Atoms and Nuclei Summary

Atomic Models, Nuclear Physics, Radioactivity, and Nuclear Reactions

Atomic Models
Bohr's Theory
Hydrogen Spectrum
Nuclear Physics
Radioactivity

1. Atomic Models

1.1 Rutherford Gold-Foil Experiment

Key Findings:

  • Most alpha particles passed through, indicating mostly empty space
  • Some alpha particles deflected at large angles, indicating concentrated positive charge
  • Very few alpha particles bounced back, indicating very small, dense nucleus

Number of Particles Scattered

\[ N \propto \sin^{-4} \left( \frac{\theta}{2} \right) \]

Distance of Closest Approach

\[ d = \frac{Z e^2}{\pi \epsilon_0 m v_i^2} \]

Distance of closest approach is of the order of \(10^{-14}\) m. Nuclear size is measured in Fermi (\(1 \, \text{Fermi} = 10^{-15}\) m).

Impact Parameter

\[ b = \frac{Ze^2 \cot \frac{\theta}{2}}{4\pi\epsilon_0 \left( \frac{1}{2} mv_i^2 \right)} \]

Where θ is the angle of scattering. For head-on collision, impact parameter is zero.

1.2 Rutherford Model of the Atom

Postulates:

  • Whole positive charge and almost entire mass concentrated in a small nucleus
  • Electrons revolve around nucleus in circular orbits
  • Atom as a whole is neutral

2. Bohr's Atomic Model

Postulates:

  • Electrons move in fixed circular orbits without radiating energy
  • Only certain orbits are permitted where angular momentum is quantized
  • Energy is radiated or absorbed only when electrons jump between orbits

2.1 Mathematical Formulation

Centripetal Force Equation
\[ \frac{1}{4\pi\epsilon_0} \frac{(Ze)(e)}{r^2} = \frac{mv^2}{r} \]
Angular Momentum Quantization
\[ mvr = n \cdot \frac{h}{2\pi} \]
Bohr's Frequency Condition
\[ E_2 - E_1 = hv \]

2.2 Bohr's Theory of Hydrogen Atom

Radius of Orbit

\[ r_n = \frac{\epsilon_0 n^2 h^2}{\pi m e^2 Z} \]

For hydrogen atom (Z=1): \( r_n \propto n^2 \)

Radii of various orbits: \( 1 : 4 : 9 : 16 \ldots \)

First orbit radius: \( r_1 = 5.3 \times 10^{-11} \, \text{m} \)

Velocity of Electron

\[ v = \frac{Ze^2}{2\varepsilon_0 n h} \]

For hydrogen atom: \( v = \alpha \frac{c}{n} \), where \( \alpha = \frac{1}{137} \) (fine structure constant)

Maximum velocity in hydrogen atom: \( 2.19 \times 10^6 \, \text{ms}^{-1} \)

Orbital Frequency

\[ v = \frac{\text{me}^4}{4\varepsilon_0^2 n^3 h^3} \]

For innermost orbit: \( v = 65.8 \times 10^{14} \, \text{Hz} \)

2.3 Energy of Electron

Kinetic Energy

\[ \text{K.E.} = \frac{\text{me}^4}{8\varepsilon_0^2 n^2 h^2} = \frac{e^2}{8\pi\varepsilon_0 r} \]

Potential Energy

\[ \text{P.E.} = \frac{-Ze^2}{4\pi\varepsilon_0 r} = \frac{-\text{me}^4}{8\varepsilon_0^2 n^2 h^2} \]

Potential energy is numerically twice the K.E.

Total Energy

\[ \text{T.E.} = -\frac{\text{me}^4}{8\epsilon_0^2 h^2} \left( \frac{1}{n^2} \right) \]

For hydrogen-like atoms: \( E_n = -R \, ch \frac{Z^2}{n^2} \)

For hydrogen atom: \( E_n = \frac{-13.6}{n^2} \, \text{eV} \)

3. Hydrogen Spectrum

3.1 Frequency and Wave Number

\[ v = \frac{me^4}{8\epsilon_0 h^3} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
\[ \bar{v} = \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

Where \( R = \frac{me^4}{8\epsilon_0 c h^3} = 10973700 \, \text{m}^{-1} \) (Rydberg's constant)

3.2 Spectral Series

S.No. Name of Series Region Origin Max. λ Min. λ
1. Lyman series UV n₂ = 2,3,4... to n₁ = 1 \(\frac{4}{3R}\) \(\frac{1}{R}\)
2. Balmer series Visible n₂ = 3,4,5... to n₁ = 2 \(\frac{36}{5R}\) \(\frac{4}{R}\)
3. Paschen series Near Infrared n₂ = 4,5,6... to n₁ = 3 \(\frac{144}{7R}\) \(\frac{9}{R}\)
4. Brackett series Middle Infrared n₂ = 5,6,7... to n₁ = 4 \(\frac{400}{9R}\) \(\frac{16}{R}\)
5. Pfund series Far Infrared n₂ = 6,7,8... to n₁ = 5 \(\frac{900}{11R}\) \(\frac{25}{R}\)

