Atoms

• Maximum VSA, SAI, SAll and LA type questions were asked from Bohr Model of Hydrogen Atom.

• Rutherford’s observations and results :
• Most of the $\alpha$$\alpha$alpha\alpha$\alpha$-particles pass through the gold foil without any deflection. This shows that most of the space in an atom is empty.
• Few $\alpha$$\alpha$alpha\alpha$\alpha$-particles got scattered, deflecting at various angles from 0 to $\pi$$\pi$pi\pi$\pi$. This shows that atom has a small positively charged core called ‘nucleus’ at centre of atom, which deflects the positively charged $\alpha$$\alpha$alpha\alpha$\alpha$-particles at different angles depending on their distance from centre of nucleus.
• Very few $\alpha$$\alpha$alpha\alpha$\alpha$-particles (1 in 8000) suffers deflection of ${180}^{\circ }$${180}^{\circ }$180^(@)180^{\circ}${180}^{\circ }$. This shows that size of nucleus is very small, nearly $1/8000$$1/8000$1//80001 / 8000$1/8000$ times the size of atom.

• Impact parameter : It is defined as the perpendicular distance of the initial velocity vector of the alpha particle from the centre of the nucleus, when the particle is far away from the nucleus of the atom.
• The scattering angle $\theta$$\theta$theta\theta$\theta$ of the $\alpha$$\alpha$alpha\alpha$\alpha$ particle and impact parameter $b$$b$bb$b$ are related as

• Smaller the impact parameter, larger the angle of scattering $\theta$$\theta$theta\theta$\theta$.

• Distance of closest approach

• An electron can revolve around the nucleus only in certain allowed circular orbits of definite energy and in these orbits it does not radiate. These orbits are known as stationary orbits.
• Angular momentum of the electron in a stationary orbit is an integral multiple of $h/2\pi$$h/2\pi$h//2pih / 2 \pi$h/2\pi$. i.e., $L=\frac{nh}{2\pi }$$L=\frac{nh}{2\pi }$L=(nh)/(2pi)L=\frac{n h}{2 \pi}$L=\frac{nh}{2\pi }$ or, $mvr=\frac{nh}{2\pi }$$mvr=\frac{nh}{2\pi }$mvr=(nh)/(2pi)m v r=\frac{n h}{2 \pi}$mvr=\frac{nh}{2\pi }$

• Velocity of the electron in the $n^{\text{th}}$$n^{\text{th }}$n^(“th “)n^{\text {th }}$n^{\text{th }}$ orbit $v_n=\frac{1}{4\pi {\epsilon }_0}\frac{2\pi Z{e}^2}{nh}=\frac{2.2\times {10}^6\mathrm{Z}}{n}\mathrm{m}/\mathrm{s}$$v_n=\frac{1}{4\pi {\epsilon }_0}\frac{2\pi Z{e}^2}{nh}=\frac{2.2\times {10}^6\mathrm{Z}}{n}\mathrm{m}/\mathrm{s}$v_(n)=(1)/(4piepsi_(0))(2pi Ze^(2))/(nh)=(2.2 xx10^(6)Z)/(n)m//sv_{n}=\frac{1}{4 \pi \varepsilon_{0}} \frac{2 \pi Z e^{2}}{n h}=\frac{2.2 \times 10^{6} \mathrm{Z}}{n} \mathrm{~m} / \mathrm{s}$v_n=\frac{1}{4\pi {\epsilon }_0}\frac{2\pi Z{e}^2}{nh}=\frac{2.2\times {10}^6\mathrm{Z}}{n}\mathrm{m}/\mathrm{s}$

• Total energy of electron in the $n^{\text{th}}$$n^{\text{th }}$n^(“th “)n^{\text {th }}$n^{\text{th }}$ orbit

• Frequency of the electron in the $n^{\text{th}}$$n^{\text{th }}$n^(“th “)n^{\text {th }}$n^{\text{th }}$ orbit

