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558 Analytical Chemistry 2.0Why does a non-linear molecule have3N – 6 vibrational modes? Consider amolecule of methane, CH 4 . Each of thefive atoms in methane can move in oneof three directions (x, y, and z) for a totalof 5 3 = 15 different ways in which themolecule can move. A molecule can movein three ways: it can move from one placeto another, what we call translational motion;it can rotate around an axis; and itsbonds can stretch and bend, what we callvibrational motion.Because the entire molecule can move inthe x, y, and z directions, three of methane’s15 different motions are translational. Inaddition, the molecule can rotate aboutits x, y, and z axes, accounting for threeadditional forms of motion. This leaves15 – 3 – 3 = 9 vibrational modes.A linear molecule, such as CO 2 , has3N – 5 vibrational modes because it canrotate around only two axes.where n is the vibrational quantum number, which has values of 0, 1, 2,…, and n 0 is the bond’s fundamental vibrational frequency. The value of n 0 ,which is determined by the bond’s strength and by the mass at each endof the bond, is a characteristic property of a bond. For example, a carboncarbonsingle bond (C–C) absorbs infrared radiation at a lower energy thana carbon-carbon double bond (C=C) because a single bond is weaker thana double bond.At room temperature most molecules are in their ground vibrationalstate (n = 0). A transition from the ground vibrational state to the firstvibrational excited state (n = 1) requires absorption of a photon with anenergy of hn 0 . Transitions in which Dn is ±1 give rise to the fundamentalabsorption lines. Weaker absorption lines, called overtones, result fromtransitions in which Dn is ±2 or ±3. The number of possible normal vibrationalmodes for a linear molecule is 3N – 5, and for a non-linear moleculeis 3N – 6, where N is the number of atoms in the molecule. Not surprisingly,infrared spectra often show a considerable number of absorption bands.Even a relatively simple molecule, such as ethanol (C 2 H 6 O), for example,has 3 9 – 6, or 21 possible normal modes of vibration, although not allof these vibrational modes give rise to an absorption. The IR spectrum forethanol is shown in Figure 10.16.UV/Vis Sp e c t r a f o r Mo l e c u l e s a n d Io n sThe valence electrons in organic molecules and polyatomic ions, such asCO 3 2– , occupy quantized sigma bonding, s, pi bonding, p, and non-bonding,n, molecular orbitals (MOs). Unoccupied sigma antibonding, s*, andpi antibonding, p*, molecular orbitals are slightly higher in energy. Becausethe difference in energy between the highest-energy occupied MOs and100percent transmittance80604020Figure 10.16 Infrared spectrum of ethanol.04000 3000 2000 1000wavenumber (cm-1)
Chapter 10 Spectroscopic Methods559Table 10.5 Electronic Transitions Involving n, s, and pMolecular OrbitalsTransition Wavelength Range Examplesσ→ σ
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(a)(b)Phase 2Phase 12.95 3.00 3.05
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Detailed Table of ContentsPreface..
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4C.4 Uncertainty for Mixed Operatio
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8B.1 Theory and Practice . . . . .
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13. Kinetic Methods .. . . . . . .
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Appendix 5: Critical Values for the
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Chapter 1 Introduction to Analytica
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