Concepts: the Bohr model and its limitations

Structure of Atom22 min readLesson 4 of 8

4.1 Why a new model was needed The Rutherford model failed on two counts. It could not explain why the atom is stable, and it could not explain the sharp lines of the hydrogen spectrum. Niels Bohr in 1913 kept the nuclear atom and added quantisation to the electron's motion. The model works well for one-electron systems and it still gives the correct energy levels of hydrogen, even though its picture of circular orbits has been replaced. 4.2 The postulates Bohr proposed that the electron in a hydrogen atom moves around the nucleus in certain allowed circular paths, called stationary states or orbits, without radiating energy. Each such orbit has a fixed radius and a fixed energy, and the electron does not lose energy while it stays in one. The model is a first step beyond classical physics: it keeps the nuclear atom and adds one quantum condition instead of discarding either idea. Energy changes only when the electron jumps between two stationary states. If the electron goes from a lower state of energy E1 to a higher state of energy E2, it absorbs a photon whose energy equals the difference. If it falls from a higher state to a lower one, it emits a photon of that same energy. The photon frequency follows from Planck's relation.

E_{2} - E_{1} = h ν

Bohr frequency rule for a transition

The third postulate says that only those orbits are allowed in which the angular momentum of the electron is an integral multiple of h divided by 2 pi.

m v r = \frac{nh}{2 π}, n = 1, 2, 3, \dots

Bohr quantisation of angular momentum

4.3 Radius, velocity and energy of the orbits Combine the quantisation condition with the Coulomb force providing the centripetal force for the circular motion. The electrostatic attraction between the nuclear charge and the electron supplies the force, and the two equations together fix the allowed radii and the corresponding energies.

\frac{m v ^{2}}{r} = \frac{k Z e ^{2}}{r ^{2}}

Force balance in the Bohr orbit, with Z the nuclear charge

Solving the pair gives the radius of the nth orbit for a hydrogen-like ion. The radius grows as the square of the principal quantum number and shrinks as the nuclear charge increases.

r_{n} = 52.9 \frac{n ^{2}}{Z} pm

Radius of the nth Bohr orbit

The energy of the electron is the sum of its kinetic energy and its electrostatic potential energy. The kinetic energy is positive and the potential energy is negative, and the total comes out negative, which is what keeps the electron bound. The expression is written as the energy of the electron relative to the free electron at rest.

E_{n} = - 2.18 \times 1 0^{- 18} \frac{Z ^{2}}{n ^{2}} J = - 13.6 \frac{Z ^{2}}{n ^{2}} eV

Energy of the nth Bohr orbit of a hydrogen-like species

For hydrogen, the first orbit has radius 52.9 pm and energy -13.6 eV. The negative sign means the atom is bound; the ground state electron needs 13.6 eV to escape completely. The spacing between levels shrinks as n increases, and the levels converge to zero energy at large n.

4.4 Explaining the spectrum A hydrogen atom in its ground state can absorb a photon only if the photon energy matches the gap to one of the higher levels. Once excited, the electron returns to a lower level in one or more steps, and each step emits a photon of a fixed energy. The set of all possible steps gives the set of observed lines, and the series are exactly the families that end on a common lower level. Putting the energy expression into the frequency rule reproduces the Rydberg equation. For an absorption from n1 to n2, the wavelength is given by the difference of 1/n squared, with the Rydberg constant appearing naturally.

\frac{1}{λ} = R Z^{2} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}}), R = \frac{m e ^{4}}{8 ε _{0}^{2} h ^{3} c}

Rydberg equation obtained from the Bohr energy levels

The shortest wavelength in a series is the jump from the highest level down to the series limit, and the longest is the jump from the nearest higher level. This is why only a few lines in a series are visible and the rest crowd together near the limit. 4.5 Successes and failures The model gives the correct radii, the correct energies and the whole hydrogen spectrum. It also explains why each element has its own spectrum, because the level spacings depend on the nuclear charge. It predicts that one-electron ions such as He+ and Li2+ should behave like scaled copies of hydrogen, and this is confirmed for the line positions. The failures are equally clear. The model works for one electron only and cannot handle helium, which has two. It cannot explain the fine splitting of spectral lines observed with better instruments. It says nothing about the intensity of lines, only their positions. It treats the electron as a particle moving in a circle with a definite path, which contradicts the uncertainty principle. Finally it gives no reason why angular momentum should be quantised in this way; that reason comes from wave mechanics.

Exam

The ground state of hydrogen needs 13.6 eV to ionise. For a hydrogen-like ion with nuclear charge Z, the ionisation energy from the ground state is 13.6 Z squared eV. Check the sign convention: bound state energies are negative, and the energy change that ejects an electron is positive.

in hinglish

Bohr model ke teen postulates yaad rakho: electron kuch fixed orbits mein bina energy radiate kiye ghoomta hai; do orbits ke beech jump par photon absorb ya emit hota hai jiski energy E2 - E1 = hν hoti hai; aur angular momentum mvr = nh/2π hota hai. Atom mein attractive force Coulomb force hoti hai jo centripetal force deti hai. Isse radius r_n = 52.9 n²/Z pm aur energy E_n = -13.6 Z²/n² eV milti hai. Hydrogen ke liye ground state ki energy -13.6 eV hai, isliye ionise karne ke liye 13.6 eV chahiye. Hydrogen spectrum ki lines tab banti hain jab electron excited state se neeche girti hai; Rydberg formula in positions ko match karta hai. Limitations : yeh sirf ek electron wale atoms ke liye sahi hai, multi-electron atoms explain nahi karta, fine structure aur line intensities nahi bataata, aur circular orbit ka path uncertainty principle se conflict karta hai. Numerical mein Z aur n sahi rakho, energy ka sign dekho, aur eV ko J mein convert karna mat bhoolo.