• Wavelength of radiation in the transition from $n_2\to n_1$$n_2\to n_1$n_(2)rarrn_(1)n_{2} \rightarrow n_{1}$n_2\to n_1$ is given by $\frac{1}{\lambda }=R{Z}^2[\frac{1}{n_1^2}-\frac{1}{n_2^2}]$$\frac{1}{\lambda }=R{Z}^2\left[\frac{1}{n_1^2}-\frac{1}{n_2^2}\right]$(1)/(lambda)=RZ^(2)[(1)/(n_(1)^(2))-(1)/(n_(2)^(2))]\frac{1}{\lambda}=R Z^{2}\left[\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right]$\frac{1}{\lambda }=R{Z}^2[\frac{1}{n_1^2}-\frac{1}{n_2^2}]$ where $R$$R$RR$R$ is called Rydberg’s constant. $R=\left(\frac{1}{4\pi {\epsilon }_0}\right)\frac{2{\pi }^{2}m{e}^4}{c{h}^3}=1.097\times {10}^7{\mathrm{m}}^{-1}$$R=\left(\frac{1}{4\pi {\epsilon }_0}\right)\frac{2{\pi }^{2}m{e}^4}{c{h}^3}=1.097\times {10}^7{\mathrm{m}}^{-1}$R=((1)/(4piepsi_(0)))(2pi^(2)me^(4))/(ch^(3))=1.097 xx10^(7)m^(-1)R=\left(\frac{1}{4 \pi \varepsilon_{0}}\right) \frac{2 \pi^{2} m e^{4}}{c h^{3}}=1.097 \times 10^{7} \mathrm{~m}^{-1}$R=\left(\frac{1}{4\pi {\epsilon }_0}\right)\frac{2{\pi }^{2}m{e}^4}{c{h}^3}=1.097\times {10}^7{\mathrm{m}}^{-1}$ (A) Spectral series of hydrogen atom: When the electron in a $\mathrm{H}$$\mathrm{H}$H\mathrm{H}$\mathrm{H}$-atom jumps from higher energy level to lower energy level, the difference of energies of the two energy levels is emitted as radiation of particular wavelength, known as spectral line. Spectral lines of different wavelengths are obtained for transition of electron between two different energy levels, which are found to fall in a number of spectral series given by

• Lyman series

• Series limit line (shortest wavelength) of Lyman series is given by

• The first line (longest wavelength) of the Lyman series is given by

• Lyman series lie in the ultraviolet region of electromagnetic spectrum.
• Lyman series is obtained in emission as well
• Balmer series as in absorption spectrum.
• Emission spectral lines corresponding to the transition of electron from higher energy levels $(n_2=3,4,\dots \mathrm{\infty })$$\left(n_2=3,4,\dots \mathrm{\infty }\right)$(n_(2)=3,4,dots oo)\left(n_{2}=3,4, \ldots \infty\right)$(n_2=3,4,\dots \mathrm{\infty })$ to second energy level $(n_1=2)$$\left(n_1=2\right)$(n_(1)=2)\left(n_{1}=2\right)$(n_1=2)$ constitute Balmer series.

• Series limit line (shortest wavelength) of Balmer series is given by

• The first line (longest wavelength) of the Balmer series is given by

• Balmer series lie in the visible region of electromagnetic spectrum.
• This series is obtained only in emission spectrum.
• Paschen series
• Emission spectral lines corresponding to the transition of electron from higher energy levels $(n_2=4,5,\dots ..,\mathrm{\infty })$$\left(n_2=4,5,\dots ..,\mathrm{\infty }\right)$(n_(2)=4,5,dots..,oo)\left(n_{2}=4,5, \ldots . ., \infty\right)$(n_2=4,5,\dots ..,\mathrm{\infty })$ to third energy level $(n_1=3)$$\left(n_1=3\right)$(n_(1)=3)\left(n_{1}=3\right)$(n_1=3)$ constitute Paschen series.

• Series limit line (shortest wavelength) of the Paschen series is given by

• Paschen series lie in the infrared region of the electromagnetic spectrum.
• This series is obtained only in the emission spectrum.
• Brackett Series
• Emission spectral lines corresponding to the transition of electron from higher energy levels $(n_2=5,6,7,\dots ..,\mathrm{\infty })$$\left(n_2=5,6,7,\dots ..,\mathrm{\infty }\right)$(n_(2)=5,6,7,dots..,oo)\left(n_{2}=5,6,7, \ldots . ., \infty\right)$(n_2=5,6,7,\dots ..,\mathrm{\infty })$ to fourth energy level $(n_1=4)$$\left(n_1=4\right)$(n_(1)=4)\left(n_{1}=4\right)$(n_1=4)$ constitute Brackett series.

