Atoms Set-1

Q1. The Balmer-$\alpha$ line ($n=3\to2$) for hydrogen and for deuterium differ slightly because the Rydberg constant depends on the reduced mass. Using $m_p\approx1836\,m_e$ and $m_d\approx3670\,m_e$, estimate the fractional change $\dfrac{\lambda_H-\lambda_D}{\lambda_H}$. (Assume the Rydberg constant scales linearly with the reduced mass.)

Q2. A hydrogen atom ($Z=1$) and a singly ionized helium ion He$^+$ ($Z=2$) both undergo the transition $n=4\to2$. If the wavelength for hydrogen is $\lambda_H$, what is the ratio $\dfrac{\lambda_{He^+}}{\lambda_H}$? (Neglect reduced-mass differences.)

Q3. Assertion (A): The energy levels of a hydrogen atom are independent of the nuclear mass.
Reason (R): Energy levels depend on the reduced mass $\displaystyle \mu=\frac{m_e M}{m_e+M}$ of the electron–nucleus system.

Q4. In an experiment on a hydrogen-like ion, measured photon energies for various electronic transitions were plotted versus $\Delta\equiv\big(\tfrac{1}{n_1^2}-\tfrac{1}{n_2^2}\big)$ and the points lie on a straight line through the origin with slope $54.4\ \mathrm{eV}$. Using $E_{\text{photon}}=13.6\,Z^2\,\Delta\ \mathrm{eV}$, the nuclear charge $Z$ of this ion is

Q5. What is the minimum wavelength $\lambda_{\min}$ (in nm) of a photon required to ionize a He$^+$ ion when its electron is in the $n=3$ level? Use $hc=1240\ \mathrm{eV\cdot nm}$ and $E_n=-13.6\,Z^2/n^2\ \mathrm{eV}$.

Q6. Assertion (A): In the Bohr model the allowed orbital angular momenta are given by $m v r = n\hbar\ (n=1,2,\dots)$.
Reason (R): This quantisation arises because the electron's mass increases relativistically with speed, and the relativistic mass increase enforces $m v r=n\hbar$.

Q7. For the hydrogen atom in the Bohr model $E_n=-13.6/n^2\ \mathrm{eV}$, kinetic energy $K$ and potential energy $V$ satisfy virial relations. Which statement is correct as $n$ increases?

Q8. Two hydrogen-like ions A and B have the same reduced mass. The wavelength of the photon emitted in the transition $3\to2$ in ion A equals the wavelength emitted in the transition $2\to1$ in ion B. The ratio $\dfrac{Z_A}{Z_B}$ equals

Q9. Assertion (A): The series limit of the Balmer series corresponds to ionisation from the $n=2$ level.
Reason (R): The series limit corresponds to the transition $n=\infty\to2$, which yields the maximum photon energy in the Balmer series.

Q10. A plot of energy levels $E_n$ (in eV) versus $1/n^2$ for an unknown hydrogen-like ion is a straight line with slope $-122.4\ \mathrm{eV}$. Which of the following correctly identifies the ion and gives the wavelength (in nm) of the photon emitted in the transition $4\to2$? (Use $hc=1240\ \mathrm{eV\cdot nm}$.)