Lecture 15: Bohr Model of the Atom • Reading: Zumdahl 12.3, 12.4 • Outline – Emission spectrum of atomic hydrogen. – The Bohr model. – Extension to higher atomic number. Light as Quantized Energy • Comparison of experiment to the “classical” prediction: Classical prediction is for significantly higher intensity as smaller wavelengths than what is observed. “The Ultraviolet Catastrophe” Light as Quantized Energy • Planck found that in order to model this behavior, one has to envision that energy (in the form of light) is lost in integer values according to: DE = nhn frequency Energy Change n = 1, 2, 3 (integers) h = Planck’s constant = 6.626 x 10-34 J.s Light as a ‘Particle’ • As frequency of incident light is increased, kinetic energy of emitted eincreases linearly. 0 1 2 men hn photon 2 n0 Frequency (n) = energy needed to release e- Interference of Light • Shine light through a crystal and look at pattern of scattering. • Diffraction can only be explained by treating light as a wave instead of a particle. Particles as waves • Electrons shine through a crystal and look at pattern of scattering. • Diffraction can only be explained by treating electrons as a wave instead of a particle. Emission Photon Emission • Relaxation from one energy level to another by emitting a photon. • With DE = hc/l • If l = 440 nm, DE= 4.5 x 10-19 J Emission spectrum of H “Quantized” spectrum DE DE “Continuous” spectrum Any DE is possible Only certain DE are allowed Emission spectrum of H (cont.) Light Bulb Hydrogen Lamp Quantized, not continuous Emission spectrum of H (cont.) We can use the emission spectrum to determine the energy levels for the hydrogen atom. Balmer Model • Joseph Balmer (1885) first noticed that the frequency of visible lines in the H atom spectrum could be reproduced by: 1 1 n 2 2 2 n n = 3, 4, 5, ….. • The above equation predicts that as n increases, the frequencies become more closely spaced. Rydberg Model • Johann Rydberg extends the Balmer model by finding more emission lines outside the visible region of the spectrum: 1 1 n Ry 2 2 n1 n 2 n1 = 1, 2, 3, ….. n2 = n1+1, n1+2, … Ry = 3.29 x 1015 1/s • This suggests that the energy levels of the H atom are proportional to 1/n2 The Bohr Model • Niels Bohr uses the emission spectrum of hydrogen to develop a quantum model for H. • Central idea: electron circles the “nucleus” in only certain allowed circular orbitals. • Bohr postulates that there is Coulombic attraction between e- and nucleus. However, classical physics is unable to explain why an H atom doesn’t simply collapse. The Bohr Model (cont.) • Bohr model for the H atom is capable of reproducing the energy levels given by the empirical formulas of Balmer and Rydberg. 2 Z = atomic number (1 for H) Z 18 E 2.178x10 J 2 n n = integer (1, 2, ….) • Ry x h = -2.178 x 10-18 J (!) The Bohr Model (cont.) 2 Z 18 E 2.178x10 J 2 n • Energy levels get closer together as n increases • at n = infinity, E = 0 The Bohr Model (cont.) • We can use the Bohr model to predict what DE is for any two energy levels DE E final E initial 1 1 18 18 DE 2.178x10 J n 2 (2.178x10 J)n 2 initial final 1 1 DE 2.178x1018 J n 2 n 2 final initial The Bohr Model (cont.) • Example: At what wavelength will emission from n = 4 to n = 1 for the H atom be observed? 1 1 DE 2.178x1018 J n 2 n 2 final initial 1 4 1 DE 2.178x10 J1 2.04x1018 J 16 18 18 DE 2.04x10 J hc l l 9.74 x108 m 97.4nm The Bohr Model (cont.) • Example: What is the longest wavelength of light that will result in removal of the e- from H? 1 1 DE 2.178x1018 J n 2 n 2 final initial 1 DE 2.178x1018 J0 1 2.178x1018 J 18 DE 2.178x10 J hc l l 9.13x108 m 91.3nm Extension to Higher Z • The Bohr model can be extended to any single electron system….must keep track of Z (atomic number). 2 Z 18 E 2.178x10 J 2 n Z = atomic number n = integer (1, 2, ….) • Examples: He+ (Z = 2), Li+2 (Z = 3), etc. Extension to Higher Z (cont.) • Example: At what wavelength will emission from n = 4 to n = 1 for the He+ atom be observed? 1 1 DE 2.178x1018 JZ 2 n 2 n 2 final initial 2 1 4 1 DE 2.178x10 J41 8.16x1018 J 16 hc 18 l 2.43x108 m 24.3nm DE 8.16x10 J l l H l He 18 Where does this go wrong? • The Bohr model’s successes are limited: • Doesn’t work for multi-electron atoms. • The “electron racetrack” picture is incorrect. • That said, the Bohr model was a pioneering, “quantized” picture of atomic energy levels.