SCIENCE · INVESTIGATE AND EXPLAIN

Bound an inferred constant and test a changed inventory

A shared calibration scale can cancel in a ratio without verifying every assumption behind that ratio.

Learning outcome: Change the shared scale, calculate ratio bounds from separate channel errors, then predict a held-out .60 M total. Identify an independent observation needed to test response selectivity or time stability.

Starting recap

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

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Try

For nominal A=1/3 and B=2/3 M, any shared positive scale preserves B/A=2. Independent ±.01 M channel bounds instead give a finite feasible range.

Open the experiment

Predict

Does a stable ratio prove the chemical system has reached equilibrium rather than a stationary biased response?

Your prediction and reason

Experiment

Change the shared scale, calculate ratio bounds from separate channel errors, then predict a held-out .60 M total. Identify an independent observation needed to test response selectivity or time stability.

Separate ratio calibration from equilibrium evidence

Worked teaching model · synthetic values · Fictional ideal A⇌B at total1 M. True A=1/(1+ratio), B=ratio*A; indicated concentrations share the same unknown positive multiplicative factor. Independent additive channel errors are bounded ±.01 M after nominal scaling, not probabilistic confidence intervals. Held-out total inventory is .60 M.

Nominal equilibrium B/A ratio: 2 ratio Shared multiplicative signal factor: 1 ratio

Reset keeps your prediction, captures and practice history.

Predicted B at changed total: solid

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

Predicted B at changed total
new independently known total concentration (mol/L)	predicted B concentration (mol/L)
0	0
0.01666667	0.01111111
0.03333333	0.02222222
0.05	0.03333333
0.06666667	0.04444444
0.08333333	0.05555556
0.1	0.06666667
0.1166667	0.07777778
0.1333333	0.08888889
0.15	0.1
0.1666667	0.1111111
0.1833333	0.1222222
0.2	0.1333333
0.2166667	0.1444444
0.2333333	0.1555556
0.25	0.1666667
0.2666667	0.1777778
0.2833333	0.1888889
0.3	0.2
0.3166667	0.2111111
0.3333333	0.2222222
0.35	0.2333333
0.3666667	0.2444444
0.3833333	0.2555556
0.4	0.2666667
0.4166667	0.2777778
0.4333333	0.2888889
0.45	0.3
0.4666667	0.3111111
0.4833333	0.3222222
0.5	0.3333333
0.5166667	0.3444444
0.5333333	0.3555556
0.55	0.3666667
0.5666667	0.3777778
0.5833333	0.3888889
0.6	0.4
0.6166667	0.4111111
0.6333333	0.4222222
0.65	0.4333333
0.6666667	0.4444444
0.6833333	0.4555556
0.7	0.4666667
0.7166667	0.4777778
0.7333333	0.4888889
0.75	0.5
0.7666667	0.5111111
0.7833333	0.5222222
0.8	0.5333333
0.8166667	0.5444444
0.8333333	0.5555556
0.85	0.5666667
0.8666667	0.5777778
0.8833333	0.5888889
0.9	0.6
0.9166667	0.6111111
0.9333333	0.6222222
0.95	0.6333333
0.9666667	0.6444444
0.9833333	0.6555556
1	0.6666667

Calculated under the declared model
Quantity	Value	Unit
Indicated A	0.3333333	mol/L
Indicated B	0.6666667	mol/L
Shared-factor ratio	2	dimensionless
K lower feasible bound	1.912621	dimensionless
K upper feasible bound	2.092784	dimensionless
Held-out B at total .60 M	0.4	mol/L
Held-out B lower feasible bound	0.394	mol/L
Held-out B upper feasible bound	0.406	mol/L

A shared positive scale cancels in a same-order isomer ratio, but independent channel errors do not.

The quotient bounds use the lowest numerator with highest denominator and vice versa; denominators remain positive throughout this grid. They are worst-case feasible intervals under stated independent additive bounds, not confidence intervals. Ratio compatibility alone does not verify equilibrium, species selectivity or the common-scale assumption; retain a changed-inventory or time-stability check.

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

A shared positive scale cancels in a same-order isomer ratio, but independent channel errors do not.

Observation to attach to a capture

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

Understand

Separate a shared scale from separate errors

For ideal A⇌B at total1 M and K=2, concentrations are1/3 and2/3 M. If both indicated values are multiplied by one positive factor g, their ratio remains2. Their indicated sum does not remain1 M. Cancellation in one observable is not complete calibration of the instrument.

Independent additive channel errors do not cancel. For indicated readings A_s and B_s and separate ±.01 M bounds, the feasible quotient runs from (B_s−.01)/(A_s+.01) to (B_s+.01)/(A_s−.01), provided all denominator values remain positive. The model grid guarantees that condition.

Carry a bound into a held-out prediction

At nominal scale1, K lies approximately between1.91262 and2.09278 under these bounds. For a new independently known total T=.60 M, predicted B=TK/(1+K). Because this function increases with K, its endpoint predictions give the corresponding feasible output interval. This is bounded arithmetic, not a statistical confidence interval; no sampling distribution or coverage probability was specified.

Challenge the interpretation

A persistent ratio could reflect equilibrium, a stable non-equilibrium process, unequal species response factors or an unobserved species. A changed-inventory prediction or time-stability comparison adds evidence, but even agreement does not uniquely identify the microscopic mechanism. Do not reuse the fitted ratio itself as an independent validation observation.

Transfer and check

For a new packet K=3 and total .80 M, predict A=.20 and B=.60 M. A shared gain of1.2 would indicate .24 and .72 M, preserving the ratio while inflating their sum. Explain why using that inflated sum as an independently known .96 M inventory would silently reuse the same calibration assumption rather than test it.

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.

Why is this worst-case interval not a confidence interval, and why would unequal channel gains change the cancellation?

Your explanation and limitations

Continue

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

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Previous: Distinguish a catalytic time scale from an equilibrium ratio Next: Connect proton transfer, conjugate stability and Lewis roles

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