Atomic and Nuclear Physics — Practice Quiz

A Physics cheat sheet for Atomic and Nuclear Physics — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Bohr Model Radius: The radius \(r_n\) of the \(n\)-th orbit in a hydrogen-like atom with atomic number \(Z\).

$$r_n = (5.29 \times 10^{-11} m) \frac{n^2}{Z}$$

Where: \(n\) = principal quantum number

Bohr Model Energy: The energy \(E_n\) of the \(n\)-th orbit in a hydrogen-like atom.

$$E_n = -(13.6 eV) \frac{Z^2}{n^2}$$

Rydberg Equation: Calculates the wavelength \(\lambda\) of a photon emitted/absorbed during a transition between levels \(n_i\) and \(n_f\).

$$\frac{1}{\lambda} = R Z^2 (\frac{1}{n_f^2} - \frac{1}{n_i^2})$$

Where: \(R\) = Rydberg constant \((1.097 \times 10^7 m^{-1})\)

X-ray Cutoff Wavelength: The minimum wavelength \(\lambda_{min}\) of X-rays produced by electrons accelerated through voltage \(V\).

$$\lambda_{min} = \frac{hc}{eV}$$

Mass-Energy Equivalence: Energy \(E\) is equivalent to mass \(m\) multiplied by the speed of light squared \(c^2\). (Used for nuclear binding energy).

$$E = mc^2$$

Where: \(c\) = speed of light

Binding Energy per Nucleon: The total binding energy \(BE\) divided by the mass number \(A\).

$$BEN = \frac{BE}{A}$$

Radioactive Decay Law: The number of remaining radioactive nuclei \(N(t)\) after time \(t\).

$$N(t) = N_0 e^{-\lambda t}$$

Where: \(\lambda\) = decay constant

Half-Life: The time \(T_{1/2}\) required for half of the radioactive nuclei to decay.

$$T_{1/2} = \frac{0.693}{\lambda}$$

Activity: The rate of decay \(A\) (decays per second) is proportional to the number of nuclei \(N\).

$$A = \lambda N$$

Where: Unit: Becquerel (Bq)

Practice quiz

  1. What is the ratio of the radius of the $n=2$ orbit in a hydrogen atom ($Z=1$) to the radius of the $n=1$ orbit in a singly ionized helium atom ($Z=2$)?

    • A) $2$
    • B) $4$
    • C) $8$
    • D) $16$

    Answer: C) $8$

  2. Calculate the energy (in $eV$) required to ionize a hydrogen atom from its ground state ($n=1$).

    • A) $3.4 \text{ eV}$
    • B) $6.8 \text{ eV}$
    • C) $13.6 \text{ eV}$
    • D) $27.2 \text{ eV}$

    Answer: C) $13.6 \text{ eV}$

  3. A hydrogen atom undergoes a transition from the $n_i=3$ state to the $n_f=2$ state. What is the wavelength of the emitted photon? (Use $R = 1.097 \times 10^7 \text{ m}^{-1}$)

    • A) $486 \text{ nm}$
    • B) $656 \text{ nm}$
    • C) $121 \text{ nm}$
    • D) $1875 \text{ nm}$

    Answer: B) $656 \text{ nm}$

  4. What is the minimum wavelength of X-rays produced when electrons are accelerated through a potential difference of $50 \text{ kV}$? (Use $h = 6.626 \times 10^{-34} \text{ J} \cdot \text{s}$, $c = 3.00 \times 10^8 \text{ m/s}$, $e = 1.602 \times 10^{-19} \text{ C}$)

    • A) $0.0124 \text{ nm}$
    • B) $0.0248 \text{ nm}$
    • C) $0.0496 \text{ nm}$
    • D) $0.0620 \text{ nm}$

    Answer: B) $0.0248 \text{ nm}$

  5. If $1 \text{ kg}$ of mass were completely converted into energy, how much energy would be released? (Use $c = 3.00 \times 10^8 \text{ m/s}$)

    • A) $3.00 \times 10^8 \text{ J}$
    • B) $9.00 \times 10^{16} \text{ J}$
    • C) $1.00 \text{ J}$
    • D) $9.00 \times 10^{13} \text{ J}$

    Answer: B) $9.00 \times 10^{16} \text{ J}$

  6. A nucleus has a total binding energy ($BE$) of $28.3 \text{ MeV}$ and a mass number ($A$) of $4$. What is its binding energy per nucleon ($BEN$)?

    • A) $4 \text{ MeV/nucleon}$
    • B) $7.075 \text{ MeV/nucleon}$
    • C) $14.15 \text{ MeV/nucleon}$
    • D) $28.3 \text{ MeV/nucleon}$

    Answer: B) $7.075 \text{ MeV/nucleon}$

  7. A radioactive sample initially contains $N_0$ nuclei. After $3$ half-lives, how many nuclei remain?

    • A) $N_0/2$
    • B) $N_0/4$
    • C) $N_0/8$
    • D) $N_0/16$

    Answer: C) $N_0/8$

  8. A radioactive isotope has a decay constant ($\lambda$) of $0.0231 \text{ s}^{-1}$. What is its half-life ($T_{1/2}$)?

    • A) $15 \text{ s}$
    • B) $30 \text{ s}$
    • C) $45 \text{ s}$
    • D) $60 \text{ s}$

    Answer: B) $30 \text{ s}$

  9. A sample contains $1.0 \times 10^{10}$ radioactive nuclei and has a decay constant ($\lambda$) of $5.0 \times 10^{-3} \text{ s}^{-1}$. What is the activity ($A$) of the sample?

    • A) $5.0 \times 10^6 \text{ Bq}$
    • B) $5.0 \times 10^7 \text{ Bq}$
    • C) $2.0 \times 10^{12} \text{ Bq}$
    • D) $2.0 \times 10^{13} \text{ Bq}$

    Answer: B) $5.0 \times 10^7 \text{ Bq}$

  10. Which of the following statements correctly describes the relationship between the energy levels of a hydrogen atom and the wavelength of emitted photons during electron transitions?

    • A) Larger energy differences between states result in longer emitted wavelengths.
    • B) The energy of an emitted photon is inversely proportional to the difference in energy levels.
    • C) Transitions to lower principal quantum numbers ($n_f$) generally result in the emission of higher energy photons.
    • D) The Rydberg equation is only applicable for absorption spectra, not emission spectra.

    Answer: C) Transitions to lower principal quantum numbers ($n_f$) generally result in the emission of higher energy photons.

Select a subject

Select a subject from the left panel to begin exploring formulas.