Foundations — Bohr model of hydrogen — postulates, radius rₙ = 0.529 n² - Z Å, energy Eₙ = −13.6 Z² - n² eV

Before you can read a single formula in the parent note, you must know what each squiggle stands for. This page builds them one at a time, from the ground up, in the order they lean on each other. By the end, $mvr=n\hslash$, $r_n=0.529n^2/Z$ Å and $E_n=-13.6Z^2/n^2$ eV will feel like plain sentences.

0. The stage: a nucleus and one electron

Everything in the Bohr model happens on a flat picture: a tiny positive lump in the middle (the nucleus) and one negative speck going round it in a circle (the electron). That's the whole cast.

Read the figure: the amber dot at the centre is the nucleus (charge $+Ze$); the white dot on the cyan circle is the electron (charge $-e$); the dashed white line is the radius $r$; the cyan arrow shows the electron's speed $v$ pointing along the circle (tangent), not toward the centre. Keep this exact picture in mind — every symbol below lives somewhere on it.

This picture is Rutherford's leftover puzzle. See Rutherford model for why the electron must be moving (a still electron would be sucked straight in).

1. Charge and the symbol $e$

Charge is the property that makes two particles push or pull each other electrically. Same signs push apart; opposite signs pull together.

Why the topic needs it: the entire force holding the atom together is electric, so we must name how much charge each partner carries.

2. Atomic number $Z$ — how strong the pull is

Picture: a bigger $Z$ means more positive lump in the centre → a stronger tug on the electron → it gets pulled into a tighter orbit. That is exactly why the radius formula divides by $Z$ and the energy multiplies by $Z^2$.

3. The distance $r$ and the radius picture

Picture: the dashed white line in figure s01. A small $r$ = a tight orbit hugging the nucleus; a large $r$ = a wide, loose orbit. The subscript version $r_n$ (met in §9) means "the radius of the $n$-th allowed track."

4. Coulomb's law — the pull, as a formula

Now we turn "opposite charges attract" into a number. This is the Coulomb force.

Why $r^2$ (why squared, not just $r$)? Because the pull spreads out over the surface of a sphere around the charge, and a sphere's area grows as $r^2$. Twice as far → the influence is spread over four times the area → the force is four times weaker. This "inverse-square" shape is the single most important fact that makes every Bohr formula come out the way it does.

Read the figure: the amber curve is the Coulomb force plotted against distance $r$. The two white dots mark $r=1$ (full strength) and $r=2$, where the force has already dropped to one quarter — not one half. That "quarter, not half" collapse is the visual signature of the inverse-square law; carry it forward, because the same $r^2$ will reappear when we balance forces.

Why the topic needs it: Coulomb's law is postulate 1's engine — it supplies the inward force that keeps the electron circling instead of flying off.

5. Speed $v$ and why a circle needs a sideways force

Here is the key mechanical fact, from absolute zero:

Read the figure: the amber arrow points inward (toward the nucleus) — that is the centripetal pull actually bending the path. The dashed white arrow shows what would happen without it: the electron shoots off straight along the tangent. So the electron's real motion is a tug-of-war between "go straight" and "get pulled in."

The force-balance idea: the Coulomb pull (§4) is that inward centripetal pull (§5). Setting them equal is the very first line of the parent's derivation:

6. Angular momentum $mvr$ — the "spin amount"

Postulate 2 talks about angular momentum. Here is what that phrase means with a picture.

Picture: imagine a spinning skater. Arms out (big $r$) = lots of angular momentum for a given push; arms in = less. It's the rotational cousin of ordinary momentum $mv$, with the extra $r$ because we care about turning, not straight-line motion.

Why the topic needs it: Bohr's boldest idea is that this exact quantity $mvr$ is not free — it can only take special stepped values. That is postulate 2, and here it is on its own, as a formula:

7. Planck's constant $h$ and $\hslash$ — the size of one quantum step

Why divide by $2\pi$? Because angular momentum is about going around a circle, and one full trip around is $2\pi$ radians. The natural unit for "per turn" quantities carries that $2\pi$, so $\hslash$ (not $h$) is the natural step size for $mvr$.

You meet the same $h$ in Planck's quantisation and the Photoelectric effect — it is the universal "lumpiness" constant.

8. The quantum number $n$ — which track we're on

Picture: the steps of a staircase. You can stand on step 1, step 2, step 3 — never on "step 2.5." Every allowed radius $r_n$ and energy $E_n$ carries this $n$ as a subscript, meaning "the value on track number $n$."

Why the topic needs it: because $n$ is discrete (no in-betweens), the energies are discrete, and that is the entire reason a line spectrum exists.

Read the figure: each cyan horizontal line is one allowed track, labelled by its $n$ and its energy $E_n$; there is simply nothing between the lines. The amber dashed line at the top marks $E=0$ (the electron just barely free). The amber arrow shows one electron jumping down from $n=2$ to $n=1$, throwing out a photon — the mechanism behind every spectral line.

9. The two headline formulas — where every symbol lands

You have now met every ingredient. Feed the quantisation condition (§6) into the force balance (§7... rather §5's equation (1)) and out drop the parent note's two star formulas. You don't need to grind the algebra here — the point is to see each symbol you learned sitting in its final place.

These two boxes are the whole destination of the parent note; every symbol in them was defined above.

10. Energy: kinetic, potential, total — and why negative

Picture a valley: the electron sits in a dip. Down in the dip its total energy is below the flat ground outside (which we call zero). A trapped/bound electron therefore has negative total energy. Freeing it means climbing up to zero — exactly the $E=0$ line in figure s04.

Why the topic needs it: $E_n=-13.6Z^2/n^2$ is literally $KE+PE$ evaluated at radius $r_n$. Knowing the sign story tells you the ground state is the deepest well and ionisation climbs out of it — see Ionisation energy.

11. The photon and $\Delta E=h\nu$ — light in packets

Picture: the electron drops from a high step to a low step (the amber arrow in figure s04); the leftover height ($\Delta E$) is thrown out as one photon of exactly matching colour. Discrete steps → discrete colours. This is postulate 3, and it links straight to Hydrogen spectrum & Rydberg formula.

How it all feeds the topic

Equipment checklist

Test yourself — cover the right side and answer each before revealing.

Back to the parent: Bohr model of hydrogen.