Ch2 Bohr’s atomic model Four puzzles –Blackbody radiation –The photoelectric effect –Compton effect –Atomic spectra Balmer formula Bohr’s model Frank-Hertz experiment

Blackbody Absorptivity (absorptance): the ratio of the radiation absorbed by a body to that incident on the body. Blackbody: A body with a surface having an absorptivity equal to unity. A realistic blackbody: For a cavity kept at a constant temperature with the interior wall blackened, a small hole in the wall behaves like a blackbody.

Some observations Stefan's Law states that the power radiated by a body is proportional to the 4th power of the absolute temperature. For a given temperature, the radiation forms a continuous spectrum with respect to the frequency.

Wein's Displacement Law

Reyleigh-Jeans law

Ultraviolet catastrophe

Puzzles in blackbody radiation Two puzzles: –Why were not radiation above the ultraviolet region present? –Why was there a non-uniform distribution of electromagnetic radiation being emitted?

Plank’s theory Planck made an assumption that the energy of an oscillator must be an integral multiple of the product of the constant h and the frequency of the electromagnetic radiation it emitted. His assumption resulted in a formula for the blackbody radiation that was in excellent agreement with experiment at all frequencies.

Two puzzles to be explained Radiation in the high frequency region were not emitted from the blackbodies because this required large energy changes which could not occur in the atoms. Certain energy states were more probable in the atoms and therefore frequencies associated with these energy states were more likely to be emitted.

The photoelectric effect When light of a high frequency was incident on a metallic surface, electrons were emitted from the surface.

Actual observation Intensity: The high intensity of light would not cause electrons to have high KE. The actual reaction time is very short (10 -9 s). Frequency: At a certain frequency called threshold frequency, electrons were emitted. A frequency beyond it will cause the electrons to have a greater KE. Stopping voltage: The energy of the ejected electrons was proportional to the frequency of the illuminating light & had nothing to do with intensity.

Einstein’s explanation For a photoelectron, E=hf. The minimum energy required to pull electrons from inside to outside the metal is called the work function W. W=hf 0 If an electron is given an energy E larger than W, it can escape the metal and will have a maximum KE:

The Compton effect (Compton scattering) This could be explained when X rays are regards as particles (photons). The collision between a photon and an electron is regarded as an elastic collision.

Discrete spectra Atoms emit and absorb light only at specific frequencies. –Emission lines, –Absorption lines, Balmer found that the wavelengths of visible and near ultraviolet line spectra of hydrogen obey a simple formula exactly: R H =1.097x10 7 m -1 is called the Rydberg ( 里德伯 ) constant.

Bohr model There are three postulates used in Bohr’s model: –The electron moved in a certain set of stable orbits in which classical mechanics can be used to describe motion of the electron. –Moving electrons in stable states do not radiate. An electron can make a sudden quantum jump between the orbits. –The orbital angular momenta of the electrons are quantized.

Quanta in the atom The total energy of the electron is inversely proportional to the square of n, i.e. where n is called quantum number. The total energy is also found to be negative, indicating a “bound” state. The most negative state, the most tightly bound electron, occurs for n=1, referred to as the ground state of the atom, n>=2, excited states. The angular momentum of the electron moving in a circular orbit can only take discrete values:

Line spectra of the H atom Energy levels: Lyman series: n=1; Balmer series: n=2; Paschen series: n=3; Brackett series: n=4

Improvement on the Bohr model Finite nuclear mass (motion of nucleus): When taking the nuclear mass into account, the reduced mass should replace the electron mass. Relativistic correction: The effect of the relativistic mass change m(v) should be considered. Faster  massive  decrease in energy. Sommerfeld’s extension: Electrons should have elliptical orbits with the same energies as that in circular orbits. The second quantum number should be introduced.

Frank-Hertz experiment Frank & Hertz in 1913 showed the existence of discrete energy levels in atoms.

Frank-Hertz experiment results

Explanation With the increase of grid potential, more electrons move to the plate and the current rises accordingly. For mercury atoms, when V=4.9V, the electrons make inelastic collision and leave the atom jump to a high orbit (n=2). The original electrons move off with little energy and could not reach the plate and thus reduce the current. As V is increased further, the current rises again and would drop at V=9.8V. This would make more atoms to jump to n=2 state.

Limitations of Bohr model It can not be generalised to deal with systems with two more electrons as the force between the electrons can not be easily added. It can not explain the closely spaced lines. It can not be used to calculate the rate of transitions between different energy levels. The Bohr model was eventually superseded by the quantum mechanics developed by E Schrodinger, W Heisenberg and others, following the ideas of L de Broglie.