JEE Physics Atoms Notes Set 03

Bohr Model of Hydrogen-like Atoms

Spectral Lines

1. \( \dfrac{1}{\lambda}=RZ^2\left(\dfrac{1}{n_1^2}-\dfrac{1}{n_2^2}\right) \) with \( R=1.097\times10^{7}\ \text{m}^{-1} \). For the same transition, \( \lambda\propto 1/Z^2 \).
2. Series: Lyman (final n = 1, ultraviolet), Balmer (n = 2, visible), Paschen (n = 3), Brackett (n = 4), Pfund (n = 5), all infrared except the first two.
3. The longest wavelength of a series comes from the transition from the next higher level. The shortest is the series limit (from n = infinity). Ratio of the longest wavelengths of Lyman and Paschen is 7 : 108.
4. If an atom has three levels, three lines are possible and \( \dfrac{1}{\lambda_{shortest}}=\dfrac{1}{\lambda_2}+\dfrac{1}{\lambda_3} \).
5. An atom absorbs only photons whose energy equals a gap between levels. A photon with enough energy to ionise it can have any larger energy, and the extra becomes kinetic energy.
6. The excited state lasts only about \( 10^{-8} \) s, so atoms return to the ground state almost immediately.

Spacing of Levels

The gap between consecutive levels is proportional to \( \dfrac{1}{n_1^2}-\dfrac{1}{n_2^2} \). The levels crowd together as n increases, so the gap between n = 3 and 4 is about 14.8 times that between n = 8 and 9.

Characteristic X-rays

1. When an inner-shell vacancy is filled, X-rays are emitted: K lines when the electron falls to the K shell, L lines when it falls to the L shell. The K alpha line comes from n = 2 to n = 1 and K beta from n = 3 to n = 1.
2. Moseley's law: \( \sqrt{f}=a(Z-b) \), so a graph of \( \sqrt{f} \) against Z is a straight line. For K lines b is about 1, and for L lines about 7.4. The beta line has a larger slope than the alpha line of the same series.
3. The cut-off wavelength of the continuous X-ray spectrum is \( \lambda_{min}=\dfrac{12400}{V}\ \text{Å} \), with V in volts. It depends only on the tube voltage.