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Bohr's Theory and Spectra of Hydrogen And ____________________________________________________________________________________________________ Subject Chemistry Paper No and Title 8 and Physical Spectroscopy Module No and Title 7 and Bohr’s theory and spectra of hydrogen and hydrogen- like ions. Module Tag CHE_P8_M7 CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 7 (BOHR’S THEORY AND SPECTRA OF HYDROGEN AND HYDROGEN-LIKE IONS) ____________________________________________________________________________________________________ TABLE OF CONTENTS 1. Learning Outcomes 2. What is a Spectrum? 2.1 Introduction 2.2 Atomic Spectra 2.3 Hydrogen Spectrum 2.4 Balmer Series 2.5 The Rydberg Formula 3. Bohr’s Theory 3.1 Bohr Model 3.2 Line Spectra 3.3 Multi-electron Atoms 4. Summary CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 7 (BOHR’S THEORY AND SPECTRA OF HYDROGEN AND HYDROGEN-LIKE IONS) ____________________________________________________________________________________________________ 1. Learning Outcomes After studying this module, you should be able to grasp the important ideas of Bohr’s Theory of the structure of atomic hydrogen. We will first review the experimental observations of the atomic spectrum of hydrogen, which led to the birth of quantum mechanics. 2. What is a Spectrum? 2.1 Introduction In spectroscopy, we are interested in the absorption and emission of radiation by matter and its consequences. A spectrum is the distribution of photon energies coming from a light source: • How many photons of each energy are emitted by the light source? Spectra are observed by passing light through a spectrograph: • Breaks the light into its component wavelengths and spreads them apart (dispersion). • Uses either prisms or diffraction gratings. The absorption and emission of photons are governed by the Bohr condition ΔE = hν. This relation and the prism (or diffraction grating) which casts white light into its components with known wavelengths bent by known angles, give us the tools we need to peer into the inner workings of atoms! Not merely check their size (atomic microscopy) but rather determine experimentally their internal energy states! How? By giving them a kick and seeing what light falls out. And the light that falls out is not smeared out over many wavelengths but rather concentrated into several lines in atomic spectra. Those lines mean incredibly precise energy differences between atomic states since any energy the atom loses, the photon gains (by conservation of energy). Modern atomic theory arose out of studies of the interaction of radiation with matter. 2.2 Atomic Spectra Atomic spectra are line spectra. The peaks are sharp because only electronic transitions are possible in atoms, whereas in molecules, rotational and vibrational energy changes can also take place simultaneously. Molecular spectra are much more complex, even for the simplest molecule, hydrogen, as shown in Figure 1, where the emission spectrum of hydrogen molecule (top) is much more complex than that of atomic hydrogen (bottom). CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 7 (BOHR’S THEORY AND SPECTRA OF HYDROGEN AND HYDROGEN-LIKE IONS) ____________________________________________________________________________________________________ Figure 1 Emission spectrum of (a) Molecular hydrogen, (b) Atomic hydrogen 2.3 Hydrogen Spectrum Scientists had observed that hot atoms or atoms in an electric discharge emit light. Viewed in a CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 7 (BOHR’S THEORY AND SPECTRA OF HYDROGEN AND HYDROGEN-LIKE IONS) ____________________________________________________________________________________________________ spectrograph, the light appeared as a series of discrete lines characteristic of the particular element (unlike the continuous curve observed for blackbody radiation). Figure2 Partial sketch of the line spectrum of atomic hydrogen 2.4 Balmer Series In 1885, Johann Jakob Balmer analyzed the hydrogen spectrum and found that hydrogen emitted four bands of light within the visible spectrum. Wavelength (nm) Colour 656.2 red 486.1 blue 434.0 blue-violet 410.1 Violet The visible lines in the spectrum of hydrogen are shown below (The violet line is very weak). ~ 2 Balmer showed that for the visible lines of hydrogen, a plot of ν vs. 1/n ; n = 3, 4, 5, ... gave a straight line. The curved cap (~) over the ν indicates that it is expressed in units of cm-1, also called wavenumber. CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 7 (BOHR’S THEORY AND SPECTRA OF HYDROGEN AND HYDROGEN-LIKE IONS) ____________________________________________________________________________________________________ ~ 2 Figure 3 Plot of ν vs. 1/n (n = 3, 4, 5, ...) for the visible emission lines of atomic hydrogen. Balmer’s formula was: 14 ⎛ 4 ⎞ Hz (1) ν = 8.2202 ×10 ⎜1− 2 ⎟ ⎝ n ⎠ and once again n = 3, 4, 5, ... This equation may also be written as: 1 1 ~ ⎛ ⎞ cm-1 (2) ν = 109680⎜ 2 − 2 ⎟ ⎝ 2 n ⎠ The visible emission spectrum of atomic hydrogen is now called the Balmer series. 