Q.(a) Explain Rutherford model of an atom. [3M].

Scale Analogy

Why we need an analogy at all

Rutherford's gold-foil experiment revealed something almost impossible to picture: the atom's entire positive charge and nearly all of its mass sit inside a nucleus that is fantastically smaller than the atom around it. The numbers are so extreme that our everyday intuition breaks down — so physicists reach for a scale analogy: blow the atom up to a size we can imagine, and see where the nucleus ends up.

The intuition: the atom is mostly empty space

Most alpha particles fired at the gold foil passed straight through, barely deflected. Only about 1 in 8000 bounced back sharply. The only way to explain this is that the atom is overwhelmingly empty, with a tiny, dense, positively charged core that the occasional alpha particle scores a near-direct hit on.

So how tiny is "tiny"? Compare the two sizes:

• Radius of a typical atom: about $1 \times 10^{-10}\ \text{m}$ ($1\ \text{angstrom}$)
• Radius of a typical nucleus: about $1 \times 10^{-15}\ \text{m}$ ($1\ \text{femtometre}$)

The ratio is

The atom is about one hundred thousand times wider than its nucleus.

Making the number imaginable

A factor of $10^5$ is just a symbol on paper. The scale analogy converts it into something the mind can hold:

• If the nucleus were the size of a pea (about $1\ \text{cm}$ across), then the atom would be a sphere roughly $10^5$ times bigger — around $1\ \text{km}$ across. The pea would sit alone at the centre of a stadium-sized region of empty space, with the electrons whirling somewhere out near the edge.
• Equivalently, if the atom were scaled up to the size of a large sports ground, the nucleus would be no bigger than a grain of sand at the centre-spot.

Either picture drives home the same point: an atom is almost entirely empty space, which is exactly why nearly every alpha particle sailed through the foil undeflected.

Density: the flip side of the analogy

The scale analogy also warns us about density. Nearly the whole mass of the atom is squeezed into that pin-point nucleus. Because volume grows as the cube of the radius, shrinking the mass-holder by $10^5$ in radius packs it into a volume $10^{15}$ times smaller. That is why nuclear matter has an almost unimaginable density — on the order of $10^{17}\ \text{kg/m}^3$ — while the atom as a whole is light and airy. …