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(BQ) Part 2 book Physical chemistry has contents: Rotational and vibrational spectroscopy, electronic spectroscopy of molecules, magnetic resonance spectroscopy, statistical mechanics, experimental kinetics and gas reactions, chemical dynamics and photochemistry,...and other contents.
13 Rotational and Vibrational Spectroscopy 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 The Basic Ideas of Spectroscopy Einstein Coefficients and Selection Rules Schro¨dinger Equation for Nuclear Motion Rotational Spectra of Diatomic Molecules Rotational Spectra of Polyatomic Molecules Vibrational Spectra of Diatomic Molecules Vibration–Rotation Spectra of Diatomic Molecules Vibrational Spectra of Polyatomic Molecules Raman Spectra Special Topic: Fourier Transform Infrared Spectroscopy Molecular spectroscopy is a powerful tool for learning about molecular structure and molecular energy levels The study of rotational spectra gives us information about moments of inertia, interatomic distances, and angles Vibrational spectra yield fundamental vibrational frequencies and force constants Electronic spectra yield electronic energy levels and dissociation energies The types of spectroscopic transitions that can occur are limited by selection rules As in the case of atoms, the principal interactions of molecules with electromagnetic radiation are of the electric dipole type, and so we will concentrate on them Magnetic dipole transitions are about 10 times weaker than electric dipole transitions, and electric quadrupole transitions are about 108 times weaker Although the selection rules limit the radiative transitions that can occur, molecular collisions can cause many additional kinds of transitions Because of molecular collisions the populations of the various molecular energy levels are in thermal equilibrium 13.1 The Basic Ideas of Spectroscopy 13.1 THE BASIC IDEAS OF SPECTROSCOPY When an isolated molecule undergoes a transition from one quantum eigenstate with energy E1 to another with energy E2 , energy is conserved by the emission or absorption of a photon The frequency of the photon is related to the difference in energies of the two states by Bohr’s relation, h ϵ hc˜ ͉ סE1 Ϫ E2 ͉ (13.1) where we have used the symbol ˜ ( ס1/) introduced in Chapter for the transition energy in wave numbers (SI unit mϪ1 , but usually cmϪ1 is used) The wave number ˜ is the number of waves per unit length If E1 Ͼ E2 , the process is photon emission; if E1 Ͻ E2 , the process is photon absorption The frequency range of photons, or the electromagnetic spectrum, is classified into different regions according to custom and experimental methods as outlined in Table 13.1 By measuring the frequency of the photon, we can learn about the eigenstates of the molecule being studied This is called molecular spectroscopy The frequency of the photon in the absorption or emission process often tells us the kinds of molecular transitions that are involved In the radio-frequency region (very low energy), transitions between nuclear spin states can occur (see Chapter 15) In the microwave region, transitions between electron spin states in molecules with unpaired electrons (Chapter 15) and, in addition, transitions between rotational states can take place In the infrared region, transitions between vibrational states take place (with and without transitions between rotational states) In the visible and ultraviolet regions, the transitions occur between electronic states (accompanied by vibrational and rotational changes) Finally, in the far ultraviolet and X-ray regions, transitions occur that can ionize or dissociate molecules Table 13.1 ␥ rays X-rays Vacuum UV Near UV Visible Near IR Mid IR Far IR Microwaves Radio waves Regions of the Electromagnetic Spectrum Wavelength in Vacuo, 0 Wave Number in Vacuo, ˜ Frequency, Photon Energy, h Molar Energy, NA h 10 pm 10 nm 200 nm 380 nm 780 nm 2.5 m 50 m mm 100 mm 1000 mm 109 cmϪ1 106 cmϪ1 50.0 ϫ 103 cmϪ1 26.3 ϫ 103 cmϪ1 12.8 ϫ 103 cmϪ1 4.00 ϫ 103 cmϪ1 200 cmϪ1 10 cmϪ1 0.1 cmϪ1 0.01 cmϪ1 30.0 EHz 30.0 PHz 1.50 PHz 789 THz 384 THz 120 THz 6.00 THz 300 GHz 3.00 GHz 300 MHz 19.9 ϫ 10Ϫ15 J 19.9 ϫ 10Ϫ18 J 993 ϫ 10Ϫ21 J 523 ϫ 10Ϫ21 J 255 ϫ 10Ϫ21 J 79.5 ϫ 10Ϫ21 J 3.98 ϫ 10Ϫ21 J 199 ϫ 10Ϫ24 J 1.99 ϫ 10Ϫ24 J 0.199 ϫ 10Ϫ24 J 12.0 GJ/mol 12.0 MJ/mol 598 kJ/mol 315 kJ/mol 153 kJ/mol 47.9 kJ/mol 2.40 kJ/mol 120 J/mol 12.0 J/mol 1.2 J/mol IR, infrared; UV, ultraviolet The abbreviations for powers of 10 are given inside the back cover of the book Source: IUPAC Report, “Names, Symbols, Definitions, and Units for Quantities in Optical Spectroscopy,” 1984 Example 13.1 Calculation of the energy of light Calculate the energy in joules per quantum, electron volts, and joules per mole of photons of wavelength 300 nm 459 460 Chapter 13 Rotational and Vibrational Spectroscopy h ס hc (6.62 ϫ 10Ϫ34 J s)(3 ϫ 108 m sϪ1 ) ס ס6.62 ϫ 10Ϫ19 J (300 ϫ 10Ϫ9 m) ( ס6.62 ϫ 10Ϫ19 J)/(1.602 ϫ 10Ϫ19 J eVϪ1 ) ס4.13 eV NA h ( ס6.02 ϫ 1023 molϪ1 )(6.62 ϫ 10Ϫ19 J) ס398 kJ molϪ1 We shall see that the energy eigenvalues of a molecule can be written as E סEr םEv םEe (13.2) where Er is the rotational energy, Ev the vibrational energy, and Ee the electronic energy When the molecule undergoes a transition to another state with the emission or absorption of a single photon of frequency , then h ( סErЈ Ϫ ErЈЈ) ( םEvЈ Ϫ EvЈЈ) ( םEeЈ Ϫ EeЈЈ) (13.3) The primes refer to the state of higher energy and the double primes to the state of lower energy The classification of the various regions of the electromagnetic spectrum by the type of transition given above is possible because, in general, ErЈ Ϫ ErЈЈ ϽϽ EvЈ Ϫ EvЈЈ ϽϽ EeЈ Ϫ EeЈЈ (13.4) That is, electronic energy level differences are much greater than vibrational energy level differences, which are much greater than rotational energy level differences Electronic transitions are often in the visible and ultraviolet part of the spectrum; vibrational transitions are in the infrared, and rotational transitions are in the far infrared and microwave regions 13.2 EINSTEIN COEFFICIENTS AND SELECTION RULES The spectrum of a molecule consists of a series of lines at the frequencies corresponding to all the possible transitions Let us consider the transition from state to state The strength or intensity of a spectral line depends on the number of molecules per unit volume Ni that were in the initial state (the population density of that state) and the probability that the transition will take place Einstein postulated that the rate of absorption of photons is proportional to the density of the electromagnetic radiation with the right frequency The radiant energy density is the radiant energy per unit volume, so it is expressed in J mϪ3 (See Section 9.16.) The spectral radiant energy density as a function of frequency is the measure of the radiant energy of a particular frequency; it is given by סd /d (13.5) Ϫ3 Thus, is expressed