• The first line (longest wavelength) of Brackett series is given by

• Brackett series lie in the infrared region of the electromagnetic spectrum.
• This series is obtained only in the emission spectrum.
• Pfund series
• Emission spectral lines corresponding to the transition of electron from higher energy levels $(n_2=6,7,8,\dots \dots .,\mathrm{\infty })$$\left(n_2=6,7,8,\dots \dots .,\mathrm{\infty }\right)$(n_(2)=6,7,8,dots dots.,oo)\left(n_{2}=6,7,8, \ldots \ldots ., \infty\right)$(n_2=6,7,8,\dots \dots .,\mathrm{\infty })$ to fifth energy level $(n_1=5)$$\left(n_1=5\right)$(n_(1)=5)\left(n_{1}=5\right)$(n_1=5)$ constitute Pfund series.

• The first line (longest wavelength) of the Pfund series is given by

• Pfund series also lie in the infrared region of electromagnetic spectrum.
• This series is obtained only in the emission spectrum.
• Number of spectral lines due to transition of electron from $n^{\text{th}}$$n^{\text{th }}$n^(“th “)n^{\text {th }}$n^{\text{th }}$ orbit to lower orbit is

• Ionisation energy : The energy required, to knock an electron completely out of the atom.
• Ionisation potential $=\frac{13.6Z^2}{n^2}\mathrm{V}$$=\frac{13.6Z^2}{n^2}\mathrm{V}$=(13.6Z^(2))/(n^(2))V=\frac{13.6 Z^{2}}{n^{2}} \mathrm{~V}$=\frac{13.6Z^2}{n^2}\mathrm{V}$

1. Previous Years’ CBSE Board Questions

2. VSA (1 mark)

1. Why is the classical (Rutherford) model for an atom of electron orbiting around the nucleus not able to explain the atomic structure?

3. SA I ( 2 marks)

2. Using Rutherford’s model of the atom, derive the expression for the total energy of the electron in hydrogen atom. What is the significance of total negative energy possessed by the electron?

3. In an experiment on $\alpha$$\alpha$alpha\alpha$\alpha$-particle scattering by a thin foil of gold, draw a plot showing the number of particles scattered versus the scattering angle $\theta$$\theta$theta\theta$\theta$. Why is it that a very small fraction of the particles are scattered at $\theta >{90}^{\circ }$$\theta >{90}^{\circ }$theta > 90^(@)\theta>90^{\circ}$\theta >{90}^{\circ }$ ?

4. SA II (3 marks)

4. Draw a schematic arrangement of the Geiger – Marsden experiment for studying $\alpha$$\alpha$alpha\alpha$\alpha$-particle scattering by a thin foil of gold. Describe briefly, by drawing trajectories of the scattered $\alpha$$\alpha$alpha\alpha$\alpha$-particles, how this study can be used to estimate the size of the nucleus.

5. State the basic assumption of the Rutherford model of the atom. Explain, in brief, why this model cannot account for the stability of an atom.

5. LA $(5$$(5$quad(5\quad(5$(5$ marks)

6. In Rutherford scattering experiment, draw the trajectory traced by $\alpha$$\alpha$alpha\alpha$\alpha$-particles in the coulomb field of target nucleus and explain how this led to estimate the size of the nucleus.

5.1. Atomic Spectra

6. SA I (2 marks)

7. Calculate the shortest wavelength in the Balmer series of hydrogen atom. In which region (infrared, visible, ultraviolet) of hydrogen spectrum does this wavelength lie?

7. SA II (3 marks)

8. The second member of Lyman series in hydrogen spectrum has wavelength $5400\text{Å}$$5400\text{Å}$5400″Å”5400 \AA$5400\text{Å}$ A. Find the wavelength of the first member.

7.1. Bohr Model of Hydrogen Atom

8. VSA (1 mark)

9. What is the ratio of radii of the orbits corresponding to first excited state and ground state in a hydrogen atom?

10. Define ionisation energy. What is its value for a hydrogen atom?

11. State Bohr’s quantisation condition for defining stationary orbits.

9. SA I (2 marks)

12. State Bohr postulate of hydrogen atom that gives the relationship for the frequency of emitted photon in a transition.

