SCIENCE · INVESTIGATE AND EXPLAIN

Derive a sequential population account with a removable equal-rate limit

A parent population N1 converts to a daughter N2, which converts to a stable product, with branching fraction one. Initially N1=N0 and N2=0. The mean equations are dN1/dt=−lambda1 N1 and dN2/dt=lambda1 N1−lambda2 N2. Solving the first gives N1=N0 exp(−lambda1 t).

Learning outcome: Derive the integrating-factor result and take the equal-rate limit before evaluating it. Verify inventory at initial time and two later times. Compare the daughter maximum condition in population and activity language, retaining explicit time units.

Starting recap

Review finite sums, exponential and logarithmic functions, radial integration measures, first-order coupled equations and mass-energy units. Keep state counts, occupations, calibration records and unused tests distinct.

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For equal rates, initial parent1000 and u=1, parent and daughter populations are each1000/e≈367.879441; stable product is1000(1−2/e)≈264.241118. With lambda1=lambda2=0.2 per minute, each activity is about73.575888 per minute. At r=2, the daughter fraction at u=1 is exp(−1)−exp(−2), not exp(−2) alone.

Open the experiment

Predict

At the daughter population maximum, which quantities are equal?

Your prediction and reason

Experiment

Derive the integrating-factor result and take the equal-rate limit before evaluating it. Verify inventory at initial time and two later times. Compare the daughter maximum condition in population and activity language, retaining explicit time units.

Take the equal-rate limit before dividing

Worked teaching model · synthetic values · Sequential parent to daughter to stable product with branching fraction1, initial parent 1000 and daughter 0. Parent lambda1=0.2 per minute; daughter lambda2=a lambda1. Display normalized mean populations and activities separately. No real radionuclide preparation, dose or detection claim.

Daughter decay constant divided by parent: 1 ratio Elapsed time in parent mean lifetimes: 1 ratio

Reset keeps your prediction, captures and practice history.

One conserved inventory through two transitions

Graph axes rescale independently; read units and values, not screen angles. 1: parent (solid).2: daughter (dashed).3: stable (dotted). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for One conserved inventory through two transitions

parent

time in parent mean lifetimes	fraction of initial population
0	1
0.06666667	0.935507
0.1333333	0.8751733
0.2	0.8187308
0.2666667	0.7659283
0.3333333	0.7165313
0.4	0.67032
0.4666667	0.6270891
0.5333333	0.5866462
0.6	0.5488116
0.6666667	0.5134171
0.7333333	0.4803053
0.8	0.449329
0.8666667	0.4203504
0.9333333	0.3932407
1	0.3678794
1.066667	0.3441538
1.133333	0.3219583
1.2	0.3011942
1.266667	0.2817693
1.333333	0.2635971
1.4	0.246597
1.466667	0.2306932
1.533333	0.2158151
1.6	0.2018965
1.666667	0.1888756
1.733333	0.1766944
1.8	0.1652989
1.866667	0.1546383
1.933333	0.1446652
2	0.1353353
2.066667	0.1266071
2.133333	0.1184418
2.2	0.1108032
2.266667	0.1036571
2.333333	0.09697197
2.4	0.09071795
2.466667	0.08486728
2.533333	0.07939393
2.6	0.07427358
2.666667	0.06948345
2.733333	0.06500225
2.8	0.06081006
2.866667	0.05688824
2.933333	0.05321934
3	0.04978707
3.066667	0.04657615
3.133333	0.04357231
3.2	0.0407622
3.266667	0.03813333
3.333333	0.03567399
3.4	0.03337327
3.466667	0.03122093
3.533333	0.0292074
3.6	0.02732372
3.666667	0.02556153
3.733333	0.02391299
3.8	0.02237077
3.866667	0.02092801
3.933333	0.0195783
4	0.01831564

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

daughter

time in parent mean lifetimes	fraction of initial population
0	0
0.06666667	0.06236713
0.1333333	0.1166898
0.2	0.1637462
0.2666667	0.2042476
0.3333333	0.2388438
0.4	0.268128
0.4666667	0.2926416
0.5333333	0.312878
0.6	0.329287
0.6666667	0.3422781
0.7333333	0.3522239
0.8	0.3594632
0.8666667	0.3643037
0.9333333	0.3670247
1	0.3678794
1.066667	0.3670974
1.133333	0.364886
1.2	0.3614331
1.266667	0.3569078
1.333333	0.3514629
1.4	0.3452357
1.466667	0.33835
1.533333	0.3309165
1.6	0.3230344
1.666667	0.3147927
1.733333	0.3062704
1.8	0.297538
1.866667	0.2886581
1.933333	0.279686
2	0.2706706
2.066667	0.2616547
2.133333	0.2526759
2.2	0.2437669
2.266667	0.2349562
2.333333	0.2262679
2.4	0.2177231
2.466667	0.2093393
2.533333	0.2011313
2.6	0.1931113
2.666667	0.1852892
2.733333	0.1776728
2.8	0.1702682
2.866667	0.1630796
2.933333	0.1561101
3	0.1493612
3.066667	0.1428335
3.133333	0.1365266
3.2	0.1304391
3.266667	0.1245689
3.333333	0.1189133
3.4	0.1134691
3.466667	0.1082325
3.533333	0.1031995
3.6	0.0983654
3.666667	0.09372562
3.733333	0.08927517
3.8	0.08500893
3.866667	0.08092165
3.933333	0.07700799
4	0.07326256

