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Stationary States of the Hydrogen Atom

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Stationary States of the Hydrogen Atom Spectroscopy Hot self-luminous objects light the Sun or a light bulb emit a continuous spectrum of wavelengths. In contract, light emitted in low=pressure gas discharge contains only discrete individual wavelengths, a discrete spectrum . These spectral lines can be recorded and measured using a grating spectroscope . 1 The Hydrogen Spectrum The spectrum of hydrogen, as viewed in the visible region, shows the Balmer series of spectral lines. Balmer did not discover them, but he did notice a regularity in their spacing, and he found by trial and error that they could be represented by the relation: Johann Jakob Balmer (1825 – 1898) red blue violet 2 Classical Physics at the Limit Rutherford’s nuclear atom model matched experimental evidence about the structure of atoms, but it had a fundamental shortcoming: it was inconsistent with Newton’s mechanics and Maxwell’s electromagnetic theory. In particular, on electron orbiting a nucleus would represent an oscillating charge that should radiate a broad spectrum of electromagnetic radiation, not lines. Further, the loss of energy by such radiation should cause the orbiting electron to spiral into the nucleus. As the 20 th century dawned, physicists could not explain the structure of atoms, the stability of matter, discrete spectral lines, emission and absorption spectra differences, or the origins of X-rays and radioactivity. 3 Niels Bohr’s Atomic Model In 1911, after receiving his PhD, Niels Bohr went to Cambridge to work in Rutherford’s laboratory. This was about a year after Rutherford had completed his nuclear model of the atom. Bohr wanted to “fix” the model so that the orbiting electrons would not radiate away their energy. Starting from Einstein’s idea of light quanta, in 1913 he proposed a radically new nuclear model of the atom that made the following assumptions: Niels Henrik David Bohr 1. Atoms consist of negative electrons orbiting a small positive nucleus; (1885-1962) 2. Atoms can exist only in certain stationary states with a particular set 1922 Nobel Prize of electron orbits and characterized by the quantum number n = 1, 2, 3, … 3. Each state has a discrete, well-defined energy En, with E1<E 2<E 3<… 4. The lowest or ground state E1 of an atom is stable and can persist indefinitely. Other stationary states E2, E 3, … are called excited states . 5. An atom can “jump” from one stationary state to another by emitting a photon of frequency f = (((Ef−−−Ei)/h , where Ei,f are the energies of the initial and final states. 6. An atom can move from a lower to a higher energy state by absorbing energy in an inelastic collision with an electron or another atom, or by absorbing a photon. 7. Atoms will seek the lowest energy state by a series of quantum jumps between states until the ground state is reached. 4 The Bohr Model The implications of the Bohr model are: 1. Matter is stable, because there are no states lower in energy than the ground state; 2. Atoms emit and absorb a discrete spectrum of light, only photons that match the interval between stationary states can be emitted or absorbed; 3. Emission spectra can be produced by collisions; 4. Absorption wavelengths are a subset of the emission wavelengths; 5. Each element in the periodic table has a different number of electrons in orbit, and therefore each has a unique signature of spectral lines. 5 Example: Wavelength of an Emitted Photons E E An atom has stationary states with energies j = 4.0 eV and k = 6.0 eV. What is the wavelength of a photon emitted in a quantum jump from state k to state j? hc =×(4.14 10−15 eV s)(3.00 × 10 8 m/s)(10 9 nm/m) = 1,242 eV nm ∆=−=Eatom Ek E j (4.0 eV) − (6.0 eV) =− 2.0 eV; Ephoton = 2.0 eV; c hc (1,242 eV nm) λ === = 621 nm f E photon (2.0 eV) 6 Energy Level Diagrams excited states ground state It is convenient to represent the energy states of an atom using an energy level diagram. Each energy level is represented by a horizontal line at at appropriate height scaled by relative energy and labeled with the state energy and quantum numbers. De-excitation photon emissions are indicated by downward arrows. Absorption excitations are indicated by upward arrows. 