University Physics III · topic UPIII-11.02
Mass defect and binding energy per nucleon
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Mass defect and binding energy per nucleon: predict a finite change
The binding energy B of a nucleus is the energy needed to separate it into free protons and neutrons at rest; it equals the mass deficit Δm, the free nucleons' mass minus the nucleus's, times c², and B/A is its average over the A nucleons. Use the supplied topic model Y_out = 19.708+(74.736-31.537*(a1^(2/3)+a2^(2/3))-0.30026*(a1^(5/3)+a2^(5/3))+0.018209*(a1^2+a2^2))/(a1+a2), where Y_out is FRAGMENTS' MEAN B/A (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. a1 is Fragment mass number A₁, dimensionless. Keep a2 (Fragment mass number A₂) = 118 fixed. For this exercise use only 72 ≤ a1 ≤ 92, at the specified test settings separated by 1. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at a1 = 76 and 77. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δa1 across this interval. State whether the response increases or decreases; explain why a finite difference between these settings does not require interpolation between them.
FRAGMENTS' MEAN B/A against a1
Predictions from the supplied Mass defect and binding energy per nucleon model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): FRAGMENTS' MEAN B/A / MeV
Horizontal axis (x): a1: Fragment mass number A₁
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| a1: Fragment mass number A₁ | FRAGMENTS' MEAN B/A / MeV |
|---|---|
| 72 | 8.611249173442125 |
| 77 | 8.61004086165772 |
| 82 | 8.60631837853937 |
| 87 | 8.600253890894114 |
| 92 | 8.592015323288495 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of a1 separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At a1 = 76, substitute into Y_out = 19.708+(74.736-31.537*(a1^(2/3)+a2^(2/3))-0.30026*(a1^(5/3)+a2^(5/3))+0.018209*(a1^2+a2^2))/(a1+a2): Y₀ = 8.61049 MeV.
- At a1 = 77, the same fixed-parameter model gives Y₁ = 8.61004 MeV.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.000448342 MeV; the input change is Δa1 = 1.
- Average sensitivity is ΔY_out/Δa1 = -0.000448342 MeV per unit input. Its negative sign means that this response decreases over the specified interval.
- This finite difference compares two specified test settings without interpolating between them. A point derivative is not required. Any quantum numbers or particle counts must also obey their physical admissibility conditions.
Answer Y₀ = 8.61049 MeV, Y₁ = 8.61004 MeV; ΔY_out = -0.000448342 MeV; average sensitivity = -0.000448342 MeV per unit input.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Mass defect and binding energy per nucleon: infer an input with uncertainty
The binding energy B of a nucleus is the energy needed to separate it into free protons and neutrons at rest; it equals the mass deficit Δm, the free nucleons' mass minus the nucleus's, times c², and B/A is its average over the A nucleons. Use the supplied topic model Y_out = 19.708+(74.736-31.537*(a1^(2/3)+a2^(2/3))-0.30026*(a1^(5/3)+a2^(5/3))+0.018209*(a1^2+a2^2))/(a1+a2), where Y_out is FRAGMENTS' MEAN B/A (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. a1 is Fragment mass number A₁, dimensionless. Keep a2 (Fragment mass number A₂) = 118 fixed. For this exercise use only 72 ≤ a1 ≤ 92, at the specified test settings separated by 1. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 8.6041641 MeV, with a bounded reading uncertainty ±0.0015387 MeV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of a1 consistent with that bound. Compare with a straight-line interpolation between the two endpoint predictions, and explain why a fractional setting may be inadmissible. Do not assume that the supplied calibration points prove monotonicity outside this branch.
FRAGMENTS' MEAN B/A against a1
Predictions from the supplied Mass defect and binding energy per nucleon model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): FRAGMENTS' MEAN B/A / MeV
Horizontal axis (x): a1: Fragment mass number A₁
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| a1: Fragment mass number A₁ | FRAGMENTS' MEAN B/A / MeV |
|---|---|
| 72 | 8.611249173442125 |
| 77 | 8.61004086165772 |
| 82 | 8.60631837853937 |
| 87 | 8.600253890894114 |
| 92 | 8.592015323288495 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Enumerate the permitted settings in the stated range, evaluate Y_out at each, and retain every setting satisfying |Y_out − Y_measured| ≤ u. The plotted predictions are supplied calibration evidence.
Reveal the worked solution
- Endpoint predictions are Y_out(72) = 8.61125 MeV and Y_out(92) = 8.59202 MeV. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives a1_line = 72 + (8.60416 − 8.61125)(92 − 72)/(8.59202 − 8.61125) = 79.3673. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Evaluate the model at each permitted setting. The central reading corresponds to a1 = 84; only the allowed settings 83, 84, 85 lie inside the stated reading bound.
- The interpolation error in the input is |a1_line − a1| = 4.6327.
- The compatible allowed values span 83 to 85. Values between specified test settings are outside this exercise's input domain even if an interpolation passes through them.
- The quoted output bound restricts which specified settings remain compatible. It does not change physical restrictions on quantum numbers or counts, and the chosen grid alone does not establish physical quantisation.
Answer Compatible a1: 83, 84, 85. Endpoint interpolation predicts 79.3673; check admissibility against the discrete settings.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.