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

stable

time in parent mean lifetimes	fraction of initial population
0	0
0.06666667	0.002125883
0.1333333	0.008136905
0.2	0.0175231
0.2666667	0.0298241
0.3333333	0.04462492
0.4	0.06155194
0.4666667	0.08026934
0.5333333	0.1004758
0.6	0.1219014
0.6666667	0.1443048
0.7333333	0.1674708
0.8	0.1912079
0.8666667	0.2153459
0.9333333	0.2397346
1	0.2642411
1.066667	0.2887488
1.133333	0.3131557
1.2	0.3373727
1.266667	0.3613229
1.333333	0.38494
1.4	0.4081673
1.466667	0.4309568
1.533333	0.4532685
1.6	0.4750691
1.666667	0.4963317
1.733333	0.5170352
1.8	0.5371631
1.866667	0.5567036
1.933333	0.5756488
2	0.5939942
2.066667	0.6117382
2.133333	0.6288823
2.2	0.6454299
2.266667	0.6613867
2.333333	0.6767601
2.4	0.691559
2.466667	0.7057934
2.533333	0.7194748
2.6	0.7326151
2.666667	0.7452273
2.733333	0.7573249
2.8	0.7689218
2.866667	0.7800321
2.933333	0.7906706
3	0.8008517
3.066667	0.8105903
3.133333	0.8199011
3.2	0.8287987
3.266667	0.8372978
3.333333	0.8454127
3.4	0.8531576
3.466667	0.8605465
3.533333	0.8675931
3.6	0.8743109
3.666667	0.8807128
3.733333	0.8868118
3.8	0.8926203
3.866667	0.8981503
3.933333	0.9034137
4	0.9084218

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Parent mean population	367.8794	nuclei
Daughter mean population	367.8794	nuclei
Stable product mean population	264.2411	nuclei
Parent activity	73.57589	per minute
Daughter activity	73.57589	per minute
Population conservation total	1000	nuclei
Time of daughter population maximum in parent lifetimes	1	ratio

Daughter population rises and falls; daughter activity additionally contains its own decay constant.

Solve dN1/dt=−lambda1 N 1 and dN2/dt=lambda1 N 1−lambda2 N 2 with an integrating factor. For u=lambda1 t and r=lambda2/lambda1, N 2/N 0=(exp(−u)−exp(−ru))/(r−1); its equal-rate limit is u exp(−u). Stable product is the remaining conserved inventory. The daughter maximum occurs where production equals daughter decay, not where the two populations necessarily match.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Daughter population rises and falls; daughter activity additionally contains its own decay constant.

Observation to attach to a capture

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

Understand

Define the state and evidence

Derive the quantitative relation

Multiply the second equation by exp(lambda2 t) and integrate. For unequal rates, N2=N0 lambda1[exp(−lambda1 t)−exp(−lambda2 t)]/(lambda2−lambda1). In normalized time u=lambda1 t and ratio r=lambda2/lambda1, the daughter fraction is [exp(−u)−exp(−ru)]/(r−1). The apparent singularity at r=1 is removable.

Test a changed model or boundary

Take the equal-rate limit by differentiating the exponential numerator with respect to r, or solve the equal-rate integrating factor directly: N2/N0=u exp(−u). The stable fraction is 1−N1/N0−N2/N0, preserving total inventory. Mean populations may be nonintegers even though individual nuclei are discrete.

Limit the inference

Activity is lambda times population, not population itself. Daughter population peaks when lambda1 N1=lambda2 N2; this gives uPeak=ln(r)/(r−1), with equal-rate limit1. The two activities then match, but the populations need not. No detector efficiency, observed count rate, isotope identity or physical source operation is inferred.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

For r=2 and u=ln(2), find the daughter fraction of the initial population.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and virtual evidence checked 31 October 2026 against the primary explanation. All questions and numerical packets are original. Supplied theoretical distributions and synthetic records are not actual laboratory measurements. No nuclear, electrical, heating, beam or material-handling procedures are requested. Retain first reasoning, actual criterion critique, revision and a separately assigned unfamiliar task in the course evidence workspace; public keys do not authenticate mastery.

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Your explanation and limitations

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