Change particle counts, pin the differences, and discover what names an element. Then prepare one-electron hydrogen states, collect possible position outcomes and match energy gaps to light.
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Protons name the element. Neutrons change the isotope. Electrons change the charge. Where an electron may be found is a different question—and a cloud of possible outcomes is not an orbit.
Six protons, seven neutrons and six electrons. What is it?
Six protons names carbon. Six plus seven nucleons gives mass number 13. Equal proton and electron counts give zero charge.
A nucleus with six protons belongs to carbon. Eight protons identifies oxygen. Carbon occurs in a pencil’s graphite; oxygen is part of water molecules. Those familiar materials contain interacting atoms, not the isolated hydrogen atom used later in this lesson.
Carbon-12 has six protons and six neutrons. Add one neutron to make carbon-13. Remove one electron instead, and carbon-12 gains charge +1 while its nucleus stays the same. Pin a card and change one column to see exactly what changed.
The probability bench prepares isolated hydrogen-1: one proton and one electron. It does not use the carbon inventory to guess a carbon cloud. Choose a state, then collect simulated position outcomes from many separate identical preparations.
Every dot is one possible measurement outcome in a new preparation. The dots accumulate; they do not race around the center or join into a route. More samples reveal a distribution without adding electrons to any one atom.
The radius probe asks what fraction of outcomes lie inside a sphere. It is an inspection boundary, not an atom’s shell. Compare the exact enclosed probability with the sample fraction. A bigger sample changes the estimate’s variability, not the state’s exact probability.
For the selected permitted emission routes, the atom loses energy and a photon carries the same amount. A larger gap gives a shorter wavelength. The ultraviolet example is labeled as invisible ultraviolet, not colored as ordinary visible purple light.
Atomic number Z is proton count. Mass number A = Z + N counts nucleons; charge number is Z − Ne. Charge Q = (Z − Ne)e, with e = 1.602176634 × 10⁻¹⁹ C exactly. The eight curated ground-state entries are from IAEA LiveChart. Unlisted configurations receive no inferred stability label. Charge arithmetic does not determine which isolated ions can bind all selected electrons.
Carbon-13 has A = 13, while NIST lists its neutral-atom relative mass near 13.00335483507. Carbon’s standard atomic-weight interval is [12.0096, 12.0116]. Those are distinct quantities. Electrons have nonzero mass, but electron count does not enter the integer mass number.
We use a nonrelativistic, spin-independent Coulomb model of hydrogen-1 with reduced electron–proton mass. The relative-coordinate scale a = a₀(1 + me/mp) is about 52.9465 pm. CODATA 2022 constants give RH = R∞/(1 + me/mp), binding scale B = hcRH ≈ 13.5983 eV, and En = −B/n² relative to separated particles at rest.
With x = r/a, the normalized spatial amplitudes are ψ1s = exp(−x)/√(πa³), ψ2s = (2−x)exp(−x/2)/√(32πa³), and ψ2pz = x cosθ exp(−x/2)/√(32πa³). Probability density is |ψ|², in pm⁻³ here. Probability in a region is its volume integral. A continuous density assigns zero probability to an exact point.
Integrating over directions gives radial densities per pm: 4x²exp(−2x)/a for 1s, x²(2−x)²exp(−x)/(8a) for 2s, and x⁴exp(−x)/(24a) for 2p. The 1s spatial density peaks at the origin, while its radial-shell density peaks at r = a. The shell’s growing volume explains the difference; it does not define an orbit.
The seeded sampler draws the correct radial and angular distributions. For 2p_z, cosθ has density 3cos²θ/2; it is not uniform. Samples come from independent identical preparations. A stationary energy state has a time-independent position density in this model. Presentation time only reveals additional trials.
For 2s, density vanishes on r = 2a, but approximately 5.2653% of probability lies inside that sphere. For the chosen real 2p_z state, density vanishes in the plane z = 0. The analytic slice samples density at a plane; the projected dots are different data.
Only 2p → 1s, 3p → 2s and 4p → 2s are offered as electric-dipole emission examples. The 3p/4p cards use energies; no unsupported clouds are drawn for them. A matching energy is not a complete absorption-probability rule. The 2s → 1s route is not an ordinary allowed one-photon electric-dipole transition.
The 2011 Parthey et al. experiment measured a 1S–2S hyperfine-centroid transition frequency of 2,466,061,413,187,035(10) Hz using two-photon spectroscopy. The simple model predicts about 2,466,038,423,686,301 Hz. The discrepancy is much greater than the experimental uncertainty. This is the total transition frequency, not each probe photon’s frequency.
The number after an element name counts nucleons. It does not replace the proton number or tell you the electrical charge. Keeping those roles separate helps interpret isotope references.
