Intuition The one core ideaAn electron is held in orbit by the electric pull of the nucleus, exactly the way a whirled ball is held by its string — but nature allows only certain special orbit sizes , like fixed steps on a staircase. From just those two facts ("pull equals the force needed to circle" and "only special sizes allowed") every radius and every energy of the atom falls out.
Before you can read the parent derivation, you need to recognise every letter and picture it uses. This page unpacks them one at a time, each built on the one before it — so that when a symbol like ε 0 m v r
Before any formula, picture the physical setup. A tiny heavy positive lump sits at the centre; a light negative speck races around it in a circle.
Definition The nucleus and the electron
Nucleus — the tiny dense centre, carrying positive charge. We write its charge as + Z e Electron — the light particle circling it, carrying negative charge − e
e e e = 1.602 × 1 0 − 19 one "unit brick" of charge — you can never have a fraction of it.
Z Z Z + Z e
Hydrogen: Z = 1 + e
Helium ion He+ Z = 2 + 2 e Why the topic needs it: the same derivation then works for any one-electron atom just by changing one number. That is why the parent says "hydrogen-like ".
Intuition Why charges multiply asZ e ⋅ e both of them. Nucleus supplies Z e e Z e 2 Z e × e = Z e 2 Z e 2
r r
v v along its circular path (metres per second). It always points along the circle (tangent), never toward the centre.
m m m = 9.11 × 1 0 − 31 m
This is the "how strong is the pull?" rule. See Coulomb's Law .
ε 0 ε 0 ε 0 = 8.85 × 1 0 − 12 4 π ε 0 1 ≈ 9 × 1 0 9 4 π Why the topic needs it: to get an actual number for the radius, you must know how strong the electric pull really is — that number lives inside ε 0
The r 2 1
See Centripetal Force and Circular Motion .
Intuition Circling is constant "falling inward"Anything moving in a circle is always changing direction , and changing direction is a kind of acceleration — pointed toward the centre. To cause that inward acceleration you need a real inward force. Let go of the whirled ball's string and it flies off in a straight line; the string's pull is what bends the path into a circle.
Definition The force balance (Ingredient 1)In the atom, the Coulomb pull IS the centripetal force . Nothing else is pulling. So:
what the pull supplies 4 π ε 0 1 r 2 Z e 2 = what circling demands r m v 2
See Quantization of Angular Momentum .
Definition Angular momentumL = m v r Angular momentum measures "how much spin-motion" an orbiting object carries. For a circle it is mass × speed × radius, L = m v r
h ℏ
h Planck's constant , h = 6.626 × 1 0 − 34 ℏ ℏ = 2 π h
n n 1 , 2 , 3 , … n = 1 n
this extra rule (the deep reason)
Why should angular momentum come in chunks? Because the electron also behaves like a wave (see de Broglie Wavelength and Standing Waves ): an allowed orbit is one where a whole number of electron-wavelengths fits exactly around the circle, like a standing wave on a loop. That "whole number of waves" condition is exactly m v r = n ℏ
Definition Kinetic energyK E Kinetic energy is the energy of motion: K E = 2 1 m v 2 K E
Definition Potential energyP E Potential energy is stored energy from position in a force field. For an attractive electric pull it is negative :
P E = − 4 π ε 0 1 r Z e 2 Why negative? We agree that when the electron is infinitely far (r → ∞ 0 releases energy, so its stored energy drops below zero. Closer in = more negative.
E E = K E + P E E negative , the electron is bound — trapped in the atom, unable to escape without extra energy added. This is why the parent's E n Ionization Energy and Virial Theorem .
Mnemonic Symbol quick-glance
ε 0 π ℏ h /2 π ∝ subscript n r n E n for orbit number n ".
⇒ a 0 0.529 Å (ångström) — a length unit, 1 A ˚ = 1 0 − 10
Force balance: pull = centripetal
Circular motion needs inward force
Quantization: mvr = n h-bar
Two streams — electrostatics (left) and quantization (right) — meet to give the radii, and radii plus energy bookkeeping give the energy levels. That is the entire logic of topic 1.2.8.
Cover the right side and test yourself. If any answer surprises you, reread that section before the parent derivation.
What does Z e Z e 2 Z e Z Z e e Z e 2
In Coulomb's law, how does the force change if you double r It falls to one-quarter (force ∝ 1/ r 2
What single clump is the constant 4 π ε 0 1 One fixed number ≈ 9 × 1 0 9 ε 0
Why does circular motion need a force pointing inward? Because direction is constantly changing (an inward acceleration), which requires a real inward (centripetal) force m v 2 / r
Write the force-balance statement in words. The Coulomb pull equals the centripetal force required to keep the electron on its circle.
What is angular momentum for a circular orbit? L = m v r
State Bohr's quantization rule and what n m v r = nh /2 π n = 1 , 2 , 3 , …
Why is potential energy negative here? We set P E = 0 P E
What does a negative total energy E The electron is bound (trapped) — energy must be added to free it.
What is ℏ h ℏ = h /2 π
What deeper idea justifies the quantization rule? A whole number of electron de Broglie wavelengths must fit around the orbit (standing wave on a loop).
