The photoelectric effect and the photon theory of light

1. The photoelectric effect and the photon theory of light • the photoelectric effect had a few confusing features • Presence of a threshold frequency. There is a minimum frequency below which no electrons flow. Wave theory associates the amplitude of the wave with energy not the frequency. Wave theory predicts that once enough energy was absorbed an electron should break free, no matter what the color. • The absence of a time lag. Immediately upon shining light with the minimum frequency or higher, regardless of the intensity of the light. The wave theory predicts that in dim light there would be a time lag between shining light and current flow so that the electrons have time to absorb enough energy to break free.

2. The kinetic energy varies with the frequency of impinging light. Above the minimum frequency of light to cause electron emission the kinetic energy of the emitted electrons increases linearly with the frequency of the emitted light. kinetic energy of electrons work function frequency of light Using Planck’s idea of packeted energy, Einstein went a bit further. He proposed that light itself is particulate, occurring as quanta of electromagnetic energy or photons. A system changes its energy whenever it emits or absorbs a a photon or “particle” of light whose energy is fixed by its frequency,

3. How did Einstein’s photon theory explain the puzzles of the photoelectric effect? • Presence of a threshold frequency. According to the photon theory, a beam of light consists of an enormous number of photons. Light intensity (brightness) is related to the number of photons (amplitude of the wave) striking the surface but not the energy. So, a photon of a certain minimum energy must be absorbed for an electron to be freed. Since energy depends on frequency (hv) the theory predicts a threshold frequency. • The absence of a time lag. An electron cannot save up energy from several photons of energy lower than the threshold frequency. One electron breaks free the moment it absorbs a photon of the appropriate energy. The current is weaker in dim light because there are fewer photons and stronger in bright light because there are many photons. • The kinetic energy varies with the frequency of impinging light. Since each photon carries a specific energy according to its frequency, a photon with energy greater than that required to eject an electron, the energy manifests itself with the kinetic energy of the electron,

4. eg. a) The energy required to dislodge electrons from sodium metal via the photoelectric effect is 275 kJ mol-1. What wavelength, in nm, of light has sufficient energy per photon to dislodge an electron from the surface of sodium. b) Suppose UV light with a wavelength of 200 nm strikes the sodium metal. Assuming that the mass of the electron is 9.1x10-31 kg, what is the velocity of the ejected electron?

5. eg. For radiation of wavelength 242.4 nm, the longest wavelength that will bring about the photodissociation of O2, what is the energy of a) one photon and b) a mole of photons. Solution.

6. If 1 mole of these 242.4 nm photons were absorbed by 10.0 kg of water and all went into increasing the temperature, what would the change in temperature be?

7. Compare this value with the change in temperature if 1 mole of photons from a microwave (wavelength 12.24 cm).

8. the quantum theory and photon theory assigned properties to energy and light that had always previously been reserved for matter: fixed quantity and discrete particles. The particulate model of energy does not fit the facts of diffraction and refraction, phenomena explained only in terms of waves. In fact the particle model does not replace the wave model and as we shall see, we have to accept both to understand reality.

9. Atomic Spectra • another observation made late in the 19th century that classical physics could not explain involved the light emitted when an element is vaporized and excited by an electric discharge or just by heating. due to differences in the index of refraction for different wavelengths of light, light can be separated into its component wavelenghts.

10. the line spectra are characteristic of the element producing them.

11. For hydrogen, Balmer found that the visible atomic spectrum could be fit to an equation, integer nl 3 656.5 4 486.3 5 434.2

12. Paschen Series Balmer Series Lyman Series • the entire line spectrum of Hydrogen was found to be able to be fit by this equation, commonly known as the Rydberg equation, integers • the Rydberg equation and the constant (1.096776x107 m-1) is based on fitting the data for hydrogen rather than on any theoretical basis

13. eg. Determine the wavelength of light emitted by a hydrogen atom for the transition of an electron from n=7 to n=3 and n=3 to n=1.

