JEE Physics Atoms Notes Set 06

Bohr's Model of the Hydrogen-Like Atom

Bohr kept the nuclear picture but added quantum rules: an electron revolves only in orbits where the angular momentum is \( mvr=\frac{nh}{2\pi} \), it does not radiate while in such an orbit, and it emits or absorbs a photon of energy \( hf=E_i-E_f \) when it jumps between orbits. Equivalently, an orbit holds a whole number of de Broglie wavelengths, \( 2\pi r=n\lambda \).
For a one-electron species of atomic number Z: \( r_n=0.529\frac{n^2}{Z} \) Å, \( v_n=2.19\times10^6\frac Zn \) m/s, and \( E_n=-13.6\frac{Z^2}{n^2} \) eV. So \( r\propto n^2/Z \), \( v\propto Z/n \), the frequency of revolution \( f\propto Z^2/n^3 \), and the period \( T\propto n^3/Z^2 \).

Energies in an Orbit

The Coulomb force provides the centripetal force, giving \( K=\frac{ke^2}{2r} \), \( U=-\frac{ke^2}{r} \) and \( E=-\frac{ke^2}{2r} \). Hence \( K=-E \) and \( U=2E \). Moving to a higher orbit raises the potential and total energy and lowers the kinetic energy. The ionisation energy of the ground state is \( 13.6Z^2 \) eV, and the energy needed to ionise a state n is \( 13.6Z^2/n^2 \) eV. The angular momentum depends only on n, not on Z.

Spectral Series

The wavelength of a line is given by \( \frac1\lambda=RZ^2\left(\frac1{n_1^2}-\frac1{n_2^2}\right) \) with \( R=1.097\times10^7 \) per m. The series are named by the lower level: Lyman (n1 = 1, ultraviolet), Balmer (n1 = 2, visible), Paschen (n1 = 3), Brackett (n1 = 4) and Pfund (n1 = 5), all infrared. A sample excited to level n can show \( \frac{n(n-1)}2 \) different lines. The first Balmer line (3 to 2) is 6561 Å and the second (4 to 2) is 4860 Å. A useful shortcut is that the Lyman first line to Balmer first line wavelength ratio is 5 : 27.

Limits and Collisions

In the alpha-particle scattering experiment, an alpha particle of kinetic energy K approaching a nucleus of charge Ze head-on stops at a distance \( r=\frac{k(2e)(Ze)}{K} \), which gives the upper limit on nuclear size. In an inelastic collision between two identical atoms, one initially at rest, half of the initial kinetic energy is available to excite an atom, so the minimum speed for exciting hydrogen to n = 2 satisfies \( \frac14mv^2=10.2 \) eV. Photons absorbed by an atom must match a level gap exactly, but an electron or photon of energy above the ionisation energy can free the electron and leave the extra as kinetic energy.

Worked Examples

The 3 to 2 transition of He\( ^+ \) has energy \( 13.6\times4\left(\frac14-\frac19\right)=7.56 \) eV.
The ionisation energy of Li\( ^{2+} \) in its first excited state is \( \frac{13.6\times9}{4}=30.6 \) eV.
If the electron in a hydrogen-like ion has total energy -3.4 eV, its kinetic energy is +3.4 eV and, for hydrogen, its principal quantum number is 2.