in J s m The energy density at the frequency required to excite atoms or molecules from E1 to E2 is represented by (12 ) Thus Einstein’s postulate about the rate of absorption of photons is summarized by the rate equation dt dN1 סϪB12 (12 )N1 (13.6) abs where B12 is the Einstein coefficient for stimulated absorption The SI unit for B12 is m kgϪ1 (Note that N1 can be taken as dimensionless or expressed in mϪ3 ) There is a minus sign because N1 decreases when electromagnetic radiation is absorbed Note that dN1 /dt סϪdN2 /dt 13.2 Einstein Coefficients and Selection Rules Excited atoms or molecules not remain in excited states indefinitely, and Einstein postulated two processes for their return to the initial state, namely, spontaneous emission and stimulated emission, as illustrated in Fig 13.1 The rate of spontaneous emission is given by (here N2 is the population density of state 2) dt סϪA21 N2 where A21 is the Einstein coefficient for spontaneous emission The SI unit for A21 is sϪ1 The rate of spontaneous emission is independent of the radiation density, and the radiation is emitted in random directions with random phases Stimulated emission is quite different in that its rate is proportional to (12 ), and the electromagnetic wave that is produced adds in phase and direction (i.e., coherently) to the stimulating wave The rate of stimulated emission is indicated by the rate equation סϪB21 (12 )N2 A21N2 Spontaneous emission B21N2 ρν∼ (ν∼12) (13.8) stim where B21 is the Einstein coefficient for stimulated emission The interesting feature in stimulated emission is that it amplifies the radiation density According to equation 13.8, incident light with frequency 12 causes more radiation to be produced with exactly the same frequency and direction as long as there are molecules in state As we will discuss later in more detail, this is the basis for a laser, which is the acronym for “light amplification by stimulated emission of radiation.” Equations 13.6–13.8 have been written for the three separate processes, but of course all three can occur in a system at the same time so that the whole rate equation is dN1 dN2 סϪ סϪB12 (12 )N1 םA21 N2 םB21 (12 )N2 dt dt (13.9) This rate equation leads to several interesting conclusions The first is that the three Einstein coefficients are related to each other This can be seen by considering the equilibrium situation in which dN1 /dt סϪdN2 /dt ס0 When the system is in equilibrium, equation 13.9 can be solved for the equilibrium spectral radiant energy density (12 ) to obtain (12 ) ס B12N1 ρν∼ (ν∼12) (13.7) spont dN2 Stimulated absorption dN2 dt 461 A21 (N1 /N2 )B12 Ϫ B21 (13.10) When the system is in equilibrium, the ratio N1 /N2 is given by the Boltzmann distribution (Section 16.1) When E2 is the energy of the higher level and E1 is the energy of the lower level, the Boltzmann distribution shows that N2 סN1 eϪ(E2 ϪE1 )/kT (13.11) Since E2 Ϫ E1 is positive, most of the atoms or molecules will be in the lower energy level at thermal equilibrium If the system is exposed to electromagnetic radiation with frequency 12 , where h12 סE2 Ϫ E1 , the equilibrium distribution can be written as N2 סexp(Ϫh12 /kB T ) N1 (13.12) Stimulated emission Figure 13.1 Definition of Einstein coefficients 462 Chapter 13 Rotational and Vibrational Spectroscopy Replacing N1 /N2 in equation 13.10 with the Boltzmann distribution yields A21 (12 ) ס (13.13) B12 eh12 /kB T Ϫ B21 This equation must be in agreement with Planck’s blackbody distribution law (equation 9.2), 8h (12 /c )3 (12 ) סh /kT (13.14) Ϫ1 e 12 because they both apply to a system at equilibrium Comparison of equation 13.13 with equation 13.14 indicates that B12 סB21 (13.15) and A21 ס 8h12 B21 c3 (13.16) Thus a measurement of any one of the three Einstein coefficients yields all three The second conclusion from equation 13.9 is that the time course of the irradiation can be calculated Since B12 סB21 , these symbols can be replaced by B, and since there is no A12 , A21 can be replaced by A N1 can be replaced by Ntotal Ϫ N2 , where Ntotal סN1 םN2 , and equation 13.9 can be integrated (see Problem 13.4) to obtain N2 B (12 ) ס (13.17) Θ1 Ϫ exp ͕Ϫ[A ם2B (12 )]t ͖Ι Ntotal A ם2B (12 ) At t ס0, there are no excited atoms or molecules But if the radiation density is held constant, N2 /Ntotal rises to an asymptotic value of B (12 )/[A ם2B (12 )] The interesting thing about this asymptotic value is that it is necessarily less than 1/2 because A Ͼ This means that irradiation of a two-level system can never put more atoms or molecules in the higher level than in the lower level This may be a surprise, but the significance of the conclusion is that laser action cannot be achieved with a two-level system In order to obtain laser action, stimulated emission must be greater than the rate of absorption so that amplification of radiation of a particular frequency is obtained This requires that B21 (12 )N2 Ͼ B12 (12 )N1 (13.18) Since B12 סB21 , laser action can be obtained only when N2 Ͼ N1 This situation is referred to as a population inversion The way population inversion can be achieved is discussed in the next chapter Quantum mechanics provides the means to calculate Anm (and Bnm ) between states n and m in terms of the transition dipole moment Anm (and Bnm ) is proportional to the square of the transition dipole moment nm , defined by nm ס Ύ ˆ ء n m d (13.19) where ˆ is the quantum mechanical dipole moment operator for the molecule: ˆ סΑ qi ri (13.20) i where the sum is over all the electrons and nuclei of the molecule, qi is the charge, and ri is the position of the ith charged particle To understand how the transition 13.2 Einstein Coefficients and Selection Rules moment enters, we can think of the molecule interacting with the electric field of the radiation because of a transient or fluctuating dipole moment given by equation 13.19 From equation 13.19, we see that if the transition dipole moment vanishes (usually because of symmetry), the spectral line has no intensity The rules governing the nonvanishing of nm are called selection rules, and these allow us to make sense out of observed molecular spectra If the transition moment from state n to state m is nonzero and there is enough population in the initial state, then the spectral line will be seen in the spectrum The quantum mechanical derivation of the relationship between the Einstein coefficients and the transition probability is too advanced for this book;* however, the final results are given here When the ground state and excited states have degeneracies of g1 and g2 , the Einstein coefficient A is given by Aס 16 g1 ͉12 ͉2 3⑀0 hc g2 (13.21) This equation indicates that the rate of spontaneous emission, A12 N2 , increases rapidly with frequency; as a matter of fact, this rate is negligible in the microwave and infrared regions, and so only absorption spectra are measured In the visible and ultraviolet regions spontaneous emission is