13. Show that the radius of the orbit in hydrogen atom varies as $n^2$$n^2$n^(2)n^{2}$n^2$, where $n$$n$nn$n$ is the principal quantum number of the atom.

14. Using Bohr’s postulates of the atomic model, derive the expression for the radius of $n^{\text{th}}$$n^{\text{th }}$n^(“th “)n^{\text {th }}$n^{\text{th }}$ electron orbit. Hence obtain the expression for Bohr’s radius.

15. In the ground state of hydrogen atom, its Bohr radius is given as $5.3\times {10}^{-11}\mathrm{m}$$5.3\times {10}^{-11}\mathrm{m}$5.3 xx10^(-11)m5.3 \times 10^{-11} \mathrm{~m}$5.3\times {10}^{-11}\mathrm{m}$. The atom is excited such that the radius becomes $21.2\times {10}^{-11}\mathrm{m}$$21.2\times {10}^{-11}\mathrm{m}$21.2 xx10^(-11)m21.2 \times 10^{-11} \mathrm{~m}$21.2\times {10}^{-11}\mathrm{m}$. Find (i) the value of the principal quantum number and (ii) the total energy of the atom in this excited state.

10. SA II (3 marks)

16. (a) The radius of the innermost electron orbit of a hydrogen atom is $5.3\times {10}^{-11}\mathrm{m}$$5.3\times {10}^{-11}\mathrm{m}$5.3 xx10^(-11)m5.3 \times 10^{-11} \mathrm{~m}$5.3\times {10}^{-11}\mathrm{m}$. Calculate its radius in $n=3$$n=3$n=3n=3$n=3$ orbit.

17. Using Bohr’s postulates, obtain the expression for the total energy of the electron in the stationary states of the hydrogen atom. Hence draw the energy level diagram showing how the line spectra corresponding to Balmer series occur due to transition between energy levels.

18. Using Bohr’s postulates for hydrogen atom, show that the total energy $(E)$$(E)$(E)(E)$(E)$ of the electron in the stationary states can be expressed as the sum of kinetic energy $(K)$$(K)$(K)(K)$(K)$ and potential energy $(U)$$(U)$(U)(U)$(U)$, where $K=-2U$$K=-2U$K=-2UK=-2 U$K=-2U$. Hence deduce the expression for the total energy in the $n^{\text{th}}$$n^{\text{th }}$n^(“th “)n^{\text {th }}$n^{\text{th }}$ energy level of hydrogen atom.

19. The energy levels of a hypothetical atom are shown below. Which of the shown transitions will result in the emission of a photon of wavelength $275\mathrm{n}\mathrm{m}$$275\mathrm{n}\mathrm{m}$275nm275 \mathrm{~nm}$275\mathrm{n}\mathrm{m}$ ?

20. Using the postulates of Bohr’s model of hydrogen atom, obtain an expression for the frequency of radiation emitted when atom make a transition from the higher energy state with quantum number $n_i$$n_i$n_(i)n_{i}$n_i$ to the lower energy state with quantum number $n_f(n_f<n_i)$$n_f\left(n_f<n_i\right)$n_(f)(n_(f) < n_(i))n_{f}\left(n_{f}<n_{i}\right)$n_f(n_f<n_i)$.

21. Using the relevant Bohr’s postulates, derive the expressions for the (a) speed of the electron in the $n^{\text{th}}$$n^{\text{th }}$n^(“th “)n^{\text {th }}$n^{\text{th }}$ orbit,

22. State any two postulates of Bohr’s theory of hydrogen atom.

23. (a) The energy levels of an atom are as shown below. Which of them will result in the transition of a photon of wavelength $275\mathrm{n}\mathrm{m}$$275\mathrm{n}\mathrm{m}$275nm275 \mathrm{~nm}$275\mathrm{n}\mathrm{m}$ ?

11. LA (5 marks)

24. (a) Write two important limitations of Rutherford model which could not explain the observed features of atomic spectra. How were these explained in Bohr’s model of hydrogen atom?

25. Using Bohr’s postulates, derive the expression for the total energy of the electron in the stationary states of the hydrogen atom. (3/5, Foreign 2014)
26. (a) Using Bohr’s theory of hydrogen atom, derive the expression for the total energy of the electron in the stationary states of the atom.