3.3 Maximum Number of Spectral Lines

\[ \text{Maximum lines} = \frac{n(n-1)}{2} \]

4. Nuclear Physics

4.1 Size of the Nucleus

\[ R = R_0(A)^{1/3} \]

Where A is mass number and \( R_0 = 1.1 \times 10^{-15} \, \text{m} \)

Nuclear radii vary from 1F to 10F

Nuclear Volume

\[ V = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi R_0^3 A \]

Nuclear volume is proportional to mass number (V ∝ A)

Nuclear Density

Density of nucleus: \( 2.29 \times 10^{17} \, \text{kg/m}^3 \)

Nuclear density does not depend on A; density of all nuclei is almost same

4.2 Atomic Mass Unit

1 amu = \( \frac{1}{12} \)th of the mass of carbon atom C¹²

\[ 1 \, \text{amu} = 1.66 \times 10^{-27} \, \text{kg} = 931.5 \, \text{MeV} \]

4.3 Mass Defect

\[ \Delta M = ZM_p + (A-Z)M_n - M' \]

Where Mₚ = mass of proton, Mₙ = mass of neutron, M' = mass of nucleus

4.4 Binding Energy

\[ B = [ZM_H + (A-Z) M_n - M]c^2 \]

Where M_H = Mₚ + Mₑ = mass of hydrogen atom, M = M' + ZMₑ = mass of atom

Binding Energy per Nucleon

\[ \bar{B} = \frac{B}{A} = \frac{[ZM_H + (A-Z)M_n - M]c^2}{A} \]

Nuclear Stability:

  • A = 1 to 39: \( \bar{B} < 8.5 \, \text{MeV/nucleon} \)
  • A = 40-120: \( \bar{B} \approx 8.5 \, \text{MeV/nucleon} \)
  • Iron (A=56): \( \bar{B} \approx 8.8 \, \text{MeV/nucleon} \)
  • A > 121: Decreases slowly up to U-238 (7.6 MeV)

4.5 Nuclear Forces

Properties:

  • Due to exchange of mesons (Yukawa 1935)
  • Strongly attractive, charge independent
  • Spin dependent, short range (maximum at 1.5 Fermi)
  • 10³⁷ times stronger than gravitational forces
  • 10² times stronger than electrostatic forces

5. Radioactivity

Radioactivity is the phenomenon of spontaneous emission of radiations by the nucleus of a substance.

5.1 Radioactive Decay Law

\[ \frac{-dN}{dt} = \lambda N_t \]
\[ N_t = N_0 e^{-\lambda t} \]

Where Nₜ = number of radioactive nuclei left after time t, λ = decay constant

5.2 Decay Constant

Decay constant is the reciprocal of the time in which number of nuclei left is 1/e times the number of nuclei at t=0.

5.3 Half-Life

\[ T_{1/2} = \frac{\log_e 2}{\lambda} = \frac{0.693}{\lambda} \]

5.4 Mean Life

\[ \tau = \frac{1}{\lambda} \]

5.5 Rate of Decay

\[ R_t = \lambda N_t \]

5.6 Units of Radioactivity

  • 1 Becquerel = 1 disintegration/sec
  • 1 Curie = \( 3.7 \times 10^{10} \) disintegrations/sec
  • 1 Rutherford = \( 10^6 \) disintegrations/sec

5.7 Nuclear Decay

Type Reaction Available Energy
Alpha Decay \( z^{X^A} \rightarrow z_2 Y^{A-4} + zHe^4 \) \( Q = [m_X - m_Y - m_c]c^2 \)
Beta Decay (Minus) \( zX^A \rightarrow z + iY^A + -i\beta^0 + v \) \( Q = [m_X - m_Y]c^2 \)
Beta Decay (Plus) \( zX^A \rightarrow z_1 Y^A + i\beta^0 + v \) \( Q = [m_X - m_Y - zm_c]c^2 \)
Electron Capture \( zX^A + -1e^0 \rightarrow z_1Y^A + v \) \( Q = [m_X - m_Y]c^2 \)

5.8 Nuclear Reactions

Nuclear Fission

Process of disintegration of a heavy nucleus into two or more moderate nuclei of comparable masses.

\[ 92 \text{U}^{235} + 0\text{n}^1 \rightarrow 92 \text{U}^{236} \rightarrow 56 \text{Ba}^{141} + 36 \text{Kr}^{92} + 3 \text{on}^1 + Q(200 \, \text{MeV}) \]

Energy released per nucleon ≈ 0.85 MeV

Nuclear Fusion

Process of combining two or more lighter nuclei to form a heavy nucleus.

\[ 4_1 \text{H}^1 \rightarrow 2 \text{He}^4 + 2_1 \text{e}^0 + Q(26.7 \, \text{MeV}) \]

Energy released per nucleon = 6.68 MeV