significant, and both emission and absorption spectra are measured The Einstein coefficient B is given by B ס 2 g1 ͉12 ͉2 3h ⑀0 g2 (13.22) If the rate of spontaneous emission is negligible, the net rate of absorption is given by rate2 Y סB21 N1 ˜ (˜ 21 ) Ϫ B12 N2 ˜ (˜ 21 ) ( סN1 Ϫ N2 )B˜ (˜ 21 ) (13.23) This shows that if the populations of the two states are equal, there will be no net absorption of radiation We can also think of A12 as a measure of the lifetime of state Consider molecules in (excited) state with no radiation field present (and so no stimulated emission) The molecules will make a transition to state 1, emitting a photon frequency ˜ 21 , with a probability A12 N2 Every time this occurs, N2 decreases After a time t , the number of molecules per unit volume in state is given by N2 (t ) סN2 (0) eϪA12 t סN2 (0) eϪt / (13.24) Ϫ1 where we have defined the lifetime סA12 Actually, if a molecule in state can also make transitions to states 3, 4, (with photons of frequency ˜ 23 , ˜ 24 , ), then the total radiative lifetime is given by סΑ A 2i i (13.25) If other decay processes besides radiative transitions are possible (such as nonradiative transitions) we must add those rates to equation 13.25 to get the total decay rate (inverse lifetime) *See J Steinfeld, Molecules and Radiation Cambridge, MA: MIT Press, 1985 463 464 Chapter 13 Rotational and Vibrational Spectroscopy Example 13.2 Radiative lifetimes and transition moments The radiative lifetime of a hydrogen atom in its first excited level (2p) is 1.6 ϫ 10Ϫ9 s What is the magnitude of the electronic transition moment 21 for this transition? The degeneracy g2 of the 2p level is [˜ ( ס2.46 ϫ 1015 sϪ1 )/(2.998 ϫ 108 m sϪ1 ) ס8.21 ϫ 106 mϪ1 ] 1/2 21 ס ΄ 3h⑀0 g2 16 ˜ 21 ΅ ס ΄ (3)(6.626 ϫ 10Ϫ34 J s)(8.854 ϫ 10Ϫ12 C NϪ1 mϪ2 )(3) 16 (8.21 ϫ 106 mϪ1 )3 (1.6 ϫ 10Ϫ9 s) 1/2 ΅ ס10.9 ϫ 10Ϫ30 C m A dipole moment of this magnitude corresponds to a distance from the proton to the electron of r ס 10.9 ϫ 10Ϫ30 C m 1.6 ϫ 10Ϫ19 C ס68.1 pm This transition dipole moment can be visualized as the movement of an electron 68.1 pm/52.9 pm ס1.29 Bohr radii 13.3 ¨ SCHRODINGER EQUATION FOR NUCLEAR MOTION We saw in Chapter 11 that the Schro¨dinger equation for a molecule can be treated in the Born–Oppenheimer approximation so that the electronic Hamiltonian is that for fixed nuclei, while the Hamiltonian for nuclear motion contains the kinetic energy operator of the nuclei and the electronic energy (as a function of the nuclear coordinates) as the potential energy operator: h¯ םE(R ) Hˆ סϪ ٌR 2 (13.26) In the absence of external fields (such as magnetic or electric fields), the potential energy term E(R ) can depend only on the relative positions of the nuclei, not on where the molecule is placed or on the orientation of the molecule in space The kinetic energy operator consists of the kinetic energy of the center of mass (leading to the translational energy of the molecule), the kinetic energy associated with rotational motion, and the kinetic energy of the vibrational motion Thus, to a very good approximation, we may write H סHtr םHrot םHvib (13.27) where the translational and rotational Hamiltonians contain only kinetic energy terms, while the vibrational Hamiltonian contains E(R ), the potential energy depending on the internuclear distances These internuclear distances are the vibrational coordinates of the molecule If the Hamiltonian is the sum of three terms, one for each kind of motion, then the wavefunction can be written as a product of wavefunctions: סtr rot vib (13.28) 13.4 Rotational Spectra of Diatomic Molecules The Schro¨dinger equations for the three terms are Hˆ tr tr סEtr tr Hˆ rot rot סErot rot Hˆ vib vib סEvib vib (13.29) (13.30) (13.31) The translational wavefunction is that for a free particle (or particle in a very large box) with a mass equal to the mass of the molecule The translational eigenvalues are very closely spaced and cannot be probed in molecular spectroscopy, so we will neglect them in our discussions To understand the number of coordinates required to describe a polyatomic molecule, consider the following The total number of coordinates needed to describe the locations of the N atoms in a molecule is 3N However, to describe the internal motions in a molecule, we are not interested in its location in space, and so the three coordinates required to specify the position of the center of mass of the molecule can be subtracted, leaving 3N Ϫ coordinates To describe the rotational motions of a molecule, we are interested in its orientation in a coordinate system The orientation of a diatomic or linear molecule with respect to a coordinate system requires two angles, so this leaves 3N Ϫ coordinates to describe the internal motions The orientation of a nonlinear polyatomic molecule with respect to a coordinate system requires three angles, so this leaves 3N Ϫ coordinates to describe the internal motions These 3N Ϫ or 3N Ϫ internal motions are referred to as vibrational degrees of freedom To sum up, for a diatomic molecule, Hˆ rot depends only on two angles, and (see equation 9.153); Hˆ vib depends only on R , the internuclear separation For polyatomic molecules, Hˆ vib is more complex, depending on 3N Ϫ coordinates for nonlinear molecules and 3N Ϫ coordinates for linear molecules We will now turn to a description of the rotational and vibrational eigenstates of both diatomic and polyatomic molecules 13.4 ROTATIONAL SPECTRA OF DIATOMIC MOLECULES To a first approximation the rotational spectrum of a diatomic molecule may be understood in terms of the Schro¨dinger equation for rotational motion of the rigid rotor (equation 9.142) The wavefunctions are the spherical harmonics YJM(, ), and there are two quantum numbers J and M for molecular rotation The energy eigenvalues are given by Er ס h¯ J (J ם1) 2I J ס0, 1, 2, M סϪJ, , 0, , םJ (13.32) where I is the moment of inertia (Section 9.11) Since the energy does not depend on M , the rotational levels are (2 J ם1)-fold degenerate In spectroscopy it is standard to express the energies of various levels in wave numbersbydividing E by hc andreferringtothesevaluesas termvalues Termvalues areusuallygivenincmϪ1 ,buttheSIunitforatermvalueismϪ1 Atildewillbeusedto indicate the wave numbers in cmϪ1 Rotational term values are represented by F˜ (J ) סEr /hc , so that the rotational term values for a diatomic molecule are given by Er J (J ם1)h F˜ (J ) ס ס סJ (J ם1)B˜ hc 8 Ic (13.33) 465 466 Chapter 13 Rotational and Vibrational Spectroscopy where the rotational constant is written h 8 Ic B˜ ס (13.34) where c is the speed of light, 2.998 ϫ 1010 cm sϪ1 The rotational energy levels for a rigid diatomic molecule are given in Fig 13.2 in terms of the rotational constant According to the Born–Oppenheimer approximation (Section 11.1), the wavefunction for a molecule in the electronic state e , the vibrational state v , and having a particular set of rotational quantum numbers