27. (a) Using postulates of Bohr’s theory of hydrogen atom, show that

28. The energy level diagram of an element is given below. Identify, by doing necessary calculations, which transition corresponds to the emission of a spectral line of wavelength $102.7\mathrm{n}\mathrm{m}$$102.7\mathrm{n}\mathrm{m}$102.7nm102.7 \mathrm{~nm}$102.7\mathrm{n}\mathrm{m}$.

12. VSA (1 mark)

29. When is $H_{\alpha }$$H_{\alpha }$H_(alpha)H_{\alpha}$H_{\alpha }$ line of the Balmer series in the emission spectrum of hydrogen atom obtained?

30. What is the maximum number of spectral lines emitted by a hydrogen atom when it is in the third excited state?

13. SAI (2 marks)

31. Define ionization energy.

32. An electron jumps from fourth to first orbit in an atom. How many maximum number of spectral lines can be emitted by the atom? To which series these lines correspond?

33. The figure shows energy level diagram of hydrogen atom.

34. (i) In hydrogen atom, an electron undergoes transition from $2^{\text{nd}}$$2^{\text{nd }}$2^(“nd “)2^{\text {nd }}$2^{\text{nd }}$ excited state to the first excited state and then to the ground state. Identify the spectral series to which these transitions belong. (ii) Find out the ratio of the wavelengths of the emitted radiations in the two cases. (AI 2012C)

14. SA II (3 marks)

35. Using Rydberg formula, calculate the longest wavelength belonging to Lyman and Balmer series of hydrogen spectrum. In which region these transitions lie? $(3/5$$(3/5$quad(3//5\quad(3 / 5$(3/5$, Foreign 2015)
36. A $12.5\mathrm{e}\mathrm{V}$$12.5\mathrm{e}\mathrm{V}$12.5eV12.5 \mathrm{eV}$12.5\mathrm{e}\mathrm{V}$ electron beam is used to bombard gaseous hydrogen at room temperature. Upto which energy level the hydrogen atoms would be excited?

37. The value of ground state energy of hydrogen atom is $-13.6\mathrm{e}\mathrm{V}$$-13.6\mathrm{e}\mathrm{V}$-13.6eV-13.6 \mathrm{eV}$-13.6\mathrm{e}\mathrm{V}$.

38. (a) The energy levels of a hypothetical hydrogen-like atom are shown in the figure. Find out the transition, from the ones shown in the figure, which will result in the emission of a photon of wavelength $275\mathrm{n}\mathrm{m}$$275\mathrm{n}\mathrm{m}$275nm275 \mathrm{~nm}$275\mathrm{n}\mathrm{m}$.

40. The ground state energy of hydrogen atom is $-13.6\mathrm{e}\mathrm{V}$$-13.6\mathrm{e}\mathrm{V}$-13.6eV-13.6 \mathrm{eV}$-13.6\mathrm{e}\mathrm{V}$. If an electron makes a transition from an energy level $-0.85\mathrm{e}\mathrm{V}$$-0.85\mathrm{e}\mathrm{V}$-0.85eV-0.85 \mathrm{eV}$-0.85\mathrm{e}\mathrm{V}$ to $-3.4\mathrm{e}\mathrm{V}$$-3.4\mathrm{e}\mathrm{V}$-3.4eV-3.4 \mathrm{eV}$-3.4\mathrm{e}\mathrm{V}$, calculate the wavelength of the spectral line emitted. To which series of hydrogen spectrum does this wavelength belong?

41. The electron in a given Bohr orbit has a total energy of $-1.5\mathrm{e}\mathrm{V}$$-1.5\mathrm{e}\mathrm{V}$-1.5eV-1.5 \mathrm{eV}$-1.5\mathrm{e}\mathrm{V}$. Calculate its

15. LA (5 marks)

42. Using Rydberg formula, calculate the wavelengths of the spectral lines of the first member of the Lyman series and of the Balmer series.

43. Using Bohr’s postulates, derive the expression for the frequency of radiation emitted when electron in hydrogen atom undergoes transition from higher energy state (quantum number $n_i$$n_i$n_(i)n_{i}$n_i$ ) to the lower state, $\left(n_f\right)$$\left(n_f\right)$(n_(f))\left(n_{f}\right)$\left(n_f\right)$. When electron in hydrogen atom jumps from energy state $n_i=4$$n_i=4$n_(i)=4n_{i}=4$n_i=4$ to $n_f=3,2,1$$n_f=3,2,1$n_(f)=3,2,1n_{f}=3,2,1$n_f=3,2,1$. Identify the spectral series to which the emission lines belong.

