Class 12 Physics Notes Chapter 12 (Atoms) – Examplar Problems (English) Book

Detailed Notes with MCQs of Chapter 12, 'Atoms'. This is a crucial chapter, bridging classical and quantum ideas, and frequently appears in various government examinations. We'll cover the key models and concepts step-by-step.

Chapter 12: Atoms - Detailed Notes for Exam Preparation

1. Early Models & Rutherford's Nuclear Model

• Thomson's Model (Plum Pudding Model): Proposed atom as a sphere of positive charge with electrons embedded in it. Failed to explain large-angle scattering in Rutherford's experiment.
• Geiger-Marsden Experiment (Rutherford's Alpha Scattering Experiment):
  • Setup: Beam of alpha particles (He²⁺ nuclei) directed at a thin gold foil. Scattered particles detected by a zinc sulphide screen.
  • Observations:
    1. Most alpha particles passed straight through (atom is mostly empty space).
    2. Some deflected by small angles (positive charge concentrated in a small region).
    3. Very few (1 in 8000) deflected by > 90°, some even ~180° (strong repulsive force from a tiny, massive, positive core - the nucleus).
  • Impact Parameter (b): Perpendicular distance of the initial velocity vector of the alpha particle from the centre of the nucleus. Scattering angle (θ) decreases as 'b' increases. For head-on collision (b=0), θ = 180°.
  • Distance of Closest Approach (r₀): For head-on collision, the kinetic energy of the alpha particle is converted entirely into electrostatic potential energy at the point of closest approach.
    K.E. = P.E. => ½ mv² = (1 / 4πε₀) * (Ze)(2e) / r₀
    This gives an estimate of the upper limit of the nuclear size (~10⁻¹⁴ m to 10⁻¹⁵ m).
• Rutherford's Nuclear Model:
  • Atom consists of a small, dense, positively charged nucleus at the centre, containing most of the mass.
  • Electrons revolve around the nucleus in orbits, like planets around the sun.
  • Electrostatic force provides the necessary centripetal force.
• Limitations of Rutherford's Model:
  1. Stability: According to classical electromagnetism, accelerating electrons (in circular orbits) must radiate energy continuously. This would cause the electron's orbit to spiral inwards, eventually collapsing into the nucleus. Atoms wouldn't be stable.
  2. Discrete Spectra: If electrons spiral inwards, they should emit radiation of continuously changing frequency (continuous spectrum). However, atoms emit line spectra (discrete frequencies).

2. Bohr's Model for Hydrogen Atom (and Hydrogen-like ions)

• Bohr combined classical and early quantum concepts to explain the stability and spectrum of hydrogen.

• Bohr's Postulates:

  1. Stationary Orbits: Electrons revolve around the nucleus only in certain specific, non-radiating orbits called stationary orbits. The electrostatic force provides the centripetal force.
     mv²/r = (1 / 4πε₀) * Ze²/r²
  2. Quantization of Angular Momentum: The angular momentum (L = mvr) of an electron in a stationary orbit is an integral multiple of h/2π (where h is Planck's constant).
     L = mvr = n(h/2π), where n = 1, 2, 3, ... (principal quantum number).
  3. Frequency Condition: An atom emits or absorbs radiation only when an electron jumps from one allowed orbit to another. The frequency (ν) of the emitted/absorbed photon is given by:
     hν = |E_f - E_i|, where E_i and E_f are the energies of the initial and final states.

• Derivations (Results are important):

