Atomic and Nuclear Physics — Hard Practice Quiz

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

Practice quiz

1. For a given hydrogen-like atom, how does the energy $E_n$ of an electron in the $n$-th orbit relate to the radius $r_n$ of that orbit?

   • $E_n \propto r_n$
   • $E_n \propto r_n^2$
   • $E_n \propto 1/r_n$
   • $E_n \propto 1/r_n^2$

   Answer: $E_n \propto 1/r_n$

2. A hydrogen-like ion with atomic number $Z$ undergoes a transition from an initial state $n_i$ to a final state $n_f$. If the emitted photon has a wavelength $\lambda$, which of the following expressions correctly relates the energy of the emitted photon to the Rydberg constant $R$ and fundamental constants $h$ and $c$?

   • $E_{photon} = hc R Z^2 (\frac{1}{n_f^2} - \frac{1}{n_i^2})$
   • $E_{photon} = \frac{R Z^2}{hc} (\frac{1}{n_f^2} - \frac{1}{n_i^2})$
   • $E_{photon} = \frac{hc}{R Z^2 (\frac{1}{n_f^2} - \frac{1}{n_i^2})}$
   • $E_{photon} = (13.6 eV) Z^2 (\frac{1}{n_i^2} - \frac{1}{n_f^2})$

   Answer: $E_{photon} = hc R Z^2 (\frac{1}{n_f^2} - \frac{1}{n_i^2})$

3. A radioactive sample initially has an activity $A_0$. After a time $t$, its activity drops to $A(t)$. If the half-life of the sample is $T_{1/2}$, derive an expression for the number of nuclei $N(t)$ remaining at time $t$ in terms of $A_0$, $T_{1/2}$, and $t$.

   • $N(t) = \frac{A_0 T_{1/2}}{0.693} e^{-(\frac{0.693}{T_{1/2}}) t}$
   • $N(t) = A_0 T_{1/2} e^{-(\frac{0.693}{T_{1/2}}) t}$
   • $N(t) = \frac{A_0}{0.693 T_{1/2}} e^{-(\frac{0.693}{T_{1/2}}) t}$
   • $N(t) = A_0 e^{-(\frac{0.693}{T_{1/2}}) t}$

   Answer: $N(t) = \frac{A_0 T_{1/2}}{0.693} e^{-(\frac{0.693}{T_{1/2}}) t}$

4. The binding energy per nucleon ($BEN$) of a nucleus with mass number $A$ is given. If the mass defect of this nucleus is $\Delta m$, which of the following expressions correctly relates $\Delta m$ to $BEN$, $A$, and the speed of light $c$?

   • $\Delta m = \frac{BEN \cdot A}{c^2}$
   • $\Delta m = BEN \cdot A \cdot c^2$
   • $\Delta m = \frac{BEN}{A \cdot c^2}$
   • $\Delta m = \frac{A}{BEN \cdot c^2}$

   Answer: $\Delta m = \frac{BEN \cdot A}{c^2}$

5. An electron is accelerated through a potential difference $V$ to produce X-rays with a minimum wavelength $\lambda_{min}$. If this electron's kinetic energy is then used to excite a hydrogen atom from its ground state ($n=1$), which of the following expressions correctly determines the maximum principal quantum number $n_{max}$ that the hydrogen atom can reach? (Assume $Z=1$ for hydrogen and the electron transfers all its kinetic energy).

   • $n_{max} = \sqrt{\frac{13.6}{13.6 - V}}$
   • $n_{max} = \sqrt{\frac{V}{13.6 - V}}$
   • $n_{max} = \frac{13.6}{V}$
   • $n_{max} = \frac{13.6 - V}{13.6}$

   Answer: $n_{max} = \sqrt{\frac{13.6}{13.6 - V}}$

6. A radioactive isotope has a half-life of $T_{1/2}$. After how many half-lives, $k$, will the activity of the sample be reduced to $1/16$th of its initial value?

   • $2$
   • $3$
   • $4$
   • $8$

   Answer: $4$

7. Consider two hydrogen-like ions, Ion A with atomic number $Z_A$ and Ion B with atomic number $Z_B$. If both ions undergo the same electron transition (i.e., from $n_i$ to $n_f$), how does the wavelength of the emitted photon from Ion A, $\lambda_A$, compare to the wavelength of the emitted photon from Ion B, $\lambda_B$?

   • $\lambda_A = \lambda_B \frac{Z_A}{Z_B}$
   • $\lambda_A = \lambda_B \frac{Z_B}{Z_A}$
   • $\lambda_A = \lambda_B (\frac{Z_B}{Z_A})^2$
   • $\lambda_A = \lambda_B (\frac{Z_A}{Z_B})^2$

   Answer: $\lambda_A = \lambda_B (\frac{Z_B}{Z_A})^2$

8. Derive an expression for the initial number of radioactive nuclei $N_0$ in a sample, given its initial activity $A_0$ and its half-life $T_{1/2}$.

   • $N_0 = \frac{A_0 T_{1/2}}{0.693}$
   • $N_0 = A_0 T_{1/2} \cdot 0.693$
   • $N_0 = \frac{0.693 A_0}{T_{1/2}}$
   • $N_0 = \frac{T_{1/2}}{0.693 A_0}$

   Answer: $N_0 = \frac{A_0 T_{1/2}}{0.693}$

9. A nucleus has a binding energy per nucleon ($BEN$) and a mass defect of $\Delta m$. If the speed of light is $c$, which of the following expressions correctly represents the mass number $A$ of the nucleus?

   • $A = \frac{\Delta m c^2}{BEN}$
   • $A = \frac{BEN}{\Delta m c^2}$
   • $A = \Delta m BEN c^2$
   • $A = \frac{\Delta m}{BEN c^2}$

   Answer: $A = \frac{\Delta m c^2}{BEN}$

10. For a hydrogen-like atom with atomic number $Z$, what is the minimum wavelength $\lambda_{ion}$ of a photon required to ionize the atom from its ground state ($n=1$)? Express your answer in terms of the Rydberg constant $R$ and $Z$.

    • $\lambda_{ion} = \frac{1}{R Z^2}$
    • $\lambda_{ion} = R Z^2$
    • $\lambda_{ion} = \frac{hc}{R Z^2}$
    • $\lambda_{ion} = \frac{R Z^2}{hc}$

    Answer: $\lambda_{ion} = \frac{1}{R Z^2}$