Bohr Model of Atom - I

Fundamental Postulates of Bohr’s Model

• Electrons orbit the nucleus in fixed circular paths of definite radii. These orbits are called stationary orbits or energy levels.
• Each orbit has a specific energy associated with it, and the electron can move from one orbit to another by absorbing or emitting a photon of light.
• The angular momentum of an electron in a given orbit is quantized, i.e., it can have only certain discrete values.

Explanation of Line Spectra of Hydrogen Atom Based on Bohr’s Model

• When an electron in a hydrogen atom transitions from a higher energy orbit to a lower energy orbit, it emits a photon of light with a wavelength that corresponds to the energy difference between the two orbits.
• The line spectra of hydrogen atom are the result of these specific transitions.

Calculation of Radius of Electron Orbits, Energy of Electron in Orbit, and Frequency of Emitted Radiation

• The radius (r) of the nth orbit is given by: $$r_n = \left(\frac{4\pi\epsilon_0}{m_e}\right)n^2a_0$$

• The energy (En) of an electron in the nth orbit is given by: $$E_n = -\frac{1}{8 \pi \epsilon_0}\frac{e^2}{r_n} = -\frac{1}{8\pi \epsilon_0}\frac{m_ek^2e^2}{4\pi \epsilon_0 n^2\hbar^2}$$ $$E_n = -\frac{1}{n^2}\frac{m_ek^2e^2}{8h^2\epsilon_0}$$

• The frequency (f) of the radiation emitted when an electron transitions from the nth orbit to the mth orbit is given by: $$f = \frac{\Delta E}{h} = \frac{E_n - E_m}{h}$$

Bohr’s Formula for Calculating Ionization Energy The ionization energy (IE) of an atom is the energy required to remove an electron from the atom’s lowest energy level (n=1). Bohr’s formula for ionization energy is: $$IE = E_1 = -\frac{1}{8h^2\epsilon_0}\frac{m_ek^2e^2}{2^2}$$ $$IE = \frac{1}{8}\frac{m_ek^2e^2}{4h^2\epsilon_0}$$ $$IE = \frac{1}{8}\frac{(9.109\times10^{-31}\text{ kg})(8.99\times10^9\text{ N}\cdot\text{m}^2/\text{C}^2)(1.602\times10^{-19}\text{ C})^2}{4(6.626\times10^{-34}\text{ J}\cdot\text{s})^2(8.85\times10^{-12}\text{ C}^2/\text{N}\cdot\text{m}^2)}$$ $$IE = \frac{1}{8}(13.60569)\text{ eV} = 1.7 \text{ eV}$$

Limitations of Bohr’s Model

• Bohr’s model:
• Doesn’t explain the splitting of spectral lines observed in the presence of external magnetic fields (Zeeman effect) or electric fields (Stark effect).
• Doesn’t explain the fine structure of spectral lines, which is due to the spin of the electron.
• Can’t explain the chemical bonding between atoms.

Application of Bohr’s Model to Other Atoms and Ions Bohr’s model can be applied to other one-electron atoms and ions such as He+ and Li2+, but it is less accurate for these larger atoms because of the increased number of electrons and electron-electron interactions.