Rutherford's gold-foil experiment — nuclear model

1. The setup — WHAT was actually done

2. The observations — WHAT was seen

3. Deriving the nuclear model — HOW conclusions follow

3.1 Estimating the closest approach (size of the nucleus)

Formula Distance of closest approach

$r_0$ — derived from energy conservation Fire an α-particle head-on. It slows, stops at distance $r_0$ , then reverses. At the stopping point all its kinetic energy has become electrostatic potential energy.

Step 1 — Why energy conservation? No energy is lost (elastic Coulomb interaction), so $\text{KE}_{\text{initial}} = \text{PE at closest approach}$ Step 2 — Write each side. KE of the α is $\tfrac{1}{2}mv^2$ . The Coulomb PE between charge $q_1=2e$ (α) and $q_2=Ze$ (nucleus, $Z=79$ for gold) is $U=\frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r}.$ Step 3 — Set them equal at $r=r_0$ (where $v=0$ ): $\frac{1}{2}mv^2=\frac{1}{4\pi\varepsilon_0}\frac{(2e)(Ze)}{r_0}.$ Step 4 — Solve for $r_0$ : $\boxed{\,r_0=\frac{1}{4\pi\varepsilon_0}\frac{2Ze^2}{\tfrac12 mv^2}=\frac{1}{4\pi\varepsilon_0}\frac{4Ze^2}{mv^2}\,}$ Plugging in gold gives $r_0\sim 10^{-14}$ m — thousands of times smaller than the atom ( $\sim 10^{-10}$ m). That size gap is the nuclear model.

Worked example Worked number: order-of-magnitude of

$r_0$ Take an α with KE $= 7.7$ MeV $=1.23\times10^{-12}$ J on gold ( $Z=79$ ).

Why start from KE? Because $\tfrac12 mv^2$ equals the given KE directly. $r_0=\frac{(9\times10^{9})(4)(79)(1.6\times10^{-19})^2}{1.23\times10^{-12}}$
Why these numbers? $k=1/4\pi\varepsilon_0=9\times10^9$ , $e=1.6\times10^{-19}$ C. $r_0\approx\frac{9\times10^{9}\times 4\times79\times2.56\times10^{-38}}{1.23\times10^{-12}}\approx 5.9\times10^{-14}\ \text{m}.$
Interpretation: the α cannot get closer than ~ $6\times10^{-14}$ m; the actual nucleus is even smaller. Compare atom radius $\sim10^{-10}$ m ⇒ nucleus is ~ $10^{4}$ times smaller in radius, $10^{12}$ times smaller in volume. The atom is essentially empty.

4. Rutherford's nuclear model — WHAT it states

5. Failures of the model — Steel-manning the classical picture

Common mistake "A revolving electron is stable, just like a planet." — Why it feels right, why it's wrong

Why it feels right: planets orbit the Sun stably under an inverse-square force; the Coulomb force is also inverse-square, so the analogy seems perfect. Why it's wrong: an electron is charged and accelerating (circular motion = centripetal acceleration). Classical electromagnetism says an accelerating charge radiates energy (electromagnetic waves). Planets don't radiate gravitational waves at any appreciable rate; electrons radiate strongly. The fix / consequence: losing energy, the electron would spiral inward and crash into the nucleus. A proper estimate using the classical Larmor radiation formula gives a collapse time on the order of $\sim10^{-11}$ – $10^{-10}$ s (roughly a hundred-billionth of a second). So the classical nuclear atom should collapse almost instantly — yet atoms are stable. This "spiral death" is the fatal flaw.

6. Active recall

Recall Cover the answers and quiz yourself

Why did most α-particles go straight through? → atom is mostly empty space.
Why did a few bounce back? → head-on hit on a tiny, dense, positive nucleus.
What formula gives closest approach and from what principle? → $r_0=\frac{1}{4\pi\varepsilon_0}\frac{4Ze^2}{mv^2}$ from energy conservation (KE = Coulomb PE).
Two failures of Rutherford's model? → predicts electron spiral/collapse (in ~ $10^{-11}$ – $10^{-10}$ s); predicts continuous (not line) spectra.
What supplies centripetal force for the electron? → electrostatic (Coulomb) attraction.

Recall Feynman: explain to a 12-year-old (hidden)

Imagine throwing thousands of marbles at a wall made mostly of cling-film with a few tiny, super-heavy steel balls hidden inside. Almost all your marbles zip straight through the film — the wall is basically empty. But once in a while a marble hits one of the steel balls dead-on and comes flying back at you. That "coming back" is the shock that tells you: hidden inside all that empty space are tiny, heavy, hard lumps. In an atom those lumps are the nucleus, and everything else is empty space with light electrons buzzing around.