7 Clicker Question A photon with a wavelength of 621 nm has an energy of 2.0 eV. For an atom with the energy levels shown, would you expect to see a line with this wavelength in the absorption spectrum? In the emission spectrum? (a) yes, yes; (b) yes, no; (c) no, yes; (d) no, no; (e) cannot tell. 8 The Balmer Series The Balmer series is now understood to be one of several series in hydrogen. They are produced by electron jumps from orbit n to orbit m, with m = 2 for the Balmer series. − 1 1 1 λ =(91.18 nm) − m2 n 2 9 Clicker Question What is the quantum number n of the electron orbit in this hydrogen atom? (a) 2; (b) 3; (c) 4; (d) 6; (e) 12. 10 The Bohr Hydrogen Atom This is a version of the mathematical formulation of the Bohr Model of the atom: F 1e2 v 2 1 e 2 =elec = = ⇒ 2 = aelec 2 v m4πε0 mr r 4 πε 0 mr hh nhn h 2πrnnn= λ = = ⇒ v= =(h ≡ h /2)π p mvn 2π rm rm 1 e2 n h 2 4πε h2 ⇒ 2 0 ∴ = rn = n 2 4πε 0 mr rm me 4πε h2 Bohr radius: a ≡0 =×5.29 10−11 m = 0.0529 nm; B me 2 2 L rnarn== B ;1 0.0529 nm; r 2 = 0.212 nm; r 3 = 0.476 nm; 11 Hydrogen Atom Energy Levels 1 1 q q E=+= K U mv 2 + e p 2 4 πε 0 r 1 1 e2 =mv 2 − 2 4 πε 0 r 1e2 1 e 2 v2 = ⇒ E = − 4πε0mr 4 πε 0 2 r 1 1 e2 2 ⇒ rnan= B E n = − 2 n4πε 0 2 a B 1 e2 EB ≡ = 13.60 eV; 4πε 0 2 aB 1 13.60 eV E=− E =−; n = 1, 2, 3, K nn2 B n 2 12 Example: Stationary States of the Hydrogen Atom Can an electron in a hydrogen atom have a speed of 3.60x10 5 m/s? If so what are its energy and the radius of its orbit? h h n 1 v1 6 vn == =; v 1 =× 2.19 10 m/s rn m n ma B n 6 v1 (2.19× 10 m/s) X n1 = =5 = 6.083 (not an integer, no such state) vn (3.60× 10 m/s) What about a speed of 3.65x10 5 m/s? 6 v1 (2.19× 10 m/s) n2 = =5 = 6.000 (an integer, the state exists) OK vn (3.65× 10 m/s) E E=−=−B 0.38 eV; r =62 a = 1.90 nm 6 62 6 B 13 Binding Energy and Ionization Energy The binding energy of an electron in stationary state n is defined as the energy that would be required to remove the electron an infinite distance from the nucleus. Therefore, the binding energy of n the =1 state of hydrogen is EB = 13.60 eV. It would be necessary to supply 13.60 eV of energy to free the electron from the proton, and one would say that the electron in the ground state of hydrogen is “bound by 13.60 eV”. The ionization energy is the energy required to remove the least bound electron from an atom. For hydrogen, this energy is 13.60 eV. For other atoms it will typically be less. 14 Quantization of Angular Momentum Actually, in constructing his atomic model, Niels Bohr did not assume that an integer number of de Broglie wavelengths fitted into the circumference of the orbit. (Bohr did not know about de Broglie waves when he formulated his theory.) Instead, he assumed that the angular momentum L of the orbit was quantized in units of ħ (= h/2 πππ). h h2π h r 1 L≡ mvr = n h; λ== = = 2 π r ; ∴nλ = 2 π r . p mv nh n Thus, Bohr’s assumption that angular momentum is quantized in units of ħ is equivalent to assuming that an integer number of de Broglie wave lengths of the electron fits into the circumference of a Bohr orbit. Where does L=n ħ come from? It comes from the symmetry that if you rotate an object by 360 O, it should return to the same state. This symmetry produces an angular momentum condition on the electron’s wave function. 15 The Hydrogen Spectrum The figure shows the energy-level diagram for hydrogen. The top “rung” is the ionization limit, which corresponds to n→∞ and to completely removing the electron from the atom. The higher energy levels of hydrogen are crowded together just below the ionization limit. The arrows show a photon absorption 1 →4 transition and a photon emission 4 →2 transition. 16 Transitions ∆E E − E f =atom = n m h h 2 2 111e 11 e m=2 =−2 −− 2 hn42πε0 aB m 42 πε 0 a B 2 2 1e 1 1 4πε h m=1 = − ≡ 0 2 2 aB 2 4πε 0 2 haB m n me − c 8πε hca 1 1 1 λ = =0 B − f e2 mn 2 2 −1 8πε 0hca B −8 1 1 λ ≡ =×9.112 10 m = 91.12 nm λ→ = λ − 0 e2 n m 0 m2 n 2 17 Example: The Lyman α Forest When astronomers look at distant quasars, they find that the light has been strongly absorbed at the local wavelength of the “Lyman ααα” 1 →2 transition of the Lyman series of hydrogen.
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