JM can be written as a product e v JM The transition moment for an electric dipole transition from a rotational state JM to a rotational state J ЈM Ј of the same electronic state is therefore given by ΎΎΎ Ј ء ء ء ˆ e v JM e v J MЈ de drot dvib (13.35) where ˆ is the dipole moment operator Note that only the rotational function (e) has changed in the transition The permanent dipole moment of a molecule in this electronic state is equal to the expectation value of the operator over the wavefunction for the electronic state: Ύ ˆ d ء e (e) 0 ס e (13.36) e Thus, equation 13.35 becomes ΎΎ Ј (e) ء ء v J M Ј v JM drot dvib Quantum number Energy ∼ 20B J=4 (13.37) Relative population ∼ 9e– 20h cB/kt ∼ 8B J=3 ∼ 12B ∼ 7e– 12h cB/kt ∼ 6B ∼ 5e– 6h cB/kt ∼ 2B ∼ 3e– 2h cB/kt ∼ 1e–h cB/kt ∼ 6B E J=2 ∼ 4B ∼ 2B J=1 J=0 J=1 ∼ 2B 0 ∼ 2B ∼ 2B ν Figure 13.2 Rotational levels for a rigid diatomic molecule and the absorption spectrum that results from ⌬ J ס1 The energies and relative populations of the two levels are indicated on the right The transitions are labeled by the upper of the two J values involved Note that the degeneracies of the levels have been taken into account in the population, and the intensities of the lines depend on the relative populations 13.4 Rotational Spectra of Diatomic Molecules The integral over the vibrational coordinate yields the permanent dipole moment in that particular vibrational state For simplicity, we will write it as , so that the final result for the integral is Ύ Ј ء J M Ј JM drot (13.38) A molecule has a rotational spectrum only if this integral is nonzero Thus, the gross selection rule for rotational spectra is that a molecule must have a permanent dipole moment to emit or absorb radiation in making a transition between different states of rotation This is expected from the fact that a rotating dipole produces an oscillating electric field that can interact with the oscillating field of a light wave A homonuclear diatomic molecule such as H2 or O2 does not have a dipole moment, so it does not show a pure rotational spectrum Heteronuclear diatomic molecules have dipole moments, so they have rotational spectra Polyatomic molecules are discussed in the next section To find the specific selection rules we need to find the conditions on the quantum numbers that make the integral in equation 13.38 nonzero For a linear molecule it can be shown that the transition moment is nonzero for ⌬ J סϮ1 ⌬M ס0, Ϯ1 This selection rule may be understood in the same way as that for atoms (Section 10.14) Since a photon has one unit of angular momentum, and angular momentum must be conserved in emission or absorption, the angular momentum of a molecule must change by a compensating amount The frequencies ˜ of the absorption lines due to J y J ם1 are given by the difference between rotational term values (equation 13.33): ˜ סF˜ (J ם1) Ϫ F˜ (J ) ([ סJ ם1)(J ם2) Ϫ J (J ם1)]B˜ ס2B˜ (J ם1) J ס0 , 1, 2, (13.39) As shown in Fig 13.2, the frequencies of the successive lines in the rotational spectrum are given by 2B˜ , 4B˜ , 6B˜ , Thus, there is a series of equally spaced lines with separations of 2B˜ A separate series of lines is found for each isotopically different species of a given molecule, because the moments of inertia of isotopically substituted molecules are different We have been talking about diatomic molecules as if they are rigid rotors, but of course they are not As the rotational motion increases, the chemical bond stretches due to centrifugal forces, the moment of inertia increases, and, consequently, the rotational energy levels come closer together This may be taken into account by adding a term to equation 13.33: Er F˜ (J ) ס סB˜ J (J ם1) Ϫ D˜ J (J ם1)2 hc (13.40) The quantity D˜ is the centrifugal distortion constant in wave numbers When centrifugal distortion is taken into account, the frequencies ˜ of the absorption lines due to J y J ם1 are given by ˜ סF˜ (J ם1) Ϫ F˜ (J ) ס2B˜ (J ם1) Ϫ 4D˜ (J ם1)3 J ס0, 1, 2, (13.41) 467 934 Index Boiling point, as a function of pressure, 182 Boltzmann constant, 89, 297 Boltzmann distribution, 461, 468, 542, 569 Bond angles, table, 883 Bond energy, 62 Bond lengths, table, 883 Bond order, 412 Bonding orbital, 399, 402, 412 Born-Oppenheimer approximation, 396 Bosons, 372, 601 Bound states, 350 Boyle temperature, 14 for a van der Waals gas, 20 Brackett series, 354 Bragg equation, 811 Bragg reflection, 812 Branched chain reactions, 674 Branching ratio, 655 Bravais lattices, 807 Brønsted equation, 741 Bubble point line, 187 Bubble point surface, 190 Buckmeisterfullerene, 446 Buffer, preparation, 260 C Cage, solvent, 734 Calorie, definition, 34, 865 Calorimeter, 33 adiabatic, 63 Calorimetry, 63 Canonical ensemble, 598 Canonical partition function, 570 for an ideal gas, 575 Capacitance, 787 Carnot cycle, 95 Catalysis: acid, 740 base, 740 enzyme, 745 general acid, 740 hydrogen ion, 740 Cathode, 222 Cells: with liquid junctions, 224 without liquid junction, 224 Celsius scale, Center of symmetry, 438, 441 Centrifugal distortion constant, 467 Centrosymmetric, 441 Chain reaction: branched, 674 determination of quantum yield, 712 unbranched, 674 Change in the binding of hydrogen ions, in a biochemical reaction, 275 Character of a representation, 452 Character tables, 452 Characteristic rotational temperature, 583 Characteristic vibrational temperature, 580 Charge number, 220 Chemical dynamics, 686 Chemical equations, as matrix equations, 163 Chemical equilibrium, criteria, 139 Chemical potential, 103 as a function of pressure of an ideal gas, 120 four different definitions, 118 of a species at equilibrium, 134 of an ideal gas in a mixture, 122 of gases, liquids, and solids, 179 significance, 118 transformed, 268 when phases have different electric potentials, 222 Chemical reaction, equilibrium condition, 133 Chemical shift, 546, 548 proton, 549 Chemical thermodynamic properties, 870 Chemisorption, 842 Chiral molecules, 529 Cholesteric liquid crystals, 829 Chromophores, 517 Circular birefringence, 530 Circular dichroism, 530 Circularly polarized light, 529 Clapeyron equation, 181 Classical mechanical observables, 307 Clausius theorem, 77 Clausius-Clapeyron equation, 183 Close packing: cubic, 826 hexagonal, 826 Coexistence curve, 181 Colligative properties, 198 Collision complex, 666 Collision density, 628 Collision diameter, 626 Collision frequency, 627 Collision theory, of bimolecular reactions, 687 Collision with a surface, 624 Collisions, of hard-sphere molecules, 626 Colloid osmotic pressure, 767 Combustion of graphite, 59 Commutability, and precision of measurement, 321 Commutative, 308 Commutator, 309 Complementarity, 299 Complex conjugate, 302, 306, 894 Complex numbers, 894 Component, 156, 159 hydrogen, 268 Composition of atmosphere, as a function of height, 25 Compressibility factor, 11 for a van der Waals gas, 19 Compression of a gas, 40 irreversible, 42 Concentration gradient, 730 Conduction band, 833 