  • Radius of nth orbit (rₙ):
    rₙ = (n²h²ε₀) / (πmZe²)
    rₙ ∝ n²/Z
    For Hydrogen (Z=1), radius of the first orbit (Bohr radius, a₀) = 0.529 Å = 0.0529 nm.
    rₙ = a₀ n²/Z
  • Velocity of electron in nth orbit (vₙ):
    vₙ = (Ze²) / (2nhε₀)
    vₙ ∝ Z/n
  • Energy of electron in nth orbit (Eₙ):
    Total Energy Eₙ = K.E + P.E = (½ mvₙ²) + (- (1 / 4πε₀) * Ze²/rₙ)
    Eₙ = - (mZ²e⁴) / (8n²h²ε₀²)
    Eₙ ∝ -Z²/n²
    Eₙ = -13.6 Z²/n² eV (electron volts)
    • Negative sign indicates the electron is bound to the nucleus.
    • Energy is quantized.
    • Ground State (n=1): Lowest energy state (-13.6 eV for H).
    • Excited States (n=2, 3, ...): Higher energy states.
    • Ionization Energy: Energy required to remove an electron from the ground state (n=1) to infinity (n=∞). For H, Ionization Energy = E∞ - E₁ = 0 - (-13.6) = +13.6 eV.
    • Excitation Energy: Energy required to move an electron from the ground state to an excited state. (e.g., 1st excitation energy for H = E₂ - E₁ = -3.4 - (-13.6) = 10.2 eV).

3. Atomic Spectra - Hydrogen Spectrum

• When an electron jumps from a higher energy orbit (nᵢ) to a lower energy orbit (n<0xE1><0xB5><0x82>), a photon of energy hν = Eᵢ - E<0xE1><0xB5><0x82> is emitted.
  hν = hc/λ = (-13.6 Z²/nᵢ²) - (-13.6 Z²/n<0xE1><0xB5><0x82>²) eV
• Rydberg Formula: Gives the wavenumber (1/λ) of the emitted spectral line:
  1/λ = R Z² (1/n<0xE1><0xB5><0x82>² - 1/nᵢ²)
  where R = (me⁴) / (8ε₀²ch³) ≈ 1.097 × 10⁷ m⁻¹ is the Rydberg constant.
• Spectral Series (Important for Exams):
  • Lyman Series: (Ultraviolet region)
    Electron jumps to n<0xE1><0xB5><0x82> = 1 from nᵢ = 2, 3, 4, ...
    1/λ = R Z² (1/1² - 1/nᵢ²)
  • Balmer Series: (Visible region)
    Electron jumps to n<0xE1><0xB5><0x82> = 2 from nᵢ = 3, 4, 5, ...
    1/λ = R Z² (1/2² - 1/nᵢ²)
    (Hα line: nᵢ=3, Hβ line: nᵢ=4, etc.)
  • Paschen Series: (Infrared region)
    Electron jumps to n<0xE1><0xB5><0x82> = 3 from nᵢ = 4, 5, 6, ...
    1/λ = R Z² (1/3² - 1/nᵢ²)
  • Brackett Series: (Infrared region)
    Electron jumps to n<0xE1><0xB5><0x82> = 4 from nᵢ = 5, 6, 7, ...
    1/λ = R Z² (1/4² - 1/nᵢ²)
  • Pfund Series: (Far Infrared region)
    Electron jumps to n<0xE1><0xB5><0x82> = 5 from nᵢ = 6, 7, 8, ...
    1/λ = R Z² (1/5² - 1/nᵢ²)
• Series Limit: Shortest wavelength (highest energy) line in a series, corresponding to transition from nᵢ = ∞.
• First Line: Longest wavelength (lowest energy) line in a series, corresponding to transition from nᵢ = n<0xE1><0xB5><0x82> + 1.

4. Energy Level Diagram

• A diagram representing the discrete energy levels (horizontal lines) of an atom.
• The lowest line represents the ground state (n=1). Higher lines represent excited states (n=2, 3,...).
• The energy difference between levels decreases as n increases.
• Arrows indicate electronic transitions, corresponding to emission or absorption of photons.

5. de Broglie's Explanation of Bohr's Second Postulate

• Louis de Broglie proposed that electrons behave as waves (matter waves) with wavelength λ = h/p = h/mv.
• He suggested that an electron can exist in an orbit only if the circumference of the orbit is an integral multiple of the electron's wavelength (forming a stationary wave).
  2πrₙ = nλ = n(h/mvₙ)
• Rearranging gives: mvₙrₙ = n(h/2π), which is Bohr's quantization condition for angular momentum.

6. Limitations of Bohr's Model

• Applicable only to hydrogen and hydrogen-like ions (single electron systems like He⁺, Li²⁺). Fails for multi-electron atoms.
• Could not explain the fine structure of spectral lines (splitting of lines into closely spaced lines).
• Could not explain the relative intensities of spectral lines.
• Could not explain the Zeeman effect (splitting of spectral lines in a magnetic field) and Stark effect (splitting in an electric field).
• Could not explain how atoms combine to form molecules.

