Test a historical single-electron atom model against quantization

Begin with evidence

A nucleus contains positive protons and neutral neutrons. Proton count Z identifies an element; changing neutron count gives isotopes, whereas removing or adding electrons changes charge state. A one-electron ion with nuclear charge +Ze has net charge (Z−1)e. These bookkeeping facts do not require a literal planetary orbit for the electron.

Connect the model

The historical Bohr construction combines a circular Coulomb force balance me v²/r=ke Ze²/r² with angular momentum mevr=nℏ. Equivalently, the circumference contains n de Broglie wavelengths: 2πr=nλ. Solving yields r=ℏ²n²/(me ke Ze²) and energy E=−ke Ze²/(2r). Here n is a positive integer and the zero of total electron energy is the separated electron at rest.

Interpret the ledger

At fixed Z, radius grows as n² while binding energy magnitude decreases as 1/n². At fixed n, radius decreases as 1/Z while binding energy magnitude grows as Z². The negative total energy means that energy must be supplied to separate the electron; it is not negative kinetic energy. K is positive and the attractive electric potential energy is more negative.

Test the boundary

This model successfully organizes single-electron energy levels, but its pictured circular path is a historical construction, not an observed quantum orbit or a probability density. Multielectron shielding, fine structure and orbital shapes are outside this workbench. The level diagram and circumference check test model consistency without pretending to visualize an actual electron path.

Worked original case

For Z=1 and n=2, r≈0.211669 nm, E≈−3.40162 eV and circumference/λ=2. At the same n with Z=2, radius halves and binding magnitude quadruples; the one-electron ion has net charge +e.

Predict, compare and retain evidence

Does a more negative bound-state energy mean a negative kinetic energy? Compare n=1,2,4 at Z=1, then Z=1,2,3 at n=2. Retain radius, energy, wavelength and circumference ratio. Derive both scaling laws and identify one property this historical model does not establish.

Changed-case practice

A one-electron Z=3 ion changes from n=1 to n=3. Find radius and binding-magnitude ratios and its net charge.

Check the reasoning after your attempt

Radius ratio is 9; binding-magnitude ratio is 1/9. With three protons and one electron, net charge is +2e. Neither result specifies the isotope’s neutron count.

Read nuclear notation without confusing isotope and ion

Write nuclear notation as a symbol with mass number A above and proton number Z below: neutron count is A−Z. Most atomic mass resides in the nucleons, while electron number and arrangement strongly influence chemical interactions. Changing electron count changes ion charge, not nuclear identity; changing A at fixed Z changes isotope.

Predict and check: a hypothetical atom labeled A=23, Z=11 with ten electrons has eleven protons, twelve neutrons and net charge +e. Adding one electron makes it neutral without changing the isotope. Adding one neutron instead gives A=24 and the same element, without by itself changing electron charge. Contrast this bookkeeping with the one-electron Bohr approximation, which does not describe the multi-electron arrangement of this example. (Current clause 15.2.A.)

Reflect and source-check

Why is a circumference diagram not evidence for a measured quantum orbit?

Original prose and supplied numerical evidence; checked 30 October 2026 against the current AP Physics 2 framework, fall 2026 clarifications and OpenStax. Virtual-only: no physical source, exposure or apparatus is requested. Public examples and model captures are not independent mastery or supervised laboratory participation.

Clause-level source/action reconciliation checked 31 October 2026 against the linked current framework and Fall 2026 clarification. Added packets are original virtual comparisons; no new practical or learner-completion claim.