Conformations, of n -butane, 770 Congruently melting compound, 204 Conjugate pairs, of work, 45 Conjugated molecule, 518 absorption spectrum, 519 Consecutive first-order reactions, 651 Conservation matrix, 164 Conservation: of components, 156 of energy, 35 Conservation, of orbital angular momentum, 388 Conservative system, 324 Continuous wave, 524 Contour surfaces, for one-electron atoms, 360 Convergence limit, 506, 509 Conversion factors, 865 Cooperative binding, 279 Cooperative denaturation, 281 Coordinates, spherical, 887 Correlation diagram, 412, 413 Correlation energy, 376 Correlation spectroscopy (COSY), 560 Correspondence principle, 314 Coulomb integral, 401 Coulomb’s law, 219 Coupling constant, 550 Coupling of biochemical reactions, 276 Covalent crystals, 825 Covalent radii of atoms, 819 Cramer’s rule, 890 Criteria for irreversibility and reversibility, 107 Critical constants: Index of gases, 14 in terms of van der Waals constants, 20 Critical exponents, 21 Critical opalescence, 17 Critical phenomena, 16 Critical point, 15, 17 Critical temperature, for superconductivity, 831 Cross product, 332 Crystal structure data, 818 Crystal structures, classification, 804 Crystallographic point groups, table, 806 Crystals: covalent, 825 hydrogen-bonded, 825 ionic, 824 molecular, 825 Cubic close packing, 826 Cubic expansion coefficient, 126 Cubic lattices, 814 characteristics, table, 827 Curie temperature, 800 Curie’s law, 797 Cyclic integral, 35 D Dalton’s law, 10 de Broglie wavelength, 298, 813, 848, of the electron, 298 Debye temperature, 603 Debye wavelength, 602 Debye-Hu¨ckel constant, 230 Debye-Hu¨ckel theory, 229 Defects, in a surface, 841 Degeneracy, 319, 355 Degree of polymerization, number average, 772 Degrees of freedom, 79, 155, 161 Denaturation: cooperative, 281 DNA, 282 of proteins, 280 Desorption, rate, 843 Detailed balance, 655 Determinants, 889 Dew point line, 187 Dew point surface, 190 Diamagnetism, 800 Diamonds, 183, 816 Diatomic molecule, 520 absorption, 520 fluorescence, 520 phosphorescence, 521 table of constants, 481 vibrational and rotational levels, 484 vibration-rotation spectra, 483 Differential operators, 304 Differentials, 886 exact, 31, 36 inexact, 36 Diffraction methods, 811 Diffusion coefficient, 633, 727 Diffusion: in a liquid, 726 in gases, 633 of a spherical molecule in a liquid, 727 Diffusion-controlled reaction, 734 Dilute real solutions, 194, 198 Dipole moment operator, 453, 462, 466 Dipole moment: allowed symmetry groups, 448 induced, 493 of a transition, 462 magnetic, 363 table, 791 Dipole-induced dipole interaction, 428 Direct spin-spin coupling, 550 Dispersion curve, 795 Dispersion force, 428 Dissociation constants: of magnesium complexes, 262 macroscopic, 280 microscopic, 280 Dissociation energy, 510 equilibrium, 479 of NaCl (g), 427 spectroscopic, 479 Dissociation, of weak acids, 255 Dissociative adsorption, 845 Dissociative chemisorption, 853 Distillation, 191 Distortion polarization, 789 DNA: denaturation, 282 structure, 282 Donnan effect, 767 Doppler broadening, 300, 622 E Early barrier, 695 Efficiency of a heat engine, 95 Effusion, 624 Eigenfunction, 305 for a particle in a box, 315 harmonic oscillator, 327 of hydrogenlike atoms, 356 Eigenvalue, 305 Einstein coefficient, 460 for spontaneous emission, 461 for stimulated absorption, 460 935 for stimulated emission, 461 Einstein temperature, 602 Einstein unit, in photochemistry, 704 Electric conductivity, 731 Electric dipole moment operator, 426 Electric dipole moment, 425 Electric dipole transitions, 388, 504 Electric dipole, transition moment, 476 Electric field strength, 219, 731 Electric mobility, 731 of ions on water, 732 Electric potential, 219 Electric susceptibility, 788 Electrical work, 45 Electrochemical cell, 222 Electrochemical equilibrium, 218 Electrochemical reaction, 225 Electrode reaction, 222 Electrolytes, activity, 227 Electrolytic cell, 222 Electromagnetic spectrum, table, 459 Electromotive force, 225 Electron affinity, 381 Electron configuration, 377 of homonuclear diatomic molecules, 411 Electron density: for atomic hydrogen, 362 for hydrogen molecule ion, 404 in a crystal, 822 in a one-dimensional crystal, 821 Electron emission from surfaces, 849 Electron g factor, 366, 539 Electron in a box, 311 Electron spin resonance (ESR), 562 Electron spin, 364, 539 Electron transfer reactions, 744 Electron, formal, 67 Electroneutrality condition, 220 Electronic absorption spectra, of diatomic molecules, 505 Electronic contributions to thermodynamic properties, 586 Electronic energy levels, 503 for the hydrogen atom, 351 Electronic spectroscopy, of molecules, 502, 517 Electronic states: of hydrogen molecules, 410 of polyatomic molecules, 416 Electrophoresis, 732 Electrostatic factor, 735 Elementary reactions, 659 Elongation work, 45 936 Index Enantiomers, 529 Encounter pairs and solvent cage, 733 End-centered unit cell, 808 Endothermic process, 56 End-to-end distance, 769 Energies of hydrogenlike atoms, in a magnetic field, 367 Energy density, 296 Energy gap, 833 Energy levels: electronic, 503 for a particle in a three-dimensional box, 318 in a magnetic field, 364, 555 of a hydrogen atom, 353 of a hydrogenlike atom, 350 of an electron in a magnetic field, 562 Energy: of a dipole in an electric field, 790 of a harmonic oscillator, 323 of hydrogen molecule ion, 400 of ionization, 354 of light, 459 zero-point, 311 Ensembles, 597 canonical, 598 microcanonical, 598 Enthalpy, 48 as a criterion for spontaneous change, 106 of adsorption, 843 transformed, 269 Enthalpy of fusion, 185 Enthalpy level diagram, 59 Enthalpy of formation, 60 of aqueous hydrogen ions, 67 Enthalpy of neutralization, 65 Enthalpy of reaction, 57, 61 Enthalpy of vaporization, 184 Entropy: as a state function, 75 Boltzmann postulate, 89 calorimetric determination of, 91 in terms of the number of microscopic states, 571 of freezing supercooled water, 86 of fusion, 185 of mixing ideal gases, 87 transformed, 269 Entropy and statistical probability, 88 Entropy and the dispersal of energy, 90 Entropy change: in a process, 80 in a reversible expansion of an ideal gas, 82 in a reversible process, 81 in a system and its surroundings, 83 in an isolated system, 79 in heating an ideal gas, 82 in irreversible processes, 85 Entropy of an ideal gas, as a function of temperature and pressure, 85 Entropy of expansion, in a isolated system, 80 Entropy of vaporization, 81 Enzyme catalysis, 745 Equation of continuity, 728 Equation of state, 8, 12, 104 ideal gas, Equilibria, involving potential differences, 220 Equilibrium composition: effect of inert gases, 150 effect of initial composition, 150 effect of pressure, 150 of a complicated system, 143 Equilibrium condition, for a chemical reaction, 133 Equilibrium constant: apparent, 265 as a function of temperature, 148, 149 calculated from the standard electrode potential, 241 calculation from tables, 143, 144 definition, 134 determination of, 138 for a cell reaction, 