Atoms Set-2

Q1. (For the hydrogen‑like ion He$^+$ ($Z=2$) an electron makes a transition from $n=4$ to $n=2$. Using the Rydberg formula

and $R_H=1.097\times10^{7}\ \mathrm{m^{-1}}$, calculate the wavelength $\lambda$ of the emitted photon.)

Q2. (A: For hydrogen the frequency $\nu_{\rm ph}$ of the photon emitted in the transition $(n+1)\to n$ approaches the classical orbital frequency $\nu_{\rm orb}$ of the electron in the $n$th orbit as $n\to\infty$.
R: Using $E_n\propto-1/n^2$ one finds for large $n$ that $\Delta E\approx dE/dn$, so $\nu_{\rm ph}=\Delta E/h\approx(1/h)\,dE/dn$, which tends to the classical orbital frequency according to the correspondence principle.)

Q3. (Using Bohr model with $E_n=-13.6\,\mathrm{eV}\,\dfrac{Z^2}{n^2}$ and the fact that kinetic energy in a Bohr orbit is $K=-E_n$, compute the kinetic energy (in eV) of the electron in the $n=2$ orbit of Li$^{2+}$ ($Z=3$).)

Q4. (An experimental plot of measured energy levels $E_n$ (in eV) of a single‑electron atom versus $1/n^2$ is a straight line through the origin with slope $-122.4\ \mathrm{eV}$. Assuming Bohr's relation $E_n=-13.6\,Z^2/n^2\ \mathrm{eV}$, determine the nuclear charge $Z$ of this atom.)

Q5. (Using Bohr model the speed of the electron in the ground state ($n=1$) of hydrogen is $v_1=\alpha c$, where $\alpha\approx1/137$ is the fine‑structure constant. Take $c=3.00\times10^8\ \mathrm{m\,s^{-1}}$. Calculate $v_1$.)

Q6. (Approximate nuclear masses by $m_p=1836\,m_e$ (protium) and $m_d=3670\,m_e$ (deuterium). With reduced mass $\mu=\dfrac{m_e M}{m_e+M}$ the Rydberg constant scales as $R\propto\mu$ and hence $\lambda\propto1/\mu$ for a given transition. Calculate the ratio $\dfrac{\lambda_D}{\lambda_H}$ for corresponding Lyman lines (give numerical value to four significant digits).)

Q7. (An experiment measures frequencies $\nu_n$ for transitions $(n+1)\to n$ in hydrogen for large $n$ and finds that a plot of $\log\nu_n$ vs $\log n$ has slope approximately $-3$. Which of the following theoretical explanations accounts for this scaling?)

Q8. (A single‑electron ion produces a spectral line at $\lambda=73.0\ \mathrm{nm}$ corresponding to the transition $3\to2$. Using $R_H=1.097\times10^{7}\ \mathrm{m^{-1}}$ and the Rydberg relation $\dfrac{1}{\lambda}=R_H Z^2\left(\dfrac{1}{2^2}-\dfrac{1}{3^2}\right)$, determine the atomic number $Z$ of the ion.)

Q9. (A: The orbital magnetic moment $\mu_{\rm orb}$ of the electron in the $n$th Bohr orbit is $n\mu_B$, where $\mu_B=\dfrac{e\hbar}{2m_e}$ is the Bohr magneton.
R: Using $m_ev_nr_n=n\hbar$ and $\mu_{\rm orb}=\dfrac{e v_n r_n}{2}$ one obtains $\mu_{\rm orb}=n\dfrac{e\hbar}{2m_e}=n\mu_B$.)

Q10. (When a hydrogen atom emits the Lyman‑$\alpha$ photon ($2\to1$) of energy $10.2\ \mathrm{eV}$, the proton recoils. Using $m_p=1.673\times10^{-27}\ \mathrm{kg}$ and $c=3.00\times10^8\ \mathrm{m\,s^{-1}}$, estimate the recoil speed $v_{\rm recoil}$ of the proton (use photon momentum $p=E/c$ and $v_{\rm recoil}=p/m_p$).)