14. based on Rutherford’s nuclear model of the atom, classical physics predicts a continuous spectrum as the electron spirals into the nucleus…? The Bohr Model of the Hydrogen Atom • Danish physicist Niels Bohr, working for Rutherford, used Planck’s and Einstein’s ideas about quantized energy and came up with a model for the hydrogen atom,

15. Postulates of the Bohr model, • The H atom has only certain allowable energy levels or stationary states. Each state is associated with a fixed circular orbit of the electron around the nucleus. • The atom does not radiate energy while in one of the stationary states. This defies classical physics but this is a “new physics” pertaining to the world of the very small! The atom does not change energy while the electron moves within an orbit.

16. The atom changes from one stationary state to another by absorbing or emitting a photon whose energy equals the difference in energy between the two stationary states. • a spectral line results when a photon of specific energy (or frequency) is emitted as the electron moves from a higher energy state to a lower one. • the Bohr model explains why the spectra are not continuous because the atom’s energy has only certain discrete energy levels or states.

17. the n’s are called quantum numbers and are associated with an orbit for the electron Balmer series • when the electron is in n=1, the atom is said to be in its lowest energy state or ground state Lyman series Paschen series • if the electron in a H atom is in any other orbit, it is in an excited state

18. The Energy States of the Hydrogen Atom • a very useful result from Bohr’s work is an equation for calculating the energy levels of an atom which were derived from the classical principles of electrostatic attraction and circular motion, where Z is the nuclear charge and n is, as before, a quantum number associated with the electron’s orbit. *Note the negative value in the above equation! The zero of energy is when n is infinity (ionization threshold). E / 10-20 J • for the hydrogen atom, Z=1 so the above equation becomes, and the energy of the ground state (n=1) is

19. to find the energy difference between any two levels, , RH, the Rydberg constant, has a value 2.179 x 10-18 J Using Planck’s equation and the relationship between n and l, n=c/l or

20. looking at the constants replacing RH/hc by this value gives the empirical Rydberg equation which we saw before. • we can use this equation to calculate the energy absorbed or emitted during a transition of the electron from one orbital to another (between two values of n. We can also figure out how much energy is required to completely remove the electron from the atom (ionize) starting from the ground state, ninitial = 1 to nfinal = infinity. a positive value since energy is absorbed to ionize a mole of hydrogen atoms,

21. Let’s look at Which transition is of greater energy? ni = 2 to nf = 3 or ni = 2 to nf = 4 ni = 2 to nf = 3 or ni = 3 to nf = 4 or

22. Without the use of a calculator, indicate which of the following transitions in the hydrogen atom results in the emission of light of the greatest energy? 1. n=4 to n=3 4. n=2 to n=1 2. n=1 to n=2 5. n=1 to n=3 3. n=3 to n=2

23. n=1 n=2 n=3 Which of the following transitions produces light with the longest wavelength? 1. n=1 to n=3 2. n=1 to n=2 3. n=3 to n=2 4. n=2 to n=1 5. n=2 to n=3

24. Limitations of the Bohr Model • despite the huge success in explaining the hydrogen atom, the Bohr model failed to predict the line spectra of any other atom even helium! • the Bohr model is a one electron model. That is, it works very well for the hydrogen atom and any other species with one electron such as He+, Li2+, Be3+, B4+, C5+, N6+, O7+, etc. Each of these cations has only one electron and, so, is called a hydrogen-like ion. eg. Determine and compare the wavelengths of photons absorbed by H, He2+, Li2+ and O7+ for the transition of the electron from n=2 to n=3. Solution.

25. for O7+, for Li2+, wavelength increases energy (and frequency) increases for He for H

26. eg. Determine the wavelength of light emitted by a hydrogen atom for the transition of an electron from n=7 to n=3 and n=3 to n=1.

27. eg. What electron transition for a hydrogen atom, ending in the n=5 orbit, will produce light of wavelength 3740 nm?

28. eg. The Pfund series of the hydrogen spectrum has, as its longest wavelength component a line at 7460 nm. What is the quantum number common to this series?

29. Bohr’s theory does not work for many electron atoms because there are additional nuclear-electron attraction and electron-electron repulsions to consider. • as well, we shall see, that electrons do not travel in fixed orbits, electron motion is less clearly defined. Taken as a picture of the atom, the Bohr model is incorrect. • however, we still use the terms “ground state” and “excited state” in the context we have discussed and retain one of the central ideas of the Bohr model, the energy of an atom occurs in discrete levels