Study Notes on Rutherford and Bohr Atomic Models

RUTHERFORD'S ATOMIC MODEL

Observations:

• Majority of alpha particles:

  • Passed through the gold foil without deviation.

• Scattering of alpha particles:

  • Some particles scattered at a small angle; this angle correlates with the impact parameter being equal to the nuclear radius.

• Few retracing particles:

  • Only a few alpha particles retraced their paths; this is when the impact parameter equals zero (the closest approach).

Distance of Closest Approach for Alpha Particles:

• The formula for the distance of closest approach, represented as $r_0$, can be defined as:

  • $r<em>0 = rac{1}{4 ext{π} ext{ε}</em>0} rac{Z<em>1Z</em>2 e^2}{mv^2}$

  • Here, $Z<em>1$ and $Z</em>2$ are the atomic numbers, $m$ is the mass, and $v$ is the velocity of the incoming alpha particle.

  • The Kinetic Energy can be expressed as:

  • $K.E = rac{1}{2} mv^2$

Impact Parameter Calculation:

• The impact parameter $b$ can be represented as:

  • $b = rac{Z<em>1 Z</em>2 e^2}{mv^2} ext{cot}( heta/2)$

• Where $heta$ is the scattering angle.

BOHR ATOM MODEL

Key Postulates:

1. Electrostatic Force:

   • The force between electron and nucleus is given by:

   • $F = rac{1}{4 ext{π} ext{ε}_0} rac{Ze e}{r^2}$

2. Angular Momentum:

   • The quantization condition of angular momentum states:

   • $mvr = rac{nh}{2 ext{π}}$

Derived Concepts:

• Radius of Orbit:

  • $r_n = rac{n^2 ext{h}^2}{4 ext{π}^2 k m Z}$

  • For hydrogen-like atoms, this simplifies.

• Velocity of Electron:

  • $v<em>n = rac{Ze^2}{4 ext{π} ext{ε}</em>0 h} rac{1}{n}$

• Energy Relationships:

  • The total energy is given by:

  • $E = - rac{Z^2 ext{me}^4}{8 ext{ε}_0^2 h^2 n^2} = - rac{13.6 Z^2}{n^2} ext{ eV}$

  • Where Kinetic Energy $K.E$ is equal to the negative of the total energy.

HYDROGEN SPECTRUM

Absorption Spectrum:

• Photon Absorption:

  • Electrons absorb specific photons whose energy corresponds to the energy difference between shells.

  • If excitation occurs to the nth shell from ground state, it can absorb (n-1) different photons.

Emission Spectrum:

• Wavelength of Emitted Photon from given transitions:

  • rac{1}{ ext{λ}} = RZ^2igg( rac{1}{n1^2} - rac{1}{n2^2}igg)

  • Rydberg constant R value is approximately $1.097 imes 10^7 ext{ m}^{-1}$.

• Total number of different wavelength photons:

  • This matches all absorbed photons emitted in the emission spectrum.

Hydrogen Emission Series:

Spectral Series:

1. Lyman Series:

   • Transition to n=1; UV spectrum range 911.6 Å to 1216 Å.

2. Balmer Series:

   • Transition to n=2; Visible spectrum range 364.6 Å to 6563 Å.

3. Paschen Series:

   • Transition to n=3; Infrared spectrum range.

4. Brackett Series:

   • Transition to n=4; Infrared spectrum range.

5. Pfund Series:

   • Transition to n=5; Infrared spectrum range.

Energy Levels and Their Values:

• Energy minima and maximums for spectral transitions:

  • $E_{ ext{min}} = R rac{(n^2 + 1)}{n^2}$

  • $E_{ ext{max}}$ at last transition can be determined by absorbing energy differences occurring through transitions.

Summary of Regions for Solar Emission Series:

• Names of series, transitions, and wavelengths:

  • Lyman: $n<em>2 ightarrow n</em>1$ $ext{(Extreme UV)}$

  • Balmer: $n<em>2 ightarrow n</em>3$ $ext{(Visible)}$

  • Paschen: $n<em>3 ightarrow n</em>4$

  • Brackett: $n<em>4 ightarrow n</em>5$

  • Pfund: $n<em>5 ightarrow n</em>6$