Atoms

1. Rutherford's Nuclear Model

Based on Alpha-particle scattering experiment (Geiger-Marsden).

Conclusions:
1. Most space in atom is empty (Most α-particles passed undeviated).
2. Positive charge and mass concentrated in a tiny central core called Nucleus.
3. Size of nucleus (10^-15 m) << Size of atom (10^-10 m).
Impact Parameter (b): ⊥ distance of velocity vector from nucleus center. Small b → Large scattering angle θ.

(Asked in NEET 2016, 2019)

2. Bohr's Model of Hydrogen Atom

1. Electrons revolve in stable orbits without radiating energy.
2. Quantization Condition: Angular momentum L = mvr = nh/2π (n = 1, 2, 3...).
3. Energy emitted/absorbed during transition: hν = E_f - E_i.

Key Formulas

Radius: r_n = 0.529 n²/Z Å.
Velocity: v_n = 2.18 × 10⁶ Z/n m/s.
Total Energy: E_n = -13.6 Z²/n² eV.
K = |E| and U = 2E = -2K.

(Asked in NEET 2017, 2018, 2020)

3. Hydrogen Spectrum

Emission spectrum of atomic hydrogen consists of discrete lines.

Rydberg Formula: 1/λ = R Z² (1/n₁² - 1/n₂²)
R = 1.097 × 10⁷ m^-1.
Series:
- Balmer (Visible): n₁=2.
- First line (Max λ): Transition 3 → 2.
- Series Limit (Min λ): Transition ∞ → 2.

(Asked in NEET 2016, 2019, 2021, 2023)

Numericals: Atoms

Q1. Calculate wavelength of H-alpha line (Balmer series) for Hydrogen. (R = 1.097 × 10⁷ m^-1)

Solution:
H-alpha is transition n=3 → n=2.
1/λ = R (1/2² - 1/3²) = R (1/4 - 1/9) = R (5/36).
λ = 36 / 5R = 36 / (5 × 1.097 × 10⁷).
λ ≈ 6563 Å (Red).

Q2. Find shortest wavelength in Lyman Series.

Solution:
Shortest λ → Max Energy → Transition ∞ → 1.
1/λ = R (1/1² - 1/∞) = R.
λ = 1/R = 1/1.097 × 10^-7 m.
λ ≈ 912 Å.

Q3. Find ratio of radii of 2nd orbit of He⁺ and 3rd orbit of Li²⁺.

Solution:
r ∝ n²/Z.
For He⁺: n=2, Z=2. r₁ ∝ 2²/2 = 2.
For Li²⁺: n=3, Z=3. r₂ ∝ 3²/3 = 3.
Ratio r₁ : r₂ = 2:3.

Q4. Velocity in 1st orbit of H-atom is v. What is velocity in 2nd orbit?

Solution:
v ∝ Z/n.
v₁ ∝ 1/1.
v₂ ∝ 1/2.
v₂ = v/2.

Q5. Ionization energy of H is 13.6 eV. Find Ionization energy of Li²⁺.

Solution:
I.E. ∝ Z².
For Li²⁺, Z=3.
I.E. = 13.6 × (3)² = 13.6 × 9.
I.E. = 122.4 eV.

Q6. Electron in H-atom jumps from n=4 to ground state n=1. How many spectral lines are possible?

Solution:
N = n(n-1)/2.
Change n = 4-1 = 3? No, n here is the higher orbit number.
Possible transitions: 4→3, 4→2, 4→1, 3→2, 3→1, 2→1.
Using formula: 4(3)/2 = 6 lines.

Q7. Excitation energy of 1st excited state of Hydrogen?

Solution:
Ground state E₁ = -13.6 eV.
1st Excited state (n=2) E₂ = -3.4 eV.
Excitation Energy = E₂ - E₁ = -3.4 - (-13.6).
ΔE = 10.2 eV.

Q8. Change in angular momentum when electron jumps from 4th to 2nd orbit.

Solution:
L = nh/2π.
Change ΔL = (4h/2π) - (2h/2π) = 2h/2π = h/π.
ΔL = h/π.

Q9. Scattering angle for alpha particle is 180°. What is impact parameter?

Solution:
For head-on collision, the particle rebounds (180°).
This implies the particle was aimed directly at nucleus center.
Impact parameter b = 0.

Q10. Alpha particle of 5 MeV fired at Uranium (Z=92). Find distance of closest approach.

Solution:
K.E. = P.E. at closest approach.
K = k(2e)(Ze) / r₀.
r₀ = 2kZe² / K.
r₀ = 2 × (9×10⁹) × 92 × (1.6×10^-19)² / (5 × 1.6 × 10^-13).
≈ 5.3 × 10^-14 m.

Important Formulae

1. Bohr Model Parameters

Radius of Orbit:

r_n = 0.529 (n²/Z) Å

Velocity of Electron:

v_n = 2.18 × 10⁶ (Z/n) m/s

Total Energy:

E_n = -13.6 (Z²/n²) eV

2. Spectral Lines

Rydberg Formula:

1/λ = R Z² (1/n₁² - 1/n₂²)

Max Wavelength (First Line):

n₂ = n₁ + 1

Min Wavelength (Series Limit):

n₂ = ∞

3. Other Relations

Distance of Closest Approach:

r₀ = 2kZe² / K_α

Number of Spectral Lines:

N = n(n - 1) / 2

20 NEET Golden Facts

1. Nucleus Density: Independent of mass number A. Extremely high (~10¹⁷ kg/m³).
2. Hydrogen Energies: E₁ = -13.6 eV, E₂ = -3.4 eV, E₃ = -1.51 eV, E₄ = -0.85 eV.
3. Ionization Energy: Energy required to remove electron from ground state to ∞. For H, it is +13.6 eV.
4. Kinetic Energy: K = -E = 13.6 Z²/n² eV.
5. Potential Energy: U = 2E = -27.2 Z²/n² eV.
6. Time Period: T ∝ n³/Z².
7. Frequency: f ∝ Z²/n³.
8. Lyman Series: UV Region. Transitions to n=1.
9. Balmer Series: Visible Region. Transitions to n=2.
10. Paschen, Brackett, Pfund: Infrared Region. Transitions to n=3, 4, 5.
11. Angular Momentum: Quantized. L = nh/2π. Only discrete values allowed.
12. Shortest Wavelength: Corresponds to max energy transition (∞ → n).
13. Alpha Scattering (N): N(θ) ∝ 1/sin⁴(θ/2).
14. Impact Parameter: For head-on collision, b=0, θ=180°.
15. Fine Structure: Bohr model fails to explain fine structure (splitting) of spectral lines.
16. Magnetic Moment: M = n (eh/4πm) = n μ_B (Bohr Magneton).
17. Recoil of Atom: When photon emitted, atom recoils (Conservation of Momentum).
18. 13.6 eV: This value comes from mass and charge of electron and Planck constant.
19. Transition Limits: H-alpha line (3→2) is Red. H-beta (4→2) is Blue-Green.
20. Zeeman Effect: Splitting of lines in B-field (Not explained by Bohr).

📱 Practice MCQs for this topic inside our App