006697758
3.96	0.0008212172
4.04	0.001003827
4.12	0.001223808
4.2	0.001488609
4.28	0.001807175
4.36	0.002190243
4.44	0.002650687
4.52	0.003203944
4.6	0.003868499
4.68	0.004666469
4.76	0.005624296
4.84	0.006773541
4.92	0.008151827
5	0.009803922
5.08	0.01178298
5.16	0.01415193
5.24	0.0169851
5.32	0.02036988
5.4	0.02440858
5.48	0.02922032
5.56	0.03494287
5.64	0.04173421
5.72	0.04977374
5.8	0.05926259
5.88	0.07042281
5.96	0.08349472
6.04	0.09873202
6.12	0.1163937
6.2	0.1367324
6.28	0.1599788
6.36	0.1863216
6.44	0.2158849
6.52	0.2487028
6.6	0.2846961
6.68	0.3236526
6.76	0.3652164
6.84	0.4088895
6.92	0.4540483
7	0.499975
7.08	0.5459013
7.16	0.5910589
7.24	0.63473
7.32	0.6762912
7.4	0.7152446
7.48	0.7512346
7.56	0.784049
7.64	0.8136087
7.72	0.8399481
7.8	0.8631912
7.88	0.8835269
7.96	0.9011859
8.04	0.9164207
8.12	0.9294904
8.2	0.9406487
8.28	0.9501359
8.36	0.958174
8.44	0.9649641
8.52	0.9706857
8.6	0.9754966
8.68	0.9795346
8.76	0.9829188
8.84	0.9857515
8.92	0.9881202
9	0.990099

Calculated under the declared model
Quantity	Value	Unit
H2A fraction	0.009803922	fraction
H2A concentration	0.0001960784	mol/L
HA− fraction	0.9803922	fraction
HA− concentration	0.01960784	mol/L
A²− fraction	0.009803922	fraction
A²− concentration	0.0001960784	mol/L
Mean negative charge per family member	1	elementary-charge units

Stepwise constants distribute one analytical family; they do not imply both protons leave at once.

The common denominator h²+Ka1 h+Ka1Ka2 normalizes all three species. At pH5 the middle form dominates, but is not exactly100%. pH is imposed by an external reservoir whose charge balance is not shown; these curves must not be presented as the pH produced by dissolving pure H2A. Amphiprotic HA− can accept or donate a proton; its isolated-salt pH requires its own charge balance.

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

Stepwise constants distribute one analytical family; they do not imply both protons leave at once.

Observation to attach to a capture

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

Understand

Derive one common denominator

For a fictional diprotic family, adjacent ratios are HA−/H2A=Ka1/h and A²−/HA−=Ka2/h. Their normalized fractions have denominator D=h²+Ka1 h+Ka1Ka2: alpha0=h²/D, alpha1=Ka1 h/D and alpha2=Ka1Ka2/D.

Every fraction is nonnegative and their sum is one. Multiplying each by the total family concentration gives species concentrations. The mean negative charge is alpha1+2alpha2; total family amount and total electrical charge are not interchangeable constraints.

State who fixes the pH

Here an external buffer reservoir imposes pH. Its ions supply the missing charge balance, but are not explicitly plotted. The curves therefore answer which forms exist at a chosen pH, not what pH results from dissolving pure H2A. That second question requires a closed-system charge balance and water equilibrium.

At pKa 1=3 and pKa 2=7, imposed pH5 gives ratio 1:100:1, so HA− is about 98.0392%, not100%. It is amphiprotic: it can donate its remaining transferable proton or accept one to reform H2A.

Transfer and check

At imposed pH7, adjacent ratios give H2A:HA−:A²−=1:10000:10000. The latter two forms are equal but each is slightly less than one half. At total .0200 M, each is about.0099995 M. Explain why equal species fractions at a stepwise pKa do not imply all other family members are absent, and why a molecular formula’s total hydrogen count is not automatically its transferable-proton count.

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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Public practice checks a specific claim. It does not establish independent mastery.

Why cannot these plotted fractions alone predict the pH obtained by dissolving pure H2A?

Your explanation and limitations

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Previous: Identify which salt ion actually transfers a proton Next: Spend a finite buffer reserve before using a logarithmic ratio

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