44. Calculate the wavelength of the first spectral line in the corresponding Lyman series of this atom.

45. Calculate the wavlength of $H_{\alpha }$$H_{\alpha }$H_(alpha)H_{\alpha}$H_{\alpha }$ line in Balmer series of hydrogen atom, given Rydberg constant $R=1.097\times {10}^7{\mathrm{m}}^{-1}$$R=1.097\times {10}^7{\mathrm{m}}^{-1}$R=1.097 xx10^(7)m^(-1)R=1.097 \times 10^{7} \mathrm{~m}^{-1}$R=1.097\times {10}^7{\mathrm{m}}^{-1}$.

16. VSA (1 mark)

46. State de-Broglie hypothesis.

17. SAI (2 marks)

47. Calculate the de-Broglie wavelength of the electron orbiting in the $n=2$$n=2$n=2n=2$n=2$ state of hydrogen atom.

48. Use de-Broglie’s hypothesis to write the relation for the $n^{\text{th}}$$n^{\text{th }}$n^(“th “)n^{\text {th }}$n^{\text{th }}$ radius of Bohr orbit in terms of Bohr’s quantization condition of orbital angular momentum.

18. SA II (3 marks)

49. (i) State Bohr’s quantization condition for defining stationary orbits. How does de Broglie hypothesis explain the stationary orbits ?

50. The kinetic energy of the electron orbiting in the first excited state of hydrogen atom is $3.4\mathrm{e}\mathrm{V}$$3.4\mathrm{e}\mathrm{V}$3.4eV3.4 \mathrm{eV}$3.4\mathrm{e}\mathrm{V}$. Determine the de Broglie wavelength associated with it.

51. An electron is revolving around the nucleus with a constant speed of $2.2\times {10}^8\mathrm{m}/\mathrm{s}$$2.2\times {10}^8\mathrm{m}/\mathrm{s}$2.2 xx10^(8)m//s2.2 \times 10^{8} \mathrm{~m} / \mathrm{s}$2.2\times {10}^8\mathrm{m}/\mathrm{s}$. Find the de Broglie wavelength associated with it.

52. Using Bohr’s second postulate of quantization of orbital angular momentum show that the circumference of the electron in the $n^{\text{th}}$$n^{\text{th }}$n^(“th “)n^{\text {th }}$n^{\text{th }}$ orbital state in hydrogen atom is $n$$n$nn$n$ times the de Broglie wavelength associated with it.

53. (a) Using de Broglie’s hypothesis, explain with the help of a suitable diagram, Bohr’s second postulate of quantization of energy levels in a hydrogen atom.

19. Detailed Solutions

1. According to electromagnetic theory, electron revolving around the nucleus are continuously accelerated. Since an accelerated charge emits energy, the radius of the circular path of a revolving electron should go on decreasing and ultimately it should fall into the nucleus. So, it could not explain the structure of the atom. As matter is stable, we cannot expect the atoms to collapse.
2. An electron revolving in an orbit of $\mathrm{H}$$\mathrm{H}$H\mathrm{H}$\mathrm{H}$-atom, has both kinetic energy and electrostatic potential energy. Kinetic energy of the electron revolving in a circular orbit of radius $r$$r$rr$r$ is $E_K=\frac{1}{2}m{v}^2$$E_K=\frac{1}{2}m{v}^2$E_(K)=(1)/(2)mv^(2)E_{K}=\frac{1}{2} m v^{2}$E_K=\frac{1}{2}m{v}^2$

5. Assumptions of Rutherford’s atomic model :

7. Wavelength $(\lambda )$$(\lambda )$(lambda)(\lambda)$(\lambda )$ of Balmer series is given by

8. For Lyman series,

9. Since $r\propto n^2$$r\propto n^2$r propn^(2)r \propto n^{2}$r\propto n^2$.

10. Ionisation energy for an atom is defined as the energy required to remove an electron completely from the outermost shell of the atom.

