SCIENCE · INVESTIGATE AND EXPLAIN

Close a shared inventory across two equilibrium relations

A shared intermediate belongs to one conserved population, not to independent reaction batches.

Learning outcome: Vary K2 at fixed K1, calculate all three amounts from one total, and verify both local ratios and the overall constant.

Starting recap

Refresh dimensionless activities, component balances, admissible roots, normalized partial pressures and independent versus shared measurement errors.

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Try

With K1=2, K2=3 and A+B+C=1 M, the ratio 1:2:6 gives A=1/9, B=2/9 and C=6/9 M.

Open the experiment

Predict

Can B decrease as K2 increases even though B/A stays fixed?

Your prediction and reason

Experiment

Vary K2 at fixed K1, calculate all three amounts from one total, and verify both local ratios and the overall constant.

Conserve inventory across two coupled isomer equilibria

Worked teaching model · synthetic values · Ideal isomers with total concentration1.00 M, K1=B/A and K2=C/B, constant temperature and volume. Both equilibria share B and must use one mass balance; they are not two independent one-molar batches.

A⇌B constant: 2 dimensionless B⇌C constant: 3 dimensionless

Reset keeps your prediction, captures and practice history.

A fraction as second K varies: solid
B fraction as second K varies: long dashes
C fraction as second K varies: short dashes

Axes rescale. A quotient comparison, inventory intersection and relaxation trajectory have different horizontal meanings; inspect labels rather than treating every curve as elapsed time.

All calculated plot coordinates

A fraction as second K varies
fraction of K2 range 1 to 4	equilibrium fraction
0	0.2
0.01666667	0.1960784
0.03333333	0.1923077
0.05	0.1886792
0.06666667	0.1851852
0.08333333	0.1818182
0.1	0.1785714
0.1166667	0.1754386
0.1333333	0.1724138
0.15	0.1694915
0.1666667	0.1666667
0.1833333	0.1639344
0.2	0.1612903
0.2166667	0.1587302
0.2333333	0.15625
0.25	0.1538462
0.2666667	0.1515152
0.2833333	0.1492537
0.3	0.1470588
0.3166667	0.1449275
0.3333333	0.1428571
0.35	0.1408451
0.3666667	0.1388889
0.3833333	0.1369863
0.4	0.1351351
0.4166667	0.1333333
0.4333333	0.1315789
0.45	0.1298701
0.4666667	0.1282051
0.4833333	0.1265823
0.5	0.125
0.5166667	0.1234568
0.5333333	0.1219512
0.55	0.1204819
0.5666667	0.1190476
0.5833333	0.1176471
0.6	0.1162791
0.6166667	0.1149425
0.6333333	0.1136364
0.65	0.1123596
0.6666667	0.1111111
0.6833333	0.1098901
0.7	0.1086957
0.7166667	0.1075269
0.7333333	0.106383
0.75	0.1052632
0.7666667	0.1041667
0.7833333	0.1030928
0.8	0.1020408
0.8166667	0.1010101
0.8333333	0.1
0.85	0.0990099
0.8666667	0.09803922
0.8833333	0.09708738
0.9	0.09615385
0.9166667	0.0952381
0.9333333	0.09433962
0.95	0.09345794
0.9666667	0.09259259
0.9833333	0.09174312
1	0.09090909

B fraction as second K varies
fraction of K2 range 1 to 4	equilibrium fraction
0	0.4
0.01666667	0.3921569
0.03333333	0.3846154
0.05	0.3773585
0.06666667	0.3703704
0.08333333	0.3636364
0.1	0.3571429
0.1166667	0.3508772
0.1333333	0.3448276
0.15	0.3389831
0.1666667	0.3333333
0.1833333	0.3278689
0.2	0.3225806
0.2166667	0.3174603
0.2333333	0.3125
0.25	0.3076923
0.2666667	0.3030303
0.2833333	0.2985075
0.3	0.2941176
0.3166667	0.2898551
0.3333333	0.2857143
0.35	0.2816901
0.3666667	0.2777778
0.3833333	0.2739726
0.4	0.2702703
0.4166667	0.2666667
0.4333333	0.2631579
0.45	0.2597403
0.4666667	0.2564103
0.4833333	0.2531646
0.5	0.25
0.5166667	0.2469136
0.5333333	0.2439024
0.55	0.2409639
0.5666667	0.2380952
0.5833333	0.2352941
0.6	0.2325581
0.6166667	0.2298851
0.6333333	0.2272727
0.65	0.2247191
0.6666667	0.2222222
0.6833333	0.2197802
0.7	0.2173913
0.7166667	0.2150538
0.7333333	0.212766
0.75	0.2105263
0.7666667	0.2083333
0.7833333	0.2061856
0.8	0.2040816
0.8166667	0.2020202
0.8333333	0.2
0.85	0.1980198
0.8666667	0.1960784
0.8833333	0.1941748
0.9	0.1923077
0.9166667	0.1904762
0.9333333	0.1886792
0.95	0.1869159
0.9666667	0.1851852
0.9833333	0.1834862
1	0.1818182