A density slice, a projection of simulated outcomes and a microscope recording can look similar. Ask what quantity generated the pixels and which physical system it describes.
Energy differences connect atomic models to spectra. Preserving units, medium, uncertainties and source provenance lets a near match remain a meaningful comparison rather than an exaggerated claim.
Write counts 6/6/6, 6/7/6 and 6/6/5 in proton/neutron/electron order. Calculate element, mass number and charge. Circle the one count that changed on each comparison.
Mix the ten digit cards. Draw one for the tens place, replace it, mix, then draw one for the ones place and replace it. The ordered pair gives a number from 00 to 99. Each new trial uses the same full set.
00–31: below a. 32–75: a to below 2a. 76–93: 2a to below 3a. 94–98: 3a to below 4a. 99: 4a or farther. These paper probabilities round the hydrogen 1s model; the final bin preserves an outer tail.
Predict the most common bin, then record 30 trials. Put one tally per result. Do not join outcomes into a line or treat the cards as successive electron locations.
Optionally combine another 30 trials. Expected paper proportions are 32%, 44%, 18%, 5% and 1%. Exact counts need not match; each finite sample fluctuates. The analytic 1s probabilities differ slightly from these rounded bins.
One tally is one simulated outcome from a new identical preparation. Each preparation still has one electron. State two limits: random variation and rounding of the analytic probabilities. Save your prediction, tally and explanation.
Can a paper game reveal a probability pattern without drawing an orbit?
A fictional counting and sampling activity. No radiation source, chemical, flame, laser, ultraviolet lamp, discharge tube or electrical apparatus. Counters are symbols; the game does not measure real atoms.
6 protons, 7 neutrons, 6 electrons Protons name carbon, nucleons give 13, equal proton/electron counts give neutrality.
It becomes carbon-12 with charge +1 The nucleus and mass number stay unchanged. Removing negative charge makes the charge number more positive.
Its element, mass number and charge, while nuclear properties remain unspecified Exact bookkeeping and a limited evaluated reference table are separate.
Independent simulated outcomes from 2,048 identical preparations Each preparation contains one electron; the plot accumulates outcomes.
A larger-radius shell includes a growing volume factor Density per volume and integrated probability across a spherical shell answer different questions.
Density is zero on that surface, with probability on both sides About 5.27% lies inside the node. A nodal surface is not a physical wall.
122 nm because its atomic energy gap is larger E = hc/λ. The gaps are approximately 1.89 eV and 10.2 eV.
The model captures a scale while the discrepancy exposes omitted physics Useful models have testable limits. A hydrogen approximation does not become a multi-electron model.
U.S. Geological Survey, Communications and Publishing. The image-specific source marks public domain. Source JPEG unchanged, with a central crop in the page; not an atom image.
USGS · Graphite Lump 2DOE’s isotope explanation, including carbon examples. Nuclear-status badges use the separate evaluated subset.
U.S. DOE · IsotopesFixed H through O symbol mapping. Missing nuclear facts are not filled by an arithmetic stability rule.
NIST · Atomic-number indexLiveChart CSV ground_states queries for 1h, 2h, 12c, 13c, 14c, 16o, 17o and 18o. Stable is the reference classification with no observed radioactive decay, not a prediction about selected electron binding.
IAEA · LiveChart data APICODATA 2022. Electron mass is nonzero. Exact SI h, c and e are distinct from experimentally adjusted constants.
NIST · CODATA 2022 constantsCarbon isotope masses and standard atomic-weight interval. Only narrow cited factual values are reproduced.
NIST · Carbon isotope massesLecture notes pp. 1–7. All lesson plots are authored from equations; MIT figures are not redistributed.
MIT · Hydrogen atom notesInstitutional teaching source. A probability cloud is not an electron path.
MIT · Hydrogen wavefunctions and orbitalsSeparation of stationary-state time dependence. The lesson does not animate a trajectory.
MIT · Wavefunctions lectureThe NIST handbook’s calculated selected reference lines are distinguished from a raw measured spectrum and from this browser approximation.
NIST · Persistent hydrogen linesThe narrow 3p/2s level difference can be converted to wavelength. It is a derived reference value, not new measured data.
NIST · Hydrogen levelsParthey et al., PRL 107, 203001 (2011), two-photon spectroscopy, 5.8 K atomic beam. Attributed result, no paper figure copied.
Parthey et al. · 2011 original manuscriptThe familiar object establishes context; the app does not model graphite bonding or a material’s full properties.
Royal Society of Chemistry · CarbonAn oxygen atom’s inventory is not a water-molecule geometry or an isolated oxygen-ion stability prediction.
Royal Society of Chemistry · OxygenIndependent subject review is pending.
Read the sources and model assumptions
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