SCIENCE · INVESTIGATE AND EXPLAIN

Solve association while rejecting impossible roots

A valid equilibrium solution must satisfy both a quotient and nonnegative material inventories.

Learning outcome: Bracket x using the smaller initial reactant. Compare unequal inventories, recover Q, and check A+C and B+C separately rather than conserving the total molecule count.

Starting recap

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

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

For A0=B0=1.00 M, C0=0 and K=4, x=4(1−x)² gives x=(9−√17)/8≈.609612 M.

Open the experiment

Predict

Could the second algebraic root describe a different equilibrium of the same closed packet?

Your prediction and reason

Experiment

Bracket x using the smaller initial reactant. Compare unequal inventories, recover Q, and check A+C and B+C separately rather than conserving the total molecule count.

Solve a closed association with feasible inventories

Worked teaching model · synthetic values · Ideal concentration activities with c°=1 M, A+B⇌C, K=4, initial C=0, constant volume. Extent x has concentration units: A=A0−x, B=B0−x, C=x. Unique root is bracketed between zero and the smaller reactant inventory.

Initial A concentration: 1 mol/L Initial B concentration: 1 mol/L

Reset keeps your prediction, captures and practice history.

Product extent side: solid
K times remaining reactant product: long 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

Product extent side
concentration extent (mol/L)	normalized equation terms
0	0
0.01666667	0.01666667
0.03333333	0.03333333
0.05	0.05
0.06666667	0.06666667
0.08333333	0.08333333
0.1	0.1
0.1166667	0.1166667
0.1333333	0.1333333
0.15	0.15
0.1666667	0.1666667
0.1833333	0.1833333
0.2	0.2
0.2166667	0.2166667
0.2333333	0.2333333
0.25	0.25
0.2666667	0.2666667
0.2833333	0.2833333
0.3	0.3
0.3166667	0.3166667
0.3333333	0.3333333
0.35	0.35
0.3666667	0.3666667
0.3833333	0.3833333
0.4	0.4
0.4166667	0.4166667
0.4333333	0.4333333
0.45	0.45
0.4666667	0.4666667
0.4833333	0.4833333
0.5	0.5
0.5166667	0.5166667
0.5333333	0.5333333
0.55	0.55
0.5666667	0.5666667
0.5833333	0.5833333
0.6	0.6
0.6166667	0.6166667
0.6333333	0.6333333
0.65	0.65
0.6666667	0.6666667
0.6833333	0.6833333
0.7	0.7
0.7166667	0.7166667
0.7333333	0.7333333
0.75	0.75
0.7666667	0.7666667
0.7833333	0.7833333
0.8	0.8
0.8166667	0.8166667
0.8333333	0.8333333
0.85	0.85
0.8666667	0.8666667
0.8833333	0.8833333
0.9	0.9
0.9166667	0.9166667
0.9333333	0.9333333
0.95	0.95
0.9666667	0.9666667
0.9833333	0.9833333
1	1

K times remaining reactant product
concentration extent (mol/L)	normalized equation terms
0	4
0.01666667	3.867778
0.03333333	3.737778
0.05	3.61
0.06666667	3.484444
0.08333333	3.361111
0.1	3.24
0.1166667	3.121111
0.1333333	3.004444
0.15	2.89
0.1666667	2.777778
0.1833333	2.667778
0.2	2.56
0.2166667	2.454444
0.2333333	2.351111
0.25	2.25
0.2666667	2.151111
0.2833333	2.054444
0.3	1.96
0.3166667	1.867778
0.3333333	1.777778
0.35	1.69
0.3666667	1.604444
0.3833333	1.521111
0.4	1.44
0.4166667	1.361111
0.4333333	1.284444
0.45	1.21
0.4666667	1.137778
0.4833333	1.067778
0.5	1
0.5166667	0.9344444
0.5333333	0.8711111
0.55	0.81
0.5666667	0.7511111
0.5833333	0.6944444
0.6	0.64
0.6166667	0.5877778
0.6333333	0.5377778
0.65	0.49
0.6666667	0.4444444
0.6833333	0.4011111
0.7	0.36
0.7166667	0.3211111
0.7333333	0.2844444
0.75	0.25
0.7666667	0.2177778
0.7833333	0.1877778
0.8	0.16
0.8166667	0.1344444
0.8333333	0.1111111
0.85	0.09
0.8666667	0.07111111
0.8833333	0.05444444
0.9	0.04
0.9166667	0.02777778
0.9333333	0.01777778
0.95	0.01
0.9666667	0.004444444
0.9833333	0.001111111
1	0

Calculated under the declared model
Quantity	Value	Unit
Equilibrium concentration extent	0.6096118	mol/L
A remaining	0.3903882	mol/L
B remaining	0.3903882	mol/L
C formed	0.6096118	mol/L
Recovered dimensionless Q	4	dimensionless

The intersection satisfies both the equilibrium relation and the reactant inventories.

Only roots within 0≤x≤min(A0,B0) are admissible. The plotted equation uses concentration values numerically normalized by 1 M; its two sides are dimensionless. A+C=A0 and B+C=B0 are the conserved component inventories, while A+B+C is not conserved for association.

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

The intersection satisfies both the equilibrium relation and the reactant inventories.

Observation to attach to a capture

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

Understand

Separate equation solving from physical admissibility

For A+B⇌C at fixed volume, define concentration extent x. Initial A0 and B0 become A0−x and B0−x, while C=x. Ideal activities normalized by c°=1 M give a dimensionless quotient numerically equal to x/[(A0−x)(B0−x)] when all concentration values are in mol/L.

With K=4 and both initial reactants 1 M, rearrangement gives 4x²−9x+4=0. The roots are approximately .609612 and 1.640388. Only the first lies in the admissible interval from zero to 1 M. The second is algebraically real but physically creates negative reactant inventories.

Check what is actually conserved

At the admissible root, A=B≈.390388 and C≈.609612 M. Each A+C and B+C equals 1 M. Their sum A+B+C is smaller than the initial two-molar species total because association changes the number of molecules. Component conservation is not the same as conservation of species count.

Use a bracket rather than a lucky root choice

The numerical model finds the intersection of the two normalized equation sides within the feasible interval. At its lower endpoint the reaction term dominates; at the upper endpoint at least one reactant is exhausted. No iteration may leave that inventory bracket. The displayed curves are algebraic constraints, not time trajectories.

Transfer and check

Take A0=1.00 and B0=.50 M with the same K. The equation is 4x²−7x+2=0, giving the admissible x=(7−√17)/8≈.359612 M. Recover A≈.640388, B≈.140388 and C≈.359612. Explain why the equilibrium C is below the .50 M limiting-reactant maximum even though the forward direction initially is favored.

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.

Which conservation law rules out the larger quadratic root?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Transform equilibrium constants with exact reaction statements Next: Recalculate partial pressures after a gas-volume perturbation

Save a science backup or import one

Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.

A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.

Choose a science backup (maximum 36 MB)

Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.

Complete file sets and course backups

A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.

Export scope

The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.

Choose the manifest and all parts together

0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.

Science storage: loading. No account or remote computation is required.