Quantum Theory and Atomic Structure 2

2. Absence of a time lag. An electron is freed the instant the atom absorbs a photon of enough energy. It doesn’t accumulate energy. • Current will be less in dim light than in bright light since fewer photons with enough energy eject fewer electrons per unit time. Skill 7-2. Interconverting the energy of a photon with its frequency and/or wavelength Problem: A microwave oven has a frequency of 2.45 x 10 9 s –1 . What is the energy of one photon of this microwave radiation? Solution: • The quantum and photon theories ascribed features to radiation that had been reserved for matter: fixed amount and discrete particles. 13 Concept 7-5. How the Bohr theory explained line spectra of the H atom; the importance of discrete atomic energy levels •When an element is vaporized and thermally or electrically excited, it emits light. Hydrogen gas gives a pinkish color. • Produces a line spectrum, a series of fine lines of individual colors. 14 Atomic spectra video clip by Philip Morrison •Spectroscopists discovered several series of H spectral lines in other regions of the electromagnetic spectrum. λ • Each series showed a similar pattern. • Equations of the form below, (Rydberg equation) were found to predict the position and wavelength of any line in a given series: Where n 1 and n 2 are postive integers with n 2 > n 1 , and R is the Rydberg constant (1.096776 x 10 7 m –1 ). 15 • For the series of spectral lines in the visible range, n 1 = 2: • Rydberg equation is empirical, based on data rather than theory. Why do the spectral lines of hydrogen follow this pattern? • Rutherford’s nuclear model did not predict the existence of atomic line spectra. It had to be modified or replaced. • Niels Bohr, working in Rutherford’s laboratory, suggested a model for the hydrogen atom that predicted the observed line spectra. 16 The Bohr Model of the Hydrogen Atom • Bohr used Planck’s and Einstein’s ideas about quantized energy proposing that hydrogen atoms had only certain fixed energy states. • Each of these states was associated with a fixed circular orbit of the electron around the nucleus. • Bohr proposed that atoms do not radiate energy while in one of their fixed (stationary) energy states. • When the electron moves to a different orbit, the atom changes to another energy state. • This happens only by the atom absorbing or emitting a photon. • The photon energy equals the difference in energy between the two stationary states: 17 •Spectral lines represent emission of a photon of specific energy (and ν) as an electron drops from a higher energy state to a lower one. •An atomic spectrum consists of lines instead of a continuum because the atom’s energy has only certain discrete levels, or states. 18 • In Bohr’s model, the quantum number n determines the radius of the electron’s orbit and is directly related to the atom’s energy. • The lower the value of n, the smaller is the radius of the orbit and the lower is the energy level of the atom. •When the electron is in the n = 1 orbit, the atom is in its lowest (first) energy level, called the ground state. • The second energy level (second stationary state) and all higher levels are called excited states. •A hydrogen atom in the 2nd energy level can return to the ground state by emitting a photon of a particular frequency: 19 •When a sample of atomic hydrogen absorbs energy, different H atoms absorb different amounts. • In various atoms all the allowable energy levels (orbits) are populated by electrons. • Transitions from outer orbits to n = 3 gives the infrared series of spectral lines. • Transitions from outer orbits to n = 2 gives the visible series of spectral lines. • Transitions from outer orbits to n = 1 gives the ultraviolet series of lines. 20 The Energy States of the Hydrogen Atom • Bohr obtained an equation giving the energies of the stationary states of the hydrogen atom: • The energy of the ground state (n = 1) is: • He defined zero energy (E = 0) as the state when the electron is totally removed from the nucleus; in other words, when n = ∞. • So E < 0 (i.e. negative) for any smaller n values. • The energy difference between any two energy levels is given by: 21 How much energy is needed to completely remove the electron from the hydrogen atom; that is, what is ∆E for the following change? Calculate this by substituting n final = ∞ and n initial = 1: • ∆E is positive because energy must be absorbed to remove the electron from the positive nucleus. For one mole of H atoms, • This is the ionization energy of the hydrogen atom. 22 •Despite its success in accounting for spectral lines of the H atom, the Bohr model failed to predict the spectrum of any other element. • The model worked well for one-electron species, but not for atoms or ions with more than one electron. • The existence of discrete atomic energy levels is retained from Bohr’s model in the current atomic model. Concept 7-6. The wave-particle duality of matter and energy and the relevant theories and experiments that led to it The Wave-Particle Duality of Matter and Energy •Louis de Broglie proposed a surprising reason for the existence of fixed energy levels in Bohr’s model atoms. • He said atoms are like other systems that display only certain allowed states, such as the wave created by a plucked guitar string. 23