2.5 The Rydberg Formula Rydberg extended Balmer’s formula as: ~ 1 ⎛ 1 1 ⎞ -1 ν = = R ⎜ − ⎟cm ; (n2 > n1) (3) H ⎜ 2 2 ⎟ λ ⎝ n1 n2 ⎠ CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 7 (BOHR’S THEORY AND SPECTRA OF HYDROGEN AND HYDROGEN-LIKE IONS) ____________________________________________________________________________________________________ to cover the additional lines of the hydrogen spectrum outside of the visible region. The modern -1 value of RH, the proportionality constant, in Rydberg’s formula is 109677.57 cm . 3. Bohr’s Theory 3.1 Bohr Model The line spectrum of atoms could not be explained by classical physics. Once again it required a “quantum hypothesis”, i.e. that the energy values are discrete or quantized. Bohr was the first to realize this. Bohr noted the line spectra of certain elements and assumed the electrons were confined to specific energy states. These were called orbits. These had been proposed earlier by Rutherford, but he could not explain the stability of an electron in an orbit, because classical theory predicted that the electron would eventually fall into the nucleus. Bohr proposed that the energy states are quantized: CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 7 (BOHR’S THEORY AND SPECTRA OF HYDROGEN AND HYDROGEN-LIKE IONS) ____________________________________________________________________________________________________ Since the energy states are quantized, the light emitted from excited atoms must appear as line spectra. After lots of math, Bohr showed that −18 ⎛ 1 ⎞ E = −2.18 ×10 ⎜ ⎟ Joule (4) ⎝ n 2 ⎠ where n is the principal quantum number (i.e., n = 1, 2, 3, … and nothing else). The constant in 4 3 the expression is the Rydberg constant expressed in Joule, and is equal to µee /(8ε 0ch ) . The beauty of this expression (besides its wonderful simplicity) is that Niels Bohr was able to derive it from the attraction of an electron for a proton (coulombic attraction) and the centripetal force of the revolving electron merely by presuming that not only energy came in discrete states (n isn't continuous, but an integer) but so did the angular momentum. " L = mvr = n! Here, n has the same meaning as in equation (4). ! = h / 2π is called the reduced Planck constant or the Dirac constant, and angular momentum is expressed in units of this constant. CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 7 (BOHR’S THEORY AND SPECTRA OF HYDROGEN AND HYDROGEN-LIKE IONS) ____________________________________________________________________________________________________ The other beauty of his orbits is the wave nature of the electron makes for an elegant explanation to quantized orbits around the atom. Consider what a wave looks like around an orbit. If we require the wave to fit an integral number of times on the circumference of the orbit in order to avoid interference and apply the de Broglie hypothesis, we have nh nh 2πr = nλ = = p mv nh or mvr = = n! 2π which is the same as Bohr’s arbitrary postulate. We thus have CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 7 (BOHR’S THEORY AND SPECTRA OF HYDROGEN AND HYDROGEN-LIKE IONS) ____________________________________________________________________________________________________ Bohr’s energy level diagram for the hydrogen atom (using equation (4)) would look something like the one shown in Figure 4. The first orbit in the Bohr model has n = 1, is closest to the nucleus, and has negative energy by convention (-13.6 eV). The furthest orbit in the Bohr model has n close to ∞ and corresponds to zero energy (unbound state). Figure 4 Bohr’s energy level diagram for the hydrogen atom. 4.2 Line Spectra Colours from excited gases arise because electrons move between energy states in the atom. Electrons in the Bohr model can only move between orbits by absorbing and emitting energy in quanta (hν). The amount of energy absorbed or emitted on movement between states is given by ΔE = Ef − Ei = hv hc ⎛ 1 1 ⎞ ΔE = hv = = −2.18 ×10 −18 J⎜ − ⎟ ⎜ 2 2 ⎟ λ ⎝ nf ni ⎠ CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 7 (BOHR’S THEORY AND SPECTRA OF HYDROGEN AND HYDROGEN-LIKE IONS) ____________________________________________________________________________________________________ • When ni > nf, energy is emitted. • When nf > ni, energy is absorbed Emission and absorption spectra complement each other. Emission spectra can be used to identify elements, since every element has a unique emission spectrum. An absorption spectrum can also
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