Key Formulas Summary:

• Distance of Closest Approach: r₀ = (1 / 4πε₀) * (2Ze²) / K.E.
• Angular Momentum: L = n(h/2π)
• Radius: rₙ = a₀ n²/Z (a₀ = 0.529 Å)
• Velocity: vₙ ∝ Z/n
• Energy: Eₙ = -13.6 Z²/n² eV
• Rydberg Formula: 1/λ = R Z² (1/n<0xE1><0xB5><0x82>² - 1/nᵢ²)

Multiple Choice Questions (MCQs)

1. In the Geiger-Marsden experiment, the reason most alpha particles pass undeflected through a thin gold foil is:
   a) The nucleus is positively charged.
   b) The atom is mostly empty space.
   c) Alpha particles are very heavy.
   d) Electrons repel the alpha particles.

2. According to Bohr's model, the angular momentum of an electron in the 3rd orbit of a hydrogen atom is:
   a) h/2π
   b) 3h/π
   c) 3h/2π
   d) 9h/2π

3. The energy of an electron in the ground state of a hydrogen atom is -13.6 eV. The energy of the electron in the first excited state (n=2) is:
   a) -3.4 eV
   b) -6.8 eV
   c) -1.51 eV
   d) -13.6 eV

4. Which spectral series of the hydrogen atom lies in the visible region of the electromagnetic spectrum?
   a) Lyman series
   b) Balmer series
   c) Paschen series
   d) Pfund series

5. The ratio of the radii of the first three Bohr orbits for a hydrogen atom is:
   a) 1 : 2 : 3
   b) 1 : 4 : 9
   c) 1 : 8 : 27
   d) 3 : 2 : 1

6. The shortest wavelength in the Lyman series for hydrogen (R = Rydberg constant) is:
   a) 1/R
   b) R
   c) 4/3R
   d) 3/4R

7. Bohr's model failed to explain:
   a) The spectrum of hydrogen atom.
   b) The stability of atoms.
   c) The spectrum of multi-electron atoms.
   d) The quantization of energy levels.

8. The ionization potential of a hydrogen atom is 13.6 V. The energy required to remove an electron from the n=2 state is:
   a) 13.6 eV
   b) 10.2 eV
   c) 6.8 eV
   d) 3.4 eV

9. De Broglie explained Bohr's second postulate by assuming that:
   a) Electrons radiate energy in orbits.
   b) Electrons are point particles.
   c) The electron wave forms a standing wave around the orbit.
   d) The nucleus exerts a quantized force on the electron.

10. If the radius of the first Bohr orbit (n=1) is 'a₀', the radius of the third Bohr orbit (n=3) for a Li²⁺ ion (Z=3) is:
    a) 3 a₀
    b) a₀ / 3
    c) 9 a₀
    d) 3 a₀

Answers to MCQs:

1. b) The atom is mostly empty space.
2. c) 3h/2π (L = nh/2π, n=3)
3. a) -3.4 eV (Eₙ = -13.6/n² eV, n=2)
4. b) Balmer series
5. b) 1 : 4 : 9 (rₙ ∝ n²)
6. a) 1/R (For shortest wavelength, nᵢ = ∞, n<0xE1><0xB5><0x82> = 1. 1/λ = R(1/1² - 1/∞²) = R => λ = 1/R)
7. c) The spectrum of multi-electron atoms.
8. d) 3.4 eV (Energy in n=2 state is -13.6/2² = -3.4 eV. Ionization energy from n=2 is E∞ - E₂ = 0 - (-3.4) = +3.4 eV)
9. c) The electron wave forms a standing wave around the orbit.
10. d) 3 a₀ (rₙ = a₀ n²/Z. For n=3, Z=3, r₃ = a₀ * 3² / 3 = a₀ * 9 / 3 = 3 a₀)

Study these notes thoroughly, focusing on the postulates, formulas, spectral series, and limitations. Practice applying the formulas to different scenarios, especially for hydrogen and hydrogen-like ions. Good luck with your preparation!