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Atoms Set-4

Practice Important MCQs for Atoms — Physics, Class 12.

Atoms (Bohr model, spectra via Rydberg formula, and reduced-mass/recoil corrections) forms the core of Class 12 Physics questions on quantization and line spectra. Board and competitive exams frequently test whether you can correctly relate energy levels to wavelengths, and whether you understand how finite nuclear mass changes the effective Rydberg constant and causes small but definite shifts in spectral lines.

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Q1. Using the Bohr model $r_n = a_0\frac{n^2}{Z}$ with $a_0 = 5.29\times10^{-11}\,\text{m}$ , calculate the radius of the first Bohr orbit ( $n=1$ ) for He $^+$ ( $Z=2$ ).

(A)

$5.29\times10^{-11}\,\text{m}$

(B)

$1.06\times10^{-10}\,\text{m}$

(C)

$2.645\times10^{-11}\,\text{m}$

(D)

$1.763\times10^{-11}\,\text{m}$

Q2. An electron in hydrogen transitions from $n=4$ to $n=2$ , emitting a photon of energy $\Delta E = E_4 - E_2$ (where $E_n$ is total energy). Using Bohr relations $K_n=-E_n$ and $V_n=2E_n$ for the $n$ th orbit, which relation between $\Delta E$ , $\Delta K=K_2-K_4$ and $\Delta V=V_2-V_4$ is correct?

(A)

$\Delta E=\Delta K=-\tfrac{1}{2}\Delta V$

(B)

$\Delta E=-\Delta K=\tfrac{1}{2}\Delta V$

(C)

$\Delta E=\Delta K+\Delta V$

(D)

$\Delta E=\tfrac{1}{2}\Delta K=-\Delta V$

Q3. Incorporating reduced mass $\mu$ in the Bohr model makes the Rydberg constant proportional to $\mu$ . With proton mass $m_p\approx1836\,m_e$ , and deuteron mass $\approx2m_p$ , which statement about the Lyman- $\alpha$ ( $n=2\to1$ ) wavelength is correct?

(A)

$\lambda_{\rm D}>\lambda_{\rm H}$ because a heavier nucleus lowers the binding energy

(B)

$\lambda_{\rm D}=\lambda_{\rm H}$ (nuclear mass cancels)

(C)

$\lambda_{\rm D}<\lambda_{\rm H}$ and $(\lambda_{\rm H}-\lambda_{\rm D})/\lambda_{\rm H}\approx5.45\times10^{-4}$

(D)

$\lambda_{\rm D}<\lambda_{\rm H}$ and $(\lambda_{\rm H}-\lambda_{\rm D})/\lambda_{\rm H}\approx2.725\times10^{-4}$

Q4. Accounting for proton recoil (momentum conservation), the photon energy for a transition is reduced by the nucleus recoil kinetic energy. For the hydrogen Lyman- $\alpha$ transition ( $\Delta E\approx10.2\,$ eV), estimate the fractional increase in wavelength $\delta=(\lambda_{\rm actual}-\lambda_0)/\lambda_0$ due to recoil using the approximation $\delta\approx\dfrac{\Delta E}{2M_p c^2}$ with $M_p c^2\approx9.38\times10^{8}\,$ eV. Which is closest?

(A)

$5.4\times10^{-8}$

(B)

$5.4\times10^{-9}$

(C)

$1.1\times10^{-9}$

(D)

$5.4\times10^{-10}$

Q5. Using the Rydberg formula $\dfrac{1}{\lambda}=RZ^2\!\left(\dfrac{1}{n_f^2}-\dfrac{1}{n_i^2}\right)$ , which single-electron transition produces a spectral line with the same wavelength as hydrogen Lyman- $\alpha$ ( $n=2\to1$ in H)?

(A)

He $^+$ : $n=3\to2$

(B)

Li^{2+} : $n=4\to2$

(C)

He $^+$ : $n=4\to2$

(D)

Li^{2+} : $n=2\to1$

Q6. A hydrogen atom makes a transition from $n=3$ to $n=1$ . Using the Rydberg formula $\displaystyle \frac{1}{\lambda}=R\!\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$ with $R=1.097\times10^7\ \mathrm{m^{-1}}$ , the wavelength of the emitted photon is closest to

(A)

$656\ \mathrm{nm}$

(B)

$410\ \mathrm{nm}$

(C)

$102.6\ \mathrm{nm}$

(D)

$97.2\ \mathrm{nm}$

Q7. The Hα line corresponds to the transition $n=3\to2$ . Taking into account reduced-mass correction where the effective Rydberg constant is $R=R_\infty\frac{\mu}{m_e}$ , and using $m_p\approx1836\,m_e$ , $m_D\approx3670\,m_e$ , which statement about the wavelengths $\lambda_H$ (hydrogen) and $\lambda_D$ (deuterium) is correct?

(A)

$\lambda_H$ is longer than $\lambda_D$ by about $0.21\ \mathrm{nm}$

(B)

$\lambda_H$ is shorter than $\lambda_D$ by about $0.21\ \mathrm{nm}$

(C)

$\lambda_H$ is longer than $\lambda_D$ by about $2.13\ \mathrm{nm}$

(D)

$\lambda_H=\lambda_D$

Q8. Using the Bohr energy formula $E_n=-\dfrac{13.6\,Z^2}{n^2}\ \mathrm{eV}$ for hydrogen-like ions, which of the following pairs of levels have equal energies?