226 for a gaseous reaction, 136 for reactions in dilute aqueous solutions, 243 from equilibrium density, 141 from statistical mechanics, 591, 594 in terms of rate constants, 663 of enzyme-catalyzed reactions, 748 of gas reactions, in terms of concentrations, 152 thermodynamic, 137 Equilibrium dissociation energy, 479 Equilibrium, as a function of temperature, 145 between phases, 119, 177 thermal, vapor-liquid, 185 Equipartition, 596 Error function, 728 Ethylene, bonding, 420 Euler formula, 895 Euler’s criterion for exactness, 38 Euler’s theorem, 23, 105 Eutectic temperature, 203 Exact differential, 31, 36 Excited states, 354, 402 of hydrogen molecule ion, 407 Excited molecule, 669 lifetime, 707 Excluded volume, 771 Exothermic process, 56 Expansion of a gas, 41 irreversible, 42 Expansion, Joule-Thomson, 53 Expectation values, 309, 359 Experimental kinetics, 641 Explosion limits, 676 Exponential drop off, 25 Exponentials, 884 Extended Debye-Hu¨ckel equation, 231 Extensive variables, Extent of reaction, 57 as a function of pressure, 140 as a function of temperature, 149 dimensionless form, 140 F Face-centered unit cell, 808 Face-centered cubic lattice, 814 Falloff region, 668 Faraday constant, 220 Femtosecond dynamics of a barrier reaction, 717 Femtosecond transition-state spectroscopy, 714 Fermi energy, 830 Fermi-Dirac distribution function, 830 Fermions, 372, 601 Ferromagnetism, 800 Fick’s first law, 727 Fick’s second law, 728 First law of thermodynamics, 30, 34 First order spectrum, 552 First-order phase transition, 180 First-order reactions, 645 consecutive, 651 reversible, 650 Flash photolysis, 644 Flow chart, for determining point groups, 447 Fluctuations, 17, 603, 781 Fluidity, 726 Fluoresence, 519 quantum yield, 708 spectrum, 522 Flux, 624, 841 Force constant, 323, 475 Index Force, 31 Formal electron, 67 Fourier transform, 564, 821, 895 infrared spectroscopy, 496 NMR spectrometer, 545, 546 Fourier’s theorem, 821 Fractional coordinates, 809, 820 Fractional distillation, 191 Fractional saturation, of hemoglobin and myoglobin, 278 Franck-Condon overlap integral, 508 Franck-Condon principle, 505 Free induction decay (FID), 545, 558 Free radical polymerization, 772 Freely jointed chain model, 768 Freely rotating chain model, 769 Freezing point lowering, 199 Freezing point, as a function of pressure, 182 Frequency domain, 545 Frequency, angular, 323 Frictional coefficient, 725, 778 Fuel cells, 245 Fugacity, 114, 136 coefficient, 116 of a van der Waals gas, 117 Fundamental equation, 102 for the enthalpy, 105 for the Gibbs energy, 107, 133 for the Helmholtz energy, 107 for the internal energy, 103 for the transformed Gibbs energy, 268, 270 with other kinds of work, 110 G Galvanic cell, 222 Gas constant, in various units, 10 recommended value, 10 Gaussian curve, 729 Gaussian functions, 419 Gay-Lussac’s law, General acid catalysis, 740 Gibbs energy, 106 additivity of contributions of species, 109 and non-PV work, 111 as a criterion for spontaneous change, 108 as a function of extent of reaction, 134 dependence on pressure, 112 dependence on temperature, 111 of a reaction system, 138 of an ideal solution, 188 transformed, 267 Gibbs equation for the entropy, 571 Gibbs paradox, 88 Gibbs, 103, 106 Gibbs-Duhem equation, 124 Gibbs-Helmholtz equation, 112, 148 Glass electrode, 244 Glide plane, 808 Gradient, 892 Gravitational acceleration, 23 Greek alphabet, 897 Grotrian diagram, 388 Ground electronic state, 587 Ground state, 353 Ground states, of homonuclear diatomic molecules, table, 414 Group, 443 multiplication table, 444 H Half life, 646 Hamiltonian function, 305 Hamiltonian operator, 304 for electronic motion, 397 for helium atom, 369 for nuclear motion, 464 for the hydrogen atom, 397 vibrational, 489 Hard-sphere potential, 599 Harmonic oscillator, 322 classical, 322 eigenfunctions, 327 quantum mechanical, 325 Hartree energy, 352 Hartree-Fock calculation, 375 Heat capacity, 50 as a function of temperature, 51 at constant pressure, 49 at constant volume, 47 of ideal gases, at high temperatures (table), 597 of solids, in statistical mechanics, 601 relations between, 51 Heat engines, 94 Heat of combustion, 65 Heat of solution, 65 Heat, 31 absorbed in expansion of a gas, 48 sign convention, 34 Heisenberg uncertainty principle, 299 Helium atom: Hamiltonian operator, 369 wave function, 370 Helmholtz energy, 106 937 in terms of the canonical partition function, 570 Henderson-Hasselbalch equation, 259 Henry’s law, 193 Hermann-Mauguin symbols, 806 Hermite polynomials, 326 Hermitian operator, 306 Heterogeneous catalysis, 853 reaction rate, 855 Heterogeneous reactions, 154 Heterogeneous system, Hexagonal close packing, 826 High polymers, 763 Highest occupied molecular orbital (HOMO), 421 Hole, 833 Homogeneous functions, 23 Homogeneous system, Homonuclear diatomic molecules: electron configurations, 411 ground states, 414 Hu¨ckel molecular orbitals: for 1,3-butadiene, 422 for benzene, 424 for ethylene, 420 theory, 419 Hybrid orbitals, 416, 430 Hybrid sp3 orbitals, 417 Hydrogen atom spectrum, 353 Hydrogen bond, 192, 430 Hydrogen component, 268 Hydrogen ion catalysis, 740 Hydrogen ions, standard enthalpy of formation, 67 Hydrogen molecule ion, 398 electron density, 404 energy, 400 Hydrogen-bonded crystals, 825 Hydrogenlike atom, total wave function, 356 Hydrogenlike radial wave function, 350 Hydrogenlike wave functions, real, 357 Hydrogen-oxygen fuel cell, 245 Hydroxyl ions, standard enthalpy of formation, 67 Hyperfine splitting constant, 563 I Ideal gas mixtures, 10 Ideal gas temperature scale, Ideal gas, single-molecule partition function, 574 Ideal solution, 186 Identity element, 438 Identity matrix, 164 938 Index Impact parameter, 631, 688 Improper axis, 442 Improper rotation axis, 438 Independent intensive properties, choices, 157 Indices, of a plane, 809 Indirect spin-spin coupling, 550 Induced dipole moment, 493 Inexact differential, 38 Infrared bond regions, 490 Infrared spectroscopy, Fourier transform, 496 Inhibition, 748 Initial reaction rate, 648 Integral heat of solution, 65 Integrals, 886 Integrated absorption coefficiemt, 515 Integrating factor, 39, 76 Intensity of light absorbed, 704 Intensive state of a system, 156 Intensive variables, Intermolecular energy transfer, rates, 706 Intermolecular forces, 427 Intermolecular potential energy, 429 Internal conversion, 520 Internal energy, 33 of combustion, 65 in terms of the canonical partition function, 570 with respect to a reference state, 33 Internal pressure, 48 Internal rotation, 489 Internuclear distance, 399 from rotational spectra, 468 Internuclear spin-spin coupling, 550 Interstitial site, 827 Intersystem crossing, 520 Intramolecular processes, rates, 706 Intrinsic viscosity, 776 relation to molar mass, 777 Inversion operation, 441 Ion product of water, at several temperatures, 879 Ion product, 255 Ion radii, 815 Ionic bonding, 425 Ionic crystals, 824 Ionic strength, 229 Ionization energy, 354, 378, 480 