PHY 102 Waves Quanta Topic 12 The Bohr

PHY 102: Waves & Quanta Topic 12 The Bohr Model John Cockburn (j. cockburn@. . . Room E 15)

• More on line spectra • Orbital model of the hydrogen atom • Failure of classical model • Quantisation of orbital angular momentum: stationary states • Successes and failures of the Bohr Model

Line Spectrum of hydrogen • Hydrogen has line spectrum ranging in wavelength from the UV to the infrared • Balmer (1885) found that the wavelengths of the spectral lines in the visible region of the spectrum could be EMPIRICALLY fitted to the relationship: (The group of hydrogen spectral lines in the visible region still known as the Balmer Series)

Line Spectrum of hydrogen • Rydberg and Ritz subsequently obtained a more general expression which applies to ALL hydrogen spectral lines (not just visible), and also to certain elements (eg alkaline metals): n 2, n 1 integers, n 2 < n 1 • R is called the Rydberg constant, which changes slightly from element to element. • For hydrogen, RH = 1. 097776 x 107 m-1 • Can a model of the atom be developed that’s consistent with this nice, elegant formula? ?

Rutherford Scattering To explain results of the Rutherford scattering : 1) Atom must be mostly empty space 2) Positive charge must be concentrated in a small volume occupying a very small fraction of the total volume of the atom………… Christmas pudding model doesn’t work Nuclear model does work Atomic radius ~ 10 -10 m Nuclear radius ~ 10 -14 m

The Bohr Model (1912 -13) Bohr suggested that the electrons in an atom orbit the positively -charged nucleus, in a similar way to planets orbiting the Sun (but centripetal force provided by electrostatic attraction rather that gravitation) Hydrogen atom: single electron orbiting positive nucleus of charge +Ze, where Z =1: v r -e +Ze F

Bohr Model: electron energy v r -e +Ze F From electrostatics, the potential energy of the electron is given by:

Bohr Model: electron energy v r -e +Ze Centripetal force equation: Kinetic energy of electron: F

Bohr Model: electron energy Total energy of electron = P. E. + K. E: But this classical treatment leaves us with a big problem………

Failure of the Classical model The orbiting electron is an accelerating charge. Accelerating charges emit electromagnetic waves and therefore lose energy Classical physics predicts electron should “spiral in” to the nucleus emitting continuous spectrum of radiation as the atom “collapses” CLASSICAL PHYSICS CAN’T GIVE US STABLE ATOMS………………. .

Bohr’s postulates • Only certain electron orbits are allowed, in which the electron does not emit em radiation (STATIONARY STATES) • An atom emits radiation only when an electron makes a transition from one stationary state to another. • The frequency of the radiation emitted when an electron makes a transition from a stationary state with energy E 2 to one with energy E 1 is given by:

Transition energies Suppose an electron is initially in stationary state with energy E 1, orbital radius r 1. It then undergoes a transition to a lower energy state E 2, with (smaller) radius r 2: If Bohr’s postulates are correct, then the frequency of the radiation emitted in the transition is given by:

Rydberg-Ritz Revisited c = fλ Bohr result: Looks promising, if we can make the connection that r is somehow proportional to “integer squared”……………….

Quantisation of angular momentum Bohr now makes the bold assumption that the orbital angular momentum of the electron is quantised……… Since v is perpendicular to r, the orbital angular momentum is just given by L = mvr. Bohr suggested that this is quantised, so that: IMPLICATIONS? ? ? . . . .

Kinetic energy (earlier slide) quantisation of A. M. (last slide)

Bohr radius So, introduction of the idea that angular momentum is quantised has the desired effect: r n 2. Simplifying the expression for r a bit (Z=1 for hydrogen): a 0, the radius of the n=1 orbit, is called the BOHR RADIUS

We conclude that in the Bohr model only certain orbital radii (and electron velocities) are allowed. Rydberg-Ritz R=1. 07 x 107 m-1 How nice.

Origins of hydrogen spectral lines:

Bohr Model: Shortcomings • The Bohr model does an excellent job of explaining the “gross” features of hydrogen line spectrum BUT • Doesn’t work well for many-electron atoms (even helium) • Can’t explain fine structure of spectral lines observed at high resolution, or relative intensities of spectral lines • Can’t explain effect of magnetic field on spectral lines (Zeeman effects), although Sommerfeld’s modifications (elliptical orbits, varying orientations) help to some extent • Is fundamentally inconsistent with Heisenberg’s uncertainty principle THE BOHR MODEL IS WRONG