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Atomic Line Spectra: the Bohr model Line Spectra of Excited Atoms Niels Bohr’s great contribution to science was in building a simple model of the atom. It was based on observations of the SHARP LINE SPECTRA of excited atoms. • Excited atoms emit light of only certain wavelengths • The wavelengths of emitted light depend on the element. High E Short λ High ν Visible lines in the H atom spectrum are called the BALMER series. See jersey.uoregon.edu/vlab/elements/Elements.html for an interactive line spectrum periodic table. See screen 7.6 on the CD-ROM Feb. 22, 2006 Feb. 22, 2006 Atomic Atomic Spectra Spectra and and Bohr Bohr Atomic Atomic Spectra Spectra and and Bohr Bohr Bohr said that this classical view was wrong. He saw the need for a new theory — now called QUANTUM or WAVE MECHANICS. –An e- can only exist in certain discrete orbits — called stationary states. –An e- is restricted to QUANTIZED (discrete) energy states. –The energy of a state = - (Rhc)/n2 = - (const)/n2 One (incorrect) view of atomic structure in early 20th century was that an electron (e-) traveled about the nucleus in an orbit. + Feb. 22, 2006 Low E Long λ Low ν Electron orbit That view: 1. Any orbit should be possible and so should any energy. 2. But a charged particle moving in a circle should emit energy. The end result should be destruction! where n = quantum no. = 1, 2, 3, 4, .… and the constants R = 1.1×107 m-1, h = 6.6×10-34 J·sec and c = 3.0 ×108 m/sec Feb. 22, 2006 Page 1 Atomic Atomic Spectra Spectra and and Bohr Bohr Energy of a quantized state = - Atomic Atomic Spectra Spectra and and Bohr Bohr If e-’s are in quantized energy states, then ∆E between states can have only certain values. This explains sharp line spectra. Rhc/n2 • Only orbits where n = integer number are permitted. • Radius of allowed orbits = 0.0529·n2 nm • Results can be used to explain atomic spectra, at least for simple atoms. E = -Rhc (1/2 2 ) n=2 E = -Rhc (1/12 ) n=1 See CD-ROM screen 7.7 See www.colorado.edu/physics/2000 Feb. 22, 2006 Feb. 22, 2006 Atomic Atomic Spectra Spectra and and Bohr Bohr E = -Rhc (1/22 ) n=2 2 = -Rhc (1/1 ) n=1 E N E R G YE Atomic Atomic Spectra Spectra and and Bohr Bohr E = -C (1/22 ) n=2 2 = -C (1/1 ) n=1 E N E R G YE ∆E = -(3/4)Rhc The values of R, the Rydberg constant, h, the Planck constant and c are known from experiment. Rhc = 2.179×10-18 J/atom so, E of emitted light = (3/4)Rhc = 1.63×10-18 J. This corresponds to a frequency ν = E/h of 2.47 × 1015 sec-1 and λ = c/ν = 121.6 nm. This is exactly in agreement with experiment! Calculate ∆E for e- “falling” from higher energy level (n = 2) to lower energy level (n = 1). ∆E = Efinal - Einitial = -Rhc[(1/12) - (1/2)2] ∆E = -(3/4)Rhc The atom’s final energy is less than its initial energy, thus the atom is losing energy. Feb. 22, 2006 Feb. 22, 2006 Page 2 Atomic Atomic Spectra Spectra and and Bohr Bohr Quantum or Wave Mechanics Bohr received the Nobel Prize in 1922 for his theory. However, there are problems — • theory is successful only for H. • quantum idea artificially introduced. • So, we go on to QUANTUM or WAVE MECHANICS to understand the atom de de Broglie Broglie (1924) (1924) proposed proposed that that all moving objects all moving objects have have wave wave properties. properties. For For electrons electrons orbiting orbiting aa nucleus, nucleus, he theorized he theorized that that aa standing standing wave wave is is set set up up in in each each orbit. orbit. Because Because each each orbit orbit has has aa predicted radius, the predicted radius, the wavelength wavelength of of aa moving moving particle particle is is λλ == h/mv h/mv (v (v is is velocity). velocity). This is the quantum part This is the mechanics part Feb. 22, 2006 Feb. 22, 2006 Quantum or Wave Mechanics Quantum or Wave Mechanics Schrödinger applied the idea of an e- behaving as a wave to the problem of electrons in atoms. He developed what is called the Baseball (115 g) at 100 mph: λ = h/mv = (6.6×10-34 J·sec) / [(0.115 kg) (45 m/sec)] = 1.3×10-34 m = 1.3×10-25 nm Experimental proof of wave properties of electrons see CD-ROM screen 7.8 e- with velocity of 1.9 × 108 cm/sec: λ = 3.88×10-10 m = 0.388 nm WAVE EQUATION Feb. 22, 2006 Feb. 22, 2006 Page 3 The solution to the wave equation gives a set of mathematical expressions called WAVE FUNCTIONS, Ψ Each describes an allowed energy state of an eQuantization comes naturally from the mathematics.