Honors Physics Unit 22 Review: The Atom

An atom is a tiny, dense, positive nucleus surrounded by mostly empty space where the electrons are found. Rutherford's gold foil experiment (run by Geiger and Marsden around 1909) showed this. Most alpha particles passed straight through the foil, but a few bounced back at large angles. That only makes sense if the positive charge and nearly all the mass sit in a nucleus about 100,000 times smaller than the atom.

Electrons can only have certain energies. In the Bohr model of hydrogen, level $n$ has energy $E_n = -\frac{13.6\text{ eV}}{n^2}$. The negative sign means the electron is bound, and $n = 1$ is the ground state. When an electron drops to a lower level, it emits a photon with energy equal to the difference, $E = hf = \frac{hc}{\lambda}$. When it absorbs a photon of exactly that energy, it jumps up. That's why each element has its own line spectrum.

Worked example. An electron in hydrogen drops from $n = 3$ to $n = 2$. $E_3 = -13.6/9 = -1.51$ eV and $E_2 = -13.6/4 = -3.40$ eV, so the photon carries $1.89$ eV. Using $hc = 1240$ eV·nm, $\lambda = 1240/1.89 \approx 656$ nm, which is the red line in hydrogen's spectrum.

The Bohr model works well for hydrogen. The quantum model replaces fixed orbits with orbitals, regions where an electron is likely to be found, and it handles atoms with many electrons.

The strong nuclear force holds protons and neutrons together, but it only reaches about $10^{-15}$ m, so each nucleon only feels its nearest neighbors. The electric repulsion between protons reaches across the whole nucleus. In big nuclei, that repulsion adds up, so heavy nuclei need extra neutrons, and every nucleus heavier than lead is unstable.

The mass of a nucleus is a little less than the mass of its separate protons and neutrons. That mass defect $\Delta m$ corresponds to the binding energy, $E = \Delta m c^2$, which is the energy needed to pull the nucleus apart. A mass of $1\text{ u}$ corresponds to a rest energy of $931.5$ MeV, so $1\text{ u}\,c^2 = 931.5$ MeV.

An unstable nucleus decays by emitting radiation. In alpha decay it loses a helium-4 nucleus, so the mass number drops by 4 and the atomic number drops by 2. In beta-minus decay a neutron turns into a proton and releases an electron and an antineutrino, so the atomic number goes up by 1 and the mass number stays the same. Gamma decay releases a high-energy photon and changes neither number.

Worked example. Uranium-238 ($Z = 92$) undergoes alpha decay. The mass number becomes $238 - 4 = 234$ and the atomic number becomes $92 - 2 = 90$, which is thorium. So $^{238}_{92}\text{U} \rightarrow \, ^{234}_{90}\text{Th} + \, ^{4}_{2}\text{He}$. Check that the top numbers ($238 = 234 + 4$) and bottom numbers ($92 = 90 + 2$) balance.

The half-life of a radioactive isotope is the time in which each nucleus has a 50% chance of decaying. It predicts that about half the nuclei in a large sample will decay. Half-lives are usually unaffected by ordinary changes in temperature, pressure, or chemical conditions, though some decay modes have small exceptions. Carbon-14 has a half-life of 5,730 years, and uranium-238's is about 4.5 billion years.

After $n$ half-lives, the amount left is $N = N_0\left(\frac{1}{2}\right)^n$, where $n = t/t_{1/2}$. The same idea in exponential form is $N = N_0 e^{-\lambda t}$, with decay constant $\lambda = \ln 2 / t_{1/2}$. Both forms work for whole or fractional numbers of half-lives. You can use logarithms with either form to solve for time.

Radiometric dating compares how much parent isotope is left with how much there was at the start, or with how much daughter isotope has built up. Living things take in carbon, so their carbon-14 to carbon-12 ratio reflects their carbon sources. After death, carbon intake stops and the carbon-14 decays. The starting ratio has varied over time, so carbon dates are calibrated using other records. Carbon dating works for organic material up to about 50,000 years old, and older rocks need long half-life pairs like uranium-lead.

Worked example. A bone has 25% of the carbon-14 to stable-carbon ratio expected for a living sample. Since $0.25 = \left(\frac{1}{2}\right)^2$, two half-lives have passed, so $t = 2 \times 5730 = 11{,}460$ years.

Decay is random for any single nucleus. Half-life only predicts what large numbers of nuclei do.

Fission splits a heavy nucleus into two medium-sized ones. Fusion joins two light nuclei into a heavier one. Both release energy for the same reason. The products are more tightly bound than the reactants, so their total mass is smaller, and the missing mass comes out as energy, $E = \Delta m c^2$.

The binding energy per nucleon curve explains which reaction works for which nuclei. It peaks near iron-56. Light nuclei like hydrogen move up the curve by fusing, and heavy nuclei like uranium-235 move up it by splitting.

In fission, a neutron hits uranium-235, which splits and releases about 200 MeV plus 2 or 3 more neutrons. Those neutrons can trigger more fissions in a chain reaction. Many reactors use a moderator to slow neutrons and control rods to absorb them. In a steady chain reaction, each fission causes one more fission on average.

Fusion typically needs temperatures of roughly $10^7$ K or higher. Fast-moving nuclei can get close enough for quantum tunneling, a process that lets them pass through the electric repulsion barrier. The Sun's core supplies suitable conditions, but sustaining fusion on Earth is difficult.

Worked example. Deuterium (2.014102 u) fuses with tritium (3.016049 u) to make helium-4 (4.002603 u) and a neutron (1.008665 u). The reactants total 5.030151 u and the products total 5.011268 u, so $\Delta m = 0.018883$ u. Multiply by 931.5 MeV/u to get about 17.6 MeV released.

Radioactive tracers let doctors see how organs work, not just their shape. In a PET scan, a tracer such as fluorine-18 attached to a glucose-like molecule emits positrons. Each positron meets an electron and they annihilate, producing two 511 keV gamma photons that fly off in opposite directions. Detectors around the patient catch the pairs, and a computer rebuilds where the tracer collected. Some tumors take up lots of the glucose-like tracer and show up brightly. Other tissues can also take up the tracer, and not all tumors show up clearly. SPECT tracers like technetium-99m emit single gamma rays directly. Medical tracers have short half-lives (about 110 minutes for fluorine-18 and 6 hours for technetium-99m), so the activity fades quickly.

Ionizing radiation harms cells by breaking DNA, either directly or by creating reactive free radicals from water. Radiation therapy uses this damage to kill cancer cells or stop them from dividing. Healthy cells can also be damaged, so treatment is planned to limit their exposure. Splitting the total dose into many sessions gives healthy tissue time to recover.

Absorbed dose is energy per mass, $D = E/m$, measured in grays ($1\text{ Gy} = 1\text{ J/kg}$). Equivalent dose multiplies by a weighting factor for the radiation type, $H = D \times W_R$, measured in sieverts. $W_R$ is 1 for gamma and beta and 20 for alpha.

Worked example. A 70 kg person absorbs 0.14 J from gamma rays. $D = 0.14/70 = 0.002$ Gy, and $H = 0.002 \times 1 = 0.002$ Sv, or 2 mSv. The same energy from alpha particles would give $0.002 \times 20 = 0.04$ Sv.