(A)

H $(n=3)$ and He $^+$ $(n=4)$

(B)

Li $^{2+}$ $(n=2)$ and H $(n=1)$

(C)

Be $^{3+}$ $(n=2)$ and Li $^{2+}$ $(n=1)$

(D)

He $^+$ $(n=2)$ and H $(n=1)$

Q9. An electron at rest far from a stationary proton is captured into the hydrogen ground state; the binding energy is $B=13.6\ \mathrm{eV}$ . Accounting for momentum conservation (photon momentum $p=E_\gamma/c$ ) gives $E_\gamma+\dfrac{E_\gamma^2}{2Mc^2}=B$ , where $M$ is proton mass. Approximately what fraction of $B$ is converted into kinetic energy of the recoiling hydrogen atom? (Use $M\approx1.67\times10^{-27}\ \mathrm{kg}$ . )

(A)

$3.6\times10^{-8}$

(B)

$7.25\times10^{-9}$

(C)

$1.5\times10^{-3}$

(D)

$7.25\times10^{-8}$

Q10. In muonic hydrogen a muon of mass $m_\mu\approx206.77\,m_e$ orbits a proton. Using the reduced mass $\mu=\dfrac{m_\mu m_p}{m_\mu+m_p}$ and the Bohr-type energy $E_n=-\dfrac{13.6\,\mu}{m_e}\dfrac{1}{n^2}\ \mathrm{eV}$ (take $m_p\approx1836\,m_e$ and $hc\approx1240\ \mathrm{eV\cdot nm}$ ), the wavelength of the $n=2\to1$ transition is approximately

(A)

$0.590\ \mathrm{nm}$

(B)

$0.095\ \mathrm{nm}$

(C)

$0.653\ \mathrm{nm}$

(D)

$65.3\ \mathrm{nm}$

Q11. An electron in a hydrogen atom makes a transition from $n=4$ to $n=2$ . Using Bohr's model and Rydberg constant $R_H=1.097\times10^7\ \mathrm{m^{-1}}$ , the wavelength of the emitted photon is closest to:

(A)

$121.6\ \mathrm{nm}$

(B)

$656\ \mathrm{nm}$

(C)

$486\ \mathrm{nm}$

(D)

$434\ \mathrm{nm}$

Q12. For hydrogen (Z=1) and singly-ionized helium He $^+$ (Z=2), both undergo the transition $n=4\to n=2$ . If the emitted wavelengths are $\lambda_H$ and $\lambda_{He^+}$ respectively, which relation is correct (neglect reduced-mass and recoil effects)?

(A)

$\lambda_{He^+}=\dfrac{\lambda_H}{4}$

(B)

$\lambda_{He^+}=\dfrac{\lambda_H}{2}$

(C)

$\lambda_{He^+}=4\,\lambda_H$

(D)

$\lambda_{He^+}=\dfrac{\lambda_H}{\sqrt{2}}$

Q13. The $4\to2$ spectral line is measured for hydrogen (proton mass $M_p=1836\,m_e$ ) and deuterium ( $M_d=3670\,m_e$ ). Using the reduced-mass correction to the Rydberg constant ( $R\propto\mu$ ), which statement about the deuterium wavelength $\lambda_D$ relative to hydrogen $\lambda_H$ is most accurate (approximate)?

(A)

$\lambda_D\approx\lambda_H+0.132\ \mathrm{nm}$ (deuterium longer)

(B)

$\lambda_D\approx\lambda_H-0.132\ \mathrm{nm}$ (deuterium shorter)

(C)

$\lambda_D\approx\lambda_H$ (no measurable change)

(D)

$\lambda_D\approx\lambda_H-1.32\ \mathrm{nm}$ (much shorter)

Q14. Assertion (A): Compared to the idealized infinite-mass nucleus case, (i) the finite nuclear-mass (reduced-mass) correction and (ii) the atomic recoil during photon emission both shift emitted spectral lines to longer wavelength (redshift).
Reason (R): The reduced mass $\mu=\dfrac{m_e M}{m_e+M}$ satisfies $\mu<m_e$ , so the effective Rydberg $R=R_\infty(\mu/m_e)$ is smaller (increasing wavelength); additionally conservation of momentum gives a recoil energy $E_{rec}\approx\dfrac{E_{ph}^2}{2Mc^2}$ which reduces photon energy.

(A)

Both A and R are true, but R does not correctly explain A

(B)

A is true but R is false

(C)

A is false but R is true

(D)

Both A and R are true and R correctly explains A

Q15. A hydrogen atom (Z=1) makes the transition $4\to2$ and emits a photon of wavelength $\lambda$ . A He $^+$ ion (Z=2) makes a transition $p\to q$ that emits a photon of the same wavelength $\lambda$ . Which integer pair $(p,q)$ satisfies this equality?

(A)

$(6,3)$

(B)

$(8,4)$

(C)

$(5,2)$

(D)

$(7,4)$

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