Bohr Model of the Atom

Rutherford's Atomic Paradox: Classical physics predicted electrons orbiting a positive nucleus would continuously emit electromagnetic radiation and spiral into the nucleus.
Bohr's Postulates (1913):
- Electrons orbit the nucleus only in specific circular orbits with specific, allowed energies.
- The hydrogen atom does not emit energy while the electron is in one of these fixed orbits.
- An electron moves to a different orbit only by emitting or absorbing a photon with energy equal to the difference in energy between the two orbits.

Energy Equation: The energies of electron orbits in hydrogen are given by:
- $E_n = -\frac{k}{n^2}$
- Where $k = 2.18 \times 10^{-18} \text{ J}$ is a constant, and $n$ is the principal quantum number (orbit number).
Significance of Negative Energy: The negative sign indicates that the electron's zero energy point is when it is completely removed from the nucleus ( $n \to \infty$ ).
Energy Levels: More negative energy means lower energy. The $n=1$ orbit has the lowest energy (ground state). As $n$ increases, the electron's energy and distance from the nucleus increase.

Ground State: The lowest energy level, typically where the electron resides ( $n=1$ ).
Excited State: A higher energy level reached when the atom absorbs energy and the electron moves to a higher $n$ orbit.
Energy Change for Transitions: The energy change for an electron transition is:
- $\Delta E = -k \left( \frac{1}{n<em>{\text{final}}^2} - \frac{1}{n</em>{\text{initial}}^2} \right)$
- When energy is absorbed (n{\text{initial}} < n{\text{final}}), $\Delta E$ is positive.
- When energy is emitted (n{\text{initial}} > n{\text{final}}), $\Delta E$ is negative (e.g., electron moves from excited state back to ground state).
Photon Properties: The energy of the corresponding photon is equal to the magnitude of $\Delta E$ ( $E_{\text{photon}} = |\Delta E|$ ).
- Photon frequency ( $\nu$ ) and wavelength ( $\lambda$ ) can be calculated using $E_{\text{photon}} = h\nu$ and $\lambda\nu = c$ .