Atoms Class 12 Notes (Modern Physics)

• Most alpha particles pass straight through — most of the atom is empty.
• A few are deflected at large angles or bounce back — a small massive positive nucleus.
• Nuclear size ~10⁻¹⁵ m, atomic size ~10⁻¹⁰ m.

• Electrons occupy only stationary orbits of quantised angular momentum:
• $mvr=\frac{nh}{2\pi }$
• .
• Radius of nth orbit:
• $r_n=n^2\times 0.529\text{A˚}$
• .
• Energy of nth level:
• $E_n=-\frac{13.6}{n^2}\text{eV}$
• (ground state
• $n=1$
• is −13.6 eV).
• Photon absorbed/emitted on a transition between levels
• $n_i\to n_f$
• :
• $\Delta E=E_i-E_f=h\nu$
• .

• $\text{Lyman series}:$
• transitions to
• $n_f=1$
• (ultraviolet).
• $\text{Balmer series}:$
• transitions to
• $n_f=2$
• (visible).
• $\text{Paschen, Brackett, Pfund}:$
• transitions to
• $n_f=3,4,5$
• (infrared).

• Kinetic energy of electron:
• $KE=\frac{1}{2}m{v}^2=\frac{e^2}{8\pi {\epsilon }_0{r}_n}$
• .
• Potential energy:
• $PE=-\frac{e^2}{4\pi {\epsilon }_0{r}_n}$
• ; total
• $E=\text{KE}+\text{PE}=-\text{KE}$
• .
• Ionisation energy of hydrogen from the ground state is 13.6 eV; ionisation potential is 13.6 V.

An electron acts as a wave whose wavelength is set by its momentum. For an electron accelerated through a potential difference V: $\lambda =\frac{1.23}{\sqrt{V}}\text{nm}$. The allowed Bohr orbits correspond to integer numbers of wavelengths fitting around the orbit: $2\pi r=n\lambda$.

Example: Find the wavelength of the second line of the Balmer series (transition n = 4 → n = 2) using R_H = 1.097 × 10⁷ m⁻¹.

Solution: $\frac{1}{\lambda }=1.097\times 10^7\left(\frac{1}{4}-\frac{1}{16}\right)=1.097\times 10^7\times \frac{3}{16}$ giving $\lambda \approx 486\text{nm}$ (Balmer H-β).

Exercises (NCERT Q12.1–Q12.9)

• 12.1) Choose the correct alternative from the clues given at the end of the each statement: (a) The size of the atom in Thomson's model is .......... the atomic size in Rutherford's model. (much greater than/no different from/much less than.) (b) In the ground state of .......... electrons are in stable equilibrium, while in .......... electrons always experience a net force. (Thomson's model/ Rutherford's model.) (c) A classical atom based on .......... is doomed to collapse. (Thomson's model/ Rutherford's model.) (d) An atom has a nearly continuous mass distribution in a .......... but has a highly non-uniform mass distribution in .......... (Thomson's model/ Rutherford's model.) (e) The positively charged part of the atom possesses most of the mass in .......... (Rutherford's model/both the models.)
• 12.2) Suppose you are given a chance to repeat the alpha-particle scattering experiment using a thin sheet of solid hydrogen in place of the gold foil. (Hydrogen is a solid at temperatures below 14 K.) What results do you expect?
• 12.3) A difference of 2.3 eV separates two energy levels in an atom. What is the frequency of radiation emitted when the atom make a transition from the upper level to the lower level?
• 12.4) The ground state energy of hydrogen atom is −13.6 eV. What are the kinetic and potential energies of the electron in this state?
• 12.5) A hydrogen atom initially in the ground level absorbs a photon, which excites it to the n = 4 level. Determine the wavelength and frequency of photon.
• 12.6) (a) Using the Bohr's model calculate the speed of the electron in a hydrogen atom in the n = 1, 2, and 3 levels. (b) Calculate the orbital period in each of these levels.
• 12.7) The radius of the innermost electron orbit of a hydrogen atom is 5.3 × 10⁻¹¹ m. What are the radii of the n = 2 and n = 3 orbits?
• 12.8) A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. What series of wavelengths will be emitted?
• 12.9) In accordance with the Bohr's model, find the quantum number that characterises the earth's revolution around the sun in an orbit of radius 1.5 × 10¹¹ m with orbital speed 3 × 10⁴ m/s. (Mass of earth = 6.0 × 10²⁴ kg.)

Answers at a glance

1. 1
   Q12.1 → (a) no different from. (b) Thomson's model / Rutherford's model. (c) Rutherford's model. (d) Thomson's model / Rutherford's model. (e) both the models.
2. 2
   Q12.2 → A solid hydrogen target scatters α-particles far less than gold: most pass straight through, deflections are small and large-angle scattering is almost entirely absent.
3. 3
   Q12.3 → ν = 5.55 × 10¹⁴ Hz.
4. 4
   Q12.4 → K = +13.6 eV (2.18 × 10⁻¹⁸ J); U = −27.2 eV.
5. 5
   Q12.5 → ν = 3.08 × 10¹⁵ Hz; λ = 9.75 × 10⁻⁸ m (≈ 97 nm).
6. 6
   Q12.6 → (a) v = 2.18 × 10⁶, 1.09 × 10⁶, 7.27 × 10⁵ m/s for n = 1, 2, 3. (b) T = 1.53 × 10⁻¹⁶, 1.22 × 10⁻¹⁵, 4.12 × 10⁻¹⁵ s.
7. 7
   Q12.7 → r₂ = 2.12 × 10⁻¹⁰ m; r₃ = 4.77 × 10⁻¹⁰ m.
8. 8
   Q12.8 → Three lines are emitted: 102.6 nm and 121.6 nm (Lyman) and 656.3 nm (Balmer Hα).
9. 9
   Q12.9 → n = 2.56 × 10⁷⁴.

MCQ Questions

1 Mark Questions

1. 1
   State one drawback of Rutherford's model of the atom.
2. 2
   State the quantization condition for the angular momentum of the electron in Bohr's model.
3. 3
   Write the relation between the radius of the nth Bohr orbit and n.
4. 4
   Which series of the hydrogen spectrum lies in the ultraviolet region?

2 Mark Questions

1. 1
   Calculate the radius of the second Bohr orbit of hydrogen. (Ans: r₂ = 4 × 0.53 = 2.12 Å)
2. 2
   Find the ionization energy of the hydrogen atom in eV. (Ans: 13.6 eV)
3. 3
   Calculate the wavelength of the first line of the Balmer series of hydrogen. (Ans: from 1/λ = R(1/4 − 1/9), λ ≈ 656 nm)

3 Mark Questions

1. 1
   Using Bohr's postulates, derive expressions for the radius and the energy of the nth orbit of the hydrogen atom. (Ans: rₙ ∝ n², Eₙ = −13.6/n² eV)
2. 2
   Explain the Lyman, Balmer and Paschen series of the hydrogen spectrum using the Rydberg formula 1/λ = R(1/n₁² − 1/n₂²).
3. 3
   Draw the energy level diagram of the hydrogen atom for the first five levels and mark the ionization limit. (Ans: E₁ = −13.6, E₂ = −3.4, E₃ = −1.51 eV; ionization at 0 eV)

5 Mark Questions

1. 1
   Describe Rutherford's α-particle scattering experiment. State the observations, the conclusions drawn, and the relation of the impact parameter to the scattering angle.
2. 2
   State Bohr's postulates for the hydrogen atom. Derive the expressions for the radius and total energy of the nth orbit and hence obtain the spectral series of hydrogen.