C fraction as second K varies
fraction of K2 range 1 to 4	equilibrium fraction
0	0.4
0.01666667	0.4117647
0.03333333	0.4230769
0.05	0.4339623
0.06666667	0.4444444
0.08333333	0.4545455
0.1	0.4642857
0.1166667	0.4736842
0.1333333	0.4827586
0.15	0.4915254
0.1666667	0.5
0.1833333	0.5081967
0.2	0.516129
0.2166667	0.5238095
0.2333333	0.53125
0.25	0.5384615
0.2666667	0.5454545
0.2833333	0.5522388
0.3	0.5588235
0.3166667	0.5652174
0.3333333	0.5714286
0.35	0.5774648
0.3666667	0.5833333
0.3833333	0.5890411
0.4	0.5945946
0.4166667	0.6
0.4333333	0.6052632
0.45	0.6103896
0.4666667	0.6153846
0.4833333	0.6202532
0.5	0.625
0.5166667	0.6296296
0.5333333	0.6341463
0.55	0.6385542
0.5666667	0.6428571
0.5833333	0.6470588
0.6	0.6511628
0.6166667	0.6551724
0.6333333	0.6590909
0.65	0.6629213
0.6666667	0.6666667
0.6833333	0.6703297
0.7	0.673913
0.7166667	0.6774194
0.7333333	0.6808511
0.75	0.6842105
0.7666667	0.6875
0.7833333	0.6907216
0.8	0.6938776
0.8166667	0.6969697
0.8333333	0.7
0.85	0.7029703
0.8666667	0.7058824
0.8833333	0.7087379
0.9	0.7115385
0.9166667	0.7142857
0.9333333	0.7169811
0.95	0.7196262
0.9666667	0.7222222
0.9833333	0.7247706
1	0.7272727

Calculated under the declared model
Quantity	Value	Unit
A equilibrium concentration	0.1111111	mol/L
B equilibrium concentration	0.2222222	mol/L
C equilibrium concentration	0.6666667	mol/L
Conserved total	1	mol/L
Overall C/A constant	6	dimensionless

One shared mass balance closes the coupled network.

The ratios A:B:C are 1:K1:K1K2. Increasing K2 drains B relative to the total even though B/A remains K1. Treating each arrow as a separate one-molar batch would duplicate the shared material and give an inconsistent answer.

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

One shared mass balance closes the coupled network.

Observation to attach to a capture

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

Understand

Use ratios to reduce the unknowns

For ideal same-formula isomers A⇌B⇌C, the equilibrium relations are B=K1A and C=K2B=K1K2A. Substitute these into A+B+C=T to obtain A=T/(1+K1+K1K2). The other two concentrations follow without treating the arrows as separate chemical vessels.

At K1=2, K2=3 and T=1 M, the concentrations are 1/9, 2/9 and 6/9 M. Both B/A=2 and C/B=3 hold. The overall C/A ratio is 6, consistent with multiplication of constants for added reactions.

Interpret a controlled comparison

Holding K1 fixed while increasing K2 shifts the common equilibrium population toward C. Both A and B fractions decline, even though their ratio remains K1. This does not mean K1 became smaller. It shows why a ratio constraint and an absolute concentration are different statements.

The slider comparisons represent different supplied parameter conditions; they do not claim an experiment can change only one microscopic constant without affecting other chemistry. Each model solution is at its declared fixed conditions.

Transfer and check

Choose K1=3, K2=2 and an independently supplied total .80 M. The ratio is 1:3:6, so A=.080, B=.240 and C=.480 M. Verify all three constraints. If a reported result instead has B=.300 M with A=.080 M, identify the violated ratio before proposing any new reaction mechanism. A calibration or transcription problem is also a competing explanation.

Source and evidence boundary

Conceptual reference checked 30 October 2026: OpenStax Chemistry 2e. These original hypothetical packets use explicitly defined activities, inventories and response assumptions. No material is mixed, heated, pressurized or measured. Numerical compatibility does not certify practical technique or uniquely establish a real chemical mechanism. Retain your prediction, feasible inventory, quotient check and changed-case interpretation.

Apply

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

What double counting occurs if each equilibrium arrow is solved using its own initial 1 M supply?

Your explanation and limitations

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Previous: Recalculate partial pressures after a gas-volume perturbation Next: Validate an approximation using depletion and output error

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