E.1 Structure of the atom

Practice Worksheet — name: ______________________ date: ____________

Formulas for this topic

Photon energy

E = hf = \frac{hc}{\lambda}

Hydrogen energy levels (HL)

E_{n} = -\frac{13.6}{n^{2}}\ \text{eV}

Quantised angular momentum (HL)

mvr = \frac{nh}{2\pi}

Nuclear radius (HL)

R = R_{0}A^{\frac{1}{3}}

Section A — Multiple choice

A1. In the Geiger–Marsden experiment, the observation that a small fraction of alpha particles deflected through more than 90° implies that:

A Atoms are mostly empty space
B Electrons orbit the nucleus
C The positive charge is concentrated in a tiny, massive nucleus
D Alpha particles are positively charged

A2. An electron in hydrogen falls from $n = 3$ ( $-1.51$ eV) to $n = 2$ ( $-3.40$ eV). The emitted photon has energy:

A 1.89 eV
B 4.91 eV
C 3.40 eV
D 1.51 eV

A3. Nucleus A has 8 times as many nucleons as nucleus B. The ratio of their radii $R_A/R_B$ is:

A 8
B 4
C 2
D $\sqrt{8}$

Section B — Short answer

B1. Explain how an absorption spectrum is formed and why its dark lines match the bright lines of the same element's emission spectrum. [3 marks]

B2. Show that the density of nuclear matter is approximately the same for all nuclei. [3 marks]

B3. An alpha particle (charge $2e$ , energy 5.0 MeV) approaches a gold nucleus ( $Z = 79$ ) head-on. Calculate the distance of closest approach. [4 marks]

Answer key

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Section A

A1: The positive charge is concentrated in a tiny, massive nucleus — Large-angle scattering needs a huge repulsive force from something more massive than the alpha particle — impossible if positive charge were spread thinly (as in the plum-pudding model). The RARITY of such events shows the nucleus is tiny; most alphas passing straight through shows the atom is mostly empty.

A2: 1.89 eV — Photon energy is the level difference: $-1.51 - (-3.40) = 1.89\ \text{eV}$ — the red H $\alpha$ line of the Balmer series at 656 nm.

A3: 2 — $R = R_0A^{1/3}$ , so the ratio is $8^{1/3} = 2$ . Volume scales with $A$ — which is why nuclear density is the same for all nuclei.

Section B

B1: White light passing through a cool gas is absorbed at exactly the photon energies matching gaps between the gas's energy levels; the excited atoms re-emit in all directions, leaving dark lines in the forward beam. The same level differences set both absorption and emission energies, so the line positions coincide.

B2: Mass $\propto A$ (each nucleon contributes ~ $u$ ). Volume $= \frac{4}{3}\pi R^3 = \frac{4}{3}\pi R_0^3 A \propto A$ . Density = mass/volume $\propto A/A$ — independent of $A$ , about $2\times10^{17}\ \text{kg m}^{-3}$ for every nucleus.

B3: At closest approach all kinetic energy is electric potential energy: $E_k = \frac{kq_1q_2}{d}$ . So $d = \frac{(8.99\times10^9)(2)(79)(1.6\times10^{-19})^2}{5.0\times10^6 \times 1.6\times10^{-19}} \approx 4.5\times10^{-14}\ \text{m}$ — an upper bound on the nuclear radius.