11. Quantum condition : The stationary orbits are those in which angular momentum of electron is an integral multiple of $\frac{h}{2\pi }$$\frac{h}{2\pi }$(h)/(2pi)\frac{h}{2 \pi}$\frac{h}{2\pi }$ i.e.,

12. Frequency condition : An atom can emit or absorb radiation in the form of discrete energy photons only when an electron jumps from a higher to a lower orbit or from a lower to a higher orbit, respectively.

14. Radius of $n^{\text{th}}$$n^{\text{th }}$n^(“th “)n^{\text {th }}$n^{\text{th }}$ orbit of hydrogen atom : In $H$$H$HH$H$-atom, an electron having charge $-e$$-e$-e-e$-e$ revolves around the nucleus of charge $+e$$+e$+e+e$+e$ in a circular orbit of radius $r$$r$rr$r$, such that necessary centripetal force is provided by the electrostatic force of attraction between the electron and nucleus.

15. (i) Since, $r\propto n^2;\frac{r_n}{r_g}=\frac{n^2}{1^2}$$r\propto n^2;\frac{r_n}{r_g}=\frac{n^2}{1^2}$r propn^(2);(r_(n))/(r_(g))=(n^(2))/(1^(2))r \propto n^{2} ; \frac{r_{n}}{r_{g}}=\frac{n^{2}}{1^{2}}$r\propto n^2;\frac{r_n}{r_g}=\frac{n^2}{1^2}$

16. (a) Radius of $n^{\text{th}}$$n^{\text{th }}$n^(“th “)n^{\text {th }}$n^{\text{th }}$ orbit, $r_n\propto n^2$$r_n\propto n^2$r_(n)propn^(2)r_{n} \propto n^{2}$r_n\propto n^2$

18. According to Bohr’s postulates for hydrogen atom, electron revolves in a circular orbit around the heavy positively charged nucleus. These are the stationary (orbits) states of the atom.

20. Suppose $m$$m$mm$m$ be the mass of an electron and $v$$v$vv$v$ be its speed in $n^{\text{th}}$$n^{\text{th }}$n^(“th “)n^{\text {th }}$n^{\text{th }}$ orbit of radius $r$$r$rr$r$. The centripetal force for revolution is produced by electrostatic attraction between electron and nucleus.

21. (a) Speed of the electron in the $n^{\text{th}}$$n^{\text{th }}$n^(“th “)n^{\text {th }}$n^{\text{th }}$ orbit : The centripetal force required for revolution is provided by the electrostatic force of attraction between the electron and the nucleus.

22. Bohr’s postulates of atomic model : Bohr introduced three postulates and laid the foundations of quantum mechanics.

23. Refer to answer 19.
24. (a) Limitation of Rutherford’s model:

25. Refer to answer $17(i)$$17(i)$17(i)17(i)$17(i)$.
26. (a) Refer to answer $17(i)$$17(i)$17(i)17(i)$17(i)$

27. Refer to answer 17(i).
28. For element $D$$D$DD$D$

29. $H_{\alpha }$$H_{\alpha }$H_(alpha)H_{\alpha}$H_{\alpha }$ line of the Balmer series in the emission spectrum of hydrogen atoms obtained when the transition occurs from $n=3$$n=3$n=3n=3$n=3$ to $n=2$$n=2$n=2n=2$n=2$ state.
30. Number of spectral lines obtained due to transition of electron from $n=4(3^{\text{rd}}$$n=4\left(3^{\text{rd }}\right.$n=4(3^(“rd “):}n=4\left(3^{\text {rd }}\right.$n=4(3^{\text{rd }}$ exited state) to $n=1$$n=1$n=1n=1$n=1$ (ground state) is

31. The minimum energy, required to free the electron from the ground state of the hydrogen atom, is known as ionization energy of that atom.

32. Number of spectral lines obtained due to transition of electron from $n=4(3^{\text{rd}}$$n=4\left(3^{\text{rd }}\right.$n=4(3^(“rd “):}n=4\left(3^{\text {rd }}\right.$n=4(3^{\text{rd }}$ exited state) to $n=1$$n=1$n=1n=1$n=1$ (ground state) is

33. (a) $\lambda =496\mathrm{n}\mathrm{m}=496\times {10}^{-9}\mathrm{m}$$\lambda =496\mathrm{n}\mathrm{m}=496\times {10}^{-9}\mathrm{m}$lambda=496nm=496 xx10^(-9)m\lambda=496 \mathrm{~nm}=496 \times 10^{-9} \mathrm{~m}$\lambda =496\mathrm{n}\mathrm{m}=496\times {10}^{-9}\mathrm{m}$

34. (i) An electron undergoes transition from 2nd excited state to the first excited state is Balmer series and then to the ground state is Lyman series.