Connections

Thomson's Plum Pudding Model — the model this experiment overturned
Coulomb's Law — the inverse-square force behind the scattering
Bohr Model of the Atom — fixes the collapse & line-spectrum failures
Atomic Spectra (Line Spectra) — what Rutherford's model couldn't explain
Discovery of the Proton and Neutron — later constituents of the nucleus
Centripetal Force and Circular Motion — physics of the orbiting electron

Who directed the gold-foil scattering experiment (and who performed it)?

Ernest Rutherford directed it; Hans Geiger and Ernest Marsden performed it (1909–1911).

What were the α-particles used as projectiles?

Helium nuclei (He²⁺), charge +2e, mass ~4 u.

Why was gold foil made extremely thin?

So each α-particle undergoes essentially one interaction, giving clean single-scattering data.

Observation 1 and its conclusion?

Most particles pass straight through → atom is mostly empty space.

Observation 3 and its conclusion?

A very few (~1 in 20,000) bounce back >90° → tiny, dense, positively charged nucleus.

Principle used to find distance of closest approach?

Conservation of energy: initial KE = Coulomb PE at the stopping point.

Formula for distance of closest approach r₀?

r₀ = (1/4πε₀)(4Ze²)/(mv²).

Roughly how much smaller is the nucleus than the atom?

Radius ~10⁴ times smaller (~10⁻¹⁴ m vs ~10⁻¹⁰ m).

What provides the centripetal force in Rutherford's orbiting-electron picture?

The electrostatic (Coulomb) attraction between nucleus and electron.

First failure of Rutherford's model?

An accelerating (circling) electron radiates energy (classical Larmor formula), so it spirals into the nucleus → predicted collapse in ~10⁻¹¹–10⁻¹⁰ s.

Second failure of Rutherford's model?

It predicts a continuous spectrum, but atoms show discrete line spectra.

What did Rutherford actually discover?

The atomic nucleus (not the electron or neutron).

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, Rutherford ke experiment ka core idea bahut simple hai. Pehle log sochte the ki atom ek "plum pudding" hai — matlab positive charge poore atom me jelly ki tarah faila hua, aur electrons uske andar raisins ki tarah. Rutherford ne alpha-particles (yeh helium ke nuclei hain, charge +2e, kaafi heavy aur fast) ko patli gold foil pe maara. Umeed thi ki sab seedha nikal jaayenge. Lekin surprise — zyaadatar seedha gaye, kuch thoda mude, aur bahut thode (jaise 20,000 me 1) to wapas bounce ho gaye!

Yeh backward bounce hi asli twist hai. Agar positive charge poore atom me faila hota, to itni strong repulsion kahin bhi nahi hoti ki heavy alpha ko peeche phenk de. To conclusion: saara positive charge aur saara mass ek bahut chhoti si jagah (nucleus) me concentrated hai, aur baaki atom to khaali space hai jisme electrons ghoomte hain. Yeh naya picture "nuclear model" kehlata hai.

Size ka andaaza energy conservation se lagta hai: jab alpha head-on aata hai, uski saari kinetic energy Coulomb potential energy ban jaati hai jahan wo ruk ke wapas mudta hai. Wahan se $r_0=\frac{1}{4\pi\varepsilon_0}\frac{4Ze^2}{mv^2}$ nikalta hai, jo lagbhag $10^{-14}$ m aata hai — atom ( $10^{-10}$ m) se hazaaron guna chhota. Yani atom lagbhag khaali hai.

Par model perfect nahi tha. Classical physics kehti hai ki circular motion me electron accelerate karta hai, aur accelerating charge energy radiate karta hai (Larmor formula) — to electron spiral karke sirf $10^{-11}$ – $10^{-10}$ second me nucleus me gir jaana chahiye, lekin atoms to stable hain. Aur yeh line spectra bhi explain nahi kar paaya. Yehi failures aage Bohr model ko janam dete hain. Exam ke liye: teen observations, unke conclusions, $r_0$ ka derivation, aur do failures — bas yeh 80/20 hai.

Rutherford's gold-foil experiment — nuclear model

1. The setup — WHAT was actually done

2. The observations — WHAT was seen

3. Deriving the nuclear model — HOW conclusions follow

3.1 Estimating the closest approach (size of the nucleus)

4. Rutherford's nuclear model — WHAT it states

5. Failures of the model — Steel-manning the classical picture

6. Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Atomic Structure (Classical)

Connections