This topic covers atomic structure and nuclear physics - the physics of the very small.
Overview
graph TD
A[Atoms & Nuclei] --> B[Atomic Models]
A --> C[Nuclear Physics]
B --> B1[Rutherford]
B --> B2[Bohr]
C --> C1[Nuclear Properties]
C --> C2[Radioactivity]
C --> C3[Fission & Fusion]Rutherford’s Model
Alpha Particle Scattering
- Most α particles pass through (atom is mostly empty)
- Few scattered at large angles
- Very few bounce back (small, dense nucleus)
Nuclear Size
$$R = R_0 A^{1/3}$$where $R_0 \approx 1.2$ fm, A = mass number
Bohr’s Model
Postulates
- Electrons revolve in fixed circular orbits
- Angular momentum is quantized: $mvr = n\frac{h}{2\pi}$
- Energy is emitted/absorbed only during transitions
Results for Hydrogen-like Atoms
Radius of nth orbit:
$$\boxed{r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m Z e^2} = 0.529 \times \frac{n^2}{Z} \text{ Å}}$$Velocity:
$$v_n = \frac{Ze^2}{2\varepsilon_0 nh} = 2.18 \times 10^6 \times \frac{Z}{n} \text{ m/s}$$Energy:
$$\boxed{E_n = -\frac{13.6 Z^2}{n^2} \text{ eV}}$$Hydrogen Spectrum
graph TD
A[Spectral Series] --> B[Lyman n→1 UV]
A --> C[Balmer n→2 Visible]
A --> D[Paschen n→3 IR]
A --> E[Brackett n→4 IR]
A --> F[Pfund n→5 IR]Rydberg Formula:
$$\frac{1}{\lambda} = RZ^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$$where $R = 1.097 \times 10^7$ m⁻¹
Nuclear Composition
- Nucleons: Protons + Neutrons
- Atomic number (Z): Number of protons
- Mass number (A): Protons + Neutrons
- Isotopes: Same Z, different A
- Isobars: Same A, different Z
- Isotones: Same number of neutrons
Nuclear Properties
Mass Defect
$$\Delta m = [Zm_p + (A-Z)m_n] - M_{nucleus}$$Binding Energy
$$\boxed{BE = \Delta m \times c^2 = \Delta m \times 931.5 \text{ MeV}}$$Binding Energy per Nucleon
$$\frac{BE}{A}$$Maximum for Fe-56 (~8.8 MeV/nucleon)
Radioactivity
Types of Decay
| Radiation | Nature | Charge | Mass | Penetration |
|---|---|---|---|---|
| α | He nucleus | +2 | 4u | Low |
| β⁻ | Electron | -1 | ~0 | Medium |
| β⁺ | Positron | +1 | ~0 | Medium |
| γ | EM wave | 0 | 0 | High |
Decay Laws
$$\boxed{N = N_0 e^{-\lambda t}}$$ $$\boxed{A = A_0 e^{-\lambda t} = \lambda N}$$Half-Life
$$\boxed{t_{1/2} = \frac{0.693}{\lambda} = \frac{\ln 2}{\lambda}}$$After n half-lives: $N = \frac{N_0}{2^n}$
Mean Life
$$\tau = \frac{1}{\lambda} = \frac{t_{1/2}}{0.693}$$Decay Equations
Alpha decay:
$$^A_Z X \rightarrow ^{A-4}_{Z-2}Y + ^4_2He$$Beta minus decay:
$$^A_Z X \rightarrow ^A_{Z+1}Y + e^- + \bar{\nu}_e$$Beta plus decay:
$$^A_Z X \rightarrow ^A_{Z-1}Y + e^+ + \nu_e$$Nuclear Reactions
Q-Value
$$Q = (m_{reactants} - m_{products})c^2$$- Q > 0: Exothermic
- Q < 0: Endothermic
Nuclear Fission
Heavy nucleus splits into lighter nuclei.
$$^{235}_{92}U + ^1_0n \rightarrow ^{141}_{56}Ba + ^{92}_{36}Kr + 3^1_0n + \text{Energy}$$Energy released: ~200 MeV per fission
Nuclear Fusion
Light nuclei combine to form heavier nucleus.
$$^2_1H + ^3_1H \rightarrow ^4_2He + ^1_0n + 17.6 \text{ MeV}$$Higher energy per nucleon but requires extremely high temperature.
Practice Problems
Find the wavelength of the second line in Balmer series of hydrogen.
Calculate the binding energy of $^{16}_8O$ if its mass is 15.995 u. (mp = 1.007825 u, mn = 1.008665 u)
The half-life of a radioactive sample is 20 days. Find the fraction remaining after 60 days.
What is the Q-value of the reaction: $^2_1H + ^2_1H \rightarrow ^3_2He + ^1_0n$? (masses: D = 2.0141 u, He-3 = 3.0160 u, n = 1.0087 u)