35. For longest wavelength of Lyman series $n_i=2$$n_i=2$n_(i)=2n_{i}=2$n_i=2$

36. Here, $\mathrm{\Delta }E=12.5\mathrm{e}\mathrm{V}$$\mathrm{\Delta }E=12.5\mathrm{e}\mathrm{V}$Delta E=12.5eV\Delta E=12.5 \mathrm{eV}$\mathrm{\Delta }E=12.5\mathrm{e}\mathrm{V}$

37. (i) $\because E_n=\frac{-13.6}{n^2}\mathrm{e}\mathrm{V}$$\because E_n=\frac{-13.6}{n^2}\mathrm{e}\mathrm{V}$:’E_(n)=(-13.6)/(n^(2))eV\because E_{n}=\frac{-13.6}{n^{2}} \mathrm{eV}$\because E_n=\frac{-13.6}{n^2}\mathrm{e}\mathrm{V}$

38. Refer to answer 19.
39. Refer to answer 32.
40. $hv=\frac{hc}{\lambda }=(E_2-E_1)$$hv=\frac{hc}{\lambda }=\left(E_2-E_1\right)$hv=(hc)/(lambda)=(E_(2)-E_(1))h v=\frac{h c}{\lambda}=\left(E_{2}-E_{1}\right)$hv=\frac{hc}{\lambda }=(E_2-E_1)$ or $\lambda =\frac{hc}{(E_2-E_1)}$$\lambda =\frac{hc}{\left(E_2-E_1\right)}$lambda=(hc)/((E_(2)-E_(1)))\lambda=\frac{h c}{\left(E_{2}-E_{1}\right)}$\lambda =\frac{hc}{(E_2-E_1)}$

41. (i) The kinetic energy $\left(E_k\right)$$\left(E_k\right)$(E_(k))\left(E_{k}\right)$\left(E_k\right)$ of the electron in an orbit is equal to negative of its total energy $(E)$$(E)$(E)(E)$(E)$

42. Refer to answer 35.
43. $\because \frac{m{v}^2}{r_n}=\frac{1}{4\pi {\epsilon }_0}\frac{e^2}{r_n^2}$$\because \frac{m{v}^2}{r_n}=\frac{1}{4\pi {\epsilon }_0}\frac{e^2}{r_n^2}$:'(mv^(2))/(r_(n))=(1)/(4piepsi_(0))(e^(2))/(r_(n)^(2))\because \frac{m v^{2}}{r_{n}}=\frac{1}{4 \pi \varepsilon_{0}} \frac{e^{2}}{r_{n}^{2}}$\because \frac{m{v}^2}{r_n}=\frac{1}{4\pi {\epsilon }_0}\frac{e^2}{r_n^2}$

44. Refer to answer 35.
45. Refer to answer 35.
46. de-Broglie hypothesis : It states that a moving particle sometimes acts as a wave and sometimes as a particle or a wave is associated with moving particle which controls the particle in every respect. The wave associated with moving particle is called matter wave or de-Broglie wave whose wavelength is given by

47. Kinetic energy of the electron in the second state of hydrogen atom

48. According to Bohr’s postulates,

49. (i) Bohr’s quantization condition : The electron revolves around the nucleus only in those orbits for which the angular momentum is an integral multiple of $h/2\pi$$h/2\pi$h//2pih / 2 \pi$h/2\pi$.

50. Kinetic energy in the first excited state of hydrogen atom

51. $v=2.2\times {10}^8\mathrm{m}/\mathrm{s}$$v=2.2\times {10}^8\mathrm{m}/\mathrm{s}$v=2.2 xx10^(8)m//sv=2.2 \times 10^{8} \mathrm{~m} / \mathrm{s}$v=2.2\times {10}^8\mathrm{m}/\mathrm{s}$

52. According to Bohr’s second postulate quantization of angular momentum

53. (a) According to de-Broglie, a stationary orbit is that which contains an integral number of de-Broglie waves associated with the revolving electron.