as a function of atomic number, 380 in electron volts, 380 of hydrogenlike atoms, 354 table, 883 Ionization potential, 480 Irreducible representations, 450 Irreversible compression of a gas, 42 Irreversible expansion of a gas, 42 Isobaric process, 48 Isolated system, Isolated system, spontaneous process, 78 Isolation method, 648 Isomer group, thermodynamic properties, 161, 163 Isomorphous replacement, 824 Isotherm, Isothermal compressibility, 17, 126 Isothermal expansion: of a van der Waals gas, 44 of an ideal gas, 113 J Joule, SI unit, 34 Joule-Thomson coefficient, 53 Joule-Thomson expansion, 53 K Kelvin equation, 208 Kelvin scale, Kinetic energy, rotational, 330 Kinetic theory of gases, 613 Kinetics, 611 in the liquid phase, 724 Klystron, 474 Kronecker delta, 303 L Langmuir adsorption isotherm, 843 assumptions, 846 for a mixture, 844 Laplacian operator, 317, 330 Laplacian, 301, 892 Larmor frequency, 538, 541, 542 Laser, 461, 523 chemical, 526 gas, 525 solid state, 524 Late barrier, 695 Lattice, 804 Laws of thermodynamics: first, 30 second, 74 third, 74, 92 zeroth law, LCAO molecular orbital, 398 Le Chaˆtelier’s principle, 146, 150 Legendre transform, 105, 267, 889 complete, 124 to introduce pH as an independent variable, 267 Lennard-Jones potential, 427, 429, 600, 631 Lever rule, 187 Lifetime, 463 of an excited molecule, 707 singlet, 708 total radiative, 463 Light amplification, 523 Light scattering, 781 Light: elliptically polarized, 532 plane polarized, 532 Line integral, 32 Linear operator, 304 Linear superposition, 309 Liquid crystals, 829 Liquids, structure, 828 Liquid-vapor equilibrium, 185 Logarithms, 884 London force, 428 Lone-pair electrons, 418 Lorentzian, 558 Low energy electron diffraction (LEED), 847 Lowest unoccupied molecular orbital (LUMO), 421 M Maclaurin series, 19, 885 Macromolecules, size and shape, 763 Macroscopic dissociation constants, 280 Magic-angle spinning, 561 Magnetic dipole moment, 363, 366 transitions, 542 vector, 795 Magnetic domains, 800 Magnetic field, 540, 795 rotating, 544 Magnetic flux density, 363, 540, 795 Magnetic properties, of nuclei, 538 Magnetic quantum number, 334 Magnetic resonance, spectroscopy, 537 Magnetic susceptibility, 364, 796 table, 797 units, 798 Magnetization, 542, 795 Magnetogyric ratio, 363, 538, 540 Mark-Houwink equation, 777 Mass average molar mass, 764, 773 Mass fractions, of polymers, 775 Mathematical applications, 895 MathematicaR , 895 Matrices, 163, 892 for a reaction system, 167 Matrix representations, 448 Index of a rotation, 451 Maxwell distribution of speeds, 617 Maxwell relations, 104, 109 applications, 125 Maxwell-Boltzmann distribution, 616 Mean free path, 630 Mean ionic activity coefficient, 228, 230 of hydrochloric acid, 232 of electrolytes, 233 Mean ionic molality, 228 Mean relative speed, 627 Mean speed, 620 Mean-square end-to-end distance, 769 Mechanisms, of chemical reactions, 659 Meissner effect, 831 Melting temperature, of DNA, 283 Membrane potential, 246 Metallic bonds, 825 Metastable states, 389 Michaelis constant, 747 Michaelis-Menten equation, 746 Microcanonical ensemble, 598 Microscopic dissociation constants, 280 Microscopic reversibility, 655 Microwave spectrometer, 473 Miller indices, 809 Mirror plane, 440 Mixing ideal gases, thermodynamic properties, 123 Mobility, of an ion, 730 Molar absorption coefficient, 512 Molar magnetic susceptibilities, 797, 799 Molar mass distributions, 772 Molar mass, from osmotic pressure, 765 from sedimentation and diffusion, 780 mass average, 764 number average, 764 of hemoglobin, 780 Molar polarization, 790 Mole fraction, 11 of a pseudoismer in a pseudoisomer group, 272 of polymers, 775 Mole, Molecular beam: experiments, 702 supersonic, 703 Molecular crystals, 825 Molecular electronic structure, 396 Molecular orbitals: bonding, 399 LCAO, 398 of hydrogen molecule, 407 theory, 398 Molecular properties, of gases (table), 590 Molecularity, 659 Moment of inertia, 329, 335, 469 Momentum, of a photon, 298 Monatomic ideal gas: thermodynamic properties, 52 translational energy, 52 Monod-Wyman-Changeux model, 752 Monolayer, 843 Morse potential, 482 Most probable speed, 620 Multilayer adsorption (BET), 847 Multiple unit cells, 804 Multiplicity, 552 Mutual exclusion rule, 495 N Nanocrystal, 835 Naperian molar absorption coefficient, 512 Natural logarithm, 43 Natural variables, 103, 108 Nematic liquid crystals, 829 Nernst equation, 226 Neutron diffraction, 813 Neutrons, thermal, 813 Newton, SI unit, 31 Newton’s law, 693 Newton-Raphson calculation, 143 Nonbonding molecular orbitals, 517 Nonideal gases, in statistical mechanics, 599 Nonideal mixture, vapor pressure, 192 Nonspontaneous processes, 75 Normal coordinates, 488 Normal modes of vibration, 486, 488 Normalization factor, 303 Normalized wave function, 303 Nuclear g factor, 539 Nuclear magnetic moment, 539 Nuclear magnetic relaxation, 556 Nuclear magnetic resonance (NMR), 541, 537, 540 energy levels, 540 frequency, 541 two-dimensional, 560 Nuclear magnetism, 538 Nuclear magneton, 539 Nuclear spin, table, 538 Null space, 164, 894 Number average degree of polymerization, 772 Number average molar mass, 764 939 Number of degrees of freedom, 22, 157 Number of different phases, 22, 156 Number of independent chemical reactions, 156, 167 Number of independent properties, 158 Number of independent variables, 22, 157 Number of microscopic states, 571 Number of normal modes of vibration, 488 Number of variables, to describe a system, 22, 36 O Oblate top, 470 Octahedral complex, 518 One-electron atoms, contour surfaces, 360 Operator, 304 for the square of the angular momentum, 333 Hamiltonian, 304 Hermitian, 306 linear, 305 quantum mechanical, 306 Optical activity, 529 and symmetry, 448 Optical rotation, 529 Optical rotatory dispersion, 532 Orbital angular momentum, 355, 366, 797 of the hydrogenlike atom, 361 quantum numbers, 411 Orbital radius, 378 Orbital, 356 antibonding, 402 bonding, 402 Greek letter, 405 Order of reaction, 644 Order of a rotation, 439 Orientation polarization, 789 Orthogonal wavefunction, 303 Orthohydrogen, 584 Orthonormal wavefunction, 303 Oscillating chemical reactions, 752 Oscillator strength, 516 Oscillator, harmonic, 322 Osmotic pressure, 200 colloid, 767 of polymer solutions, 764 total, 767 virial equation, 201 Overall order, 644 Overlap integral, 399 940 Index Ozone layer: destruction by chlorine atoms, 714 in the stratosphere, 712 Ozone, decomposition of, 661 P Pair correlation function, 828 Pair interaction potential, 766 Pairs of molecular orbitals from pairs of atomic orbitals, 406 Parahydrogen, 584 Parallel reactions, 655 Paramagnetism, 563, 797 Parity, 388, 399 Partial molar entropy, 180 Partial molar Gibbs energy, 108 Partial molar properties, 23, 180 Partial molar volume, 23 Partial pressure, 11 Particle, in a one-dimensional box, 311 Particle, in a three-dimensional box, 317 Partition function: for an ideal gas, 573 single molecule, 572 Pascal unit of pressure, Pascal’s triangle, 552 Paschen series, 354 Path between two states, 36 Pauli exclusion principle, 371, 374 Pauli principle, 408 Peak speed, 703 Periodic table, 377 Permittivity: of vacuum, 219, 349, 787, 796 relative, 788 table, 788 Perovskite structure, 832 pH, 243 as an independent variable, 267 dependence on ionic strength, 257 Phase diagrams: for one-component systems, 178 for two-component systems, 202 Phase difference, 820 Phase equilibrium, 177 Phase problem, 822, 824 Phase rule, 155, 158 Phase transition: first order, 180 second order, 181 Phosphorescence, 519, 521 quantum yield, 709 spectrum, 522 Photochemical reactions, 710 Photochemistry, 686 principles, 704 Photodissociation coefficient, 713 Photoelectron spectroscopy (PES), 527, 849 Photoelectrons, 527 Photons, 296 Photosynthesis, 718 Physical adsorption, 842 Physical chemical data, list of tables, 868 Physical constants, values, 867 Physical quantities, 5, 863 pK, 255 of weak acids as a function of ionic strength, 258 table, 880 Planck’s blackbody distribution law, 462 Planck’s constant, 297 Point group symmetry, 438 Point group, of a molecule, identification, 443, 444, 445, 447 table, 806 Point of inflection, 17 Poiseuille equation, 725 Poisons, 854 Polarizability, 492 ellipsoid, 493 of a dielectric, 789 table, 791 Polarization, 787 distortion, 789 molar, 790 of a dielectric, 786 orientation, 789 Polarized light, circularly polarized, 529 Polyatomic molecules, electronic spectra, 517 Polymer chain, spatial configuration, 767 Polymer, molar mass, 202 Polymerization: free radical, 772 step-growth, 772 Polyprotic acids, statistical effects, 284 Population inversion, 462, 523 Population: of rotational states, 486 of vibrational levels, 478 Postulates of quantum mechanics, 336 Potential barrier, for adsorption, 843 Potential difference, 45 Potential energy: for a diatomic molecule, 475 for Cl2 , 477 for HCl, 482 for hydrogen molecules, 410 intermolecular, 429 surface, 690 of rotation around an axis, 771 Potentiometer, 223 Power series, 12 Precision of measurement, and commutability, 321 Precursor state, 853 Predissociation, 511 Pre-exponential factor, 657 for bimolecular reactions, 700 Pressure: in SI units, of an ideal gas, 623 partial, 11 Primary kinetic salt effect, 742 Primitive cubic lattice, 814 Primitive unit cell, 804 Principal axes, 440, 470, 471 Principal moments of inertia, 470 Principle of detailed balance, 656 Principles of photochemistry, 704 Probability density, 302, 310, 313, 326, 614 for a random walker, 768 for molecular speeds, 613 in terms of energy, 619 joint, 615 of various speeds, 618 Probability, 302 Processes: adiabatic, 32 endothermic, 56 exothermic, 56 in an isolated system, 78 irreversible, 79 nonspontaneous, 75 spontaneous, 75 types, 79 Products of inertia, 470 Prolate top, 470 Promoters, 854 Propagation reactions, 677 Proper rotation axis, 438 Proteins: denaturation, 280 structure, 280 physical constants, 780 Proton: charge, 220 chemical shift, 549 spins, 542 spin-spin coupling constants, 551 Pseudoisomer group, 272 Index Pulse length, 544 Pulse mode, 524 Pulse sequences, 560 P-V-T surface, 15 Pythagorean theorem, 627 Q Quantized energy levels, 311 Quantum confined structure, 833 Quantum dot, 835 absorption spectra, 836 Quantum mechanical harmonic oscillator, 325 Quantum mechanical operator, 306 Quantum mechanics, 296 postulates, 336 Quantum number: for angular momentum, 331, 334, 355 for spin, 365 magnetic, 334 vibrational, 325 Quantum theory, 295 Quantum well, 833 Quantum wire, 834 Quantum yield, 705 for fluorescence, 708 for phosphorescence, 709 of photochemical reactions, 710 Quasistatic, 32 Quenching process, 709 R Radial functions, 356 Radial probability densities, 376 for the hydrogen atom, 358 Radiant energy density, 297, 340, 460 Radiative lifetime, 464 Radius of curvature, 209 Radius of gyration, 771 Raman apparatus, 492 Raman spectra, 491 resonance, 496 of CO2 , 494 Raman transitions, selection rules, 503 Random walk, 768 Rank of a matrix, 165 Raoult’s law, 186 Rate constant: for a diffusion-controlled reaction, 735 in terms of molecular partition functions, 699 of enzyme-catalyzed reactions, as a function of pH, 750 of ionic reactions, 742 theoretical calculation, 692 Rate: of absorption, of photons, 460 of conversion, 642 of desorption, 843 of reaction, 642 of heterogeneous reaction, 854, 855 of intermolecular energy transfer, 706 of ionization of water, 739 Rate-determining step, 662 Reaction cross section, 688 Reaction enthalpy, 57, 136 calculated from K as a function of T, 145 Reaction entropy, 136 Reaction Gibbs energy, 133, 137 under specified conditions, 135 Reaction heat capacity, 62 Reaction order, 644 Reaction probability, 693, 695 Reaction quotient, 135 Reaction rate, 642 initial, 648 Reaction: activation-controlled, 734 bimolecular, 659 diffusion-controlled, 734 first order, 645 heterogeneous, 154 oscillating, 752 parallel, 655 second order, 646 trimolecular, 659 unimolecular, 659 zero order, 647 Reduced mass, 330, 349 Reduction potentials, 239 Reduction reaction, 222 Reference state, 60 Reflection operation, 440 Refractive index, 792 frequency dependence, 794 Relative humidity, 11 Relative permeability, 796 Relative permittivity, 219 as a function of frequency, 794 of HCl, 792 Relative velocity, 688 Relative viscosity, 776 Relaxation methods, 736 Relaxation time, 646 for a one-step reaction, 738 nuclear magnetic, 556 Renaturation of DNA, 283 Representations: in symmetry, 449 941 irreducible, 450 Resistivity, 731 Resonance integral, 401 Resonance Raman spectroscopy, 496 Resonance signal, 544 Resonate cavity, 524 Reversible adiabatic expansion, of an ideal monatomic gas, 55 Reversible first-order reactions, 650 Reversible process, 42 Rhodopsin, 719 Rigid rotor, 329 Root-mean-square speed, 620 Rotation operation, 439 Rotational constant, 466 Rotational contributions to the thermodynamic properties of an ideal gas, 583 Rotational energy, 330 linear molecule, 472 spherical top, 471 symmetric top, 472 Rotational levels of diatomic molecule, 466 Rotational spectroscopy, 458 of diatomic molecules, 465 of polyatomic molecules, 469 Rotational symmetry, 806 Rotational temperature, characteristic, 583 Rotatory dispersion, 530 RRKM theory, 670, 700 Russell-Saunders coupling, 384 Rydberg constant, 353 in electron volts, 354 S Sackur-Tetrode equation, 85, 578 Saddle point, 690, 698 Scalar product, 332, 891 Scalar sum, 381 Scanning tunneling microscopy (STM), 851 Scattering factor, 819 Scattering of X-rays from a unit cell, 819 Scattering of plane-polarized light, 782 Schoenflies groups, 443 Schoenflies symbols, table, 806 Schrodinger equation, 301 for hydrogen-like atoms, 349 for nuclear motion, 464 for the hydrogenlike atom, in atomic units, 391 time-dependent, 337 942 Index time-independent, 302 Screw axis, 808 Second law of thermodynamics, 74, 77 mathematical statement, 77 two parts, 78 Second order spectrum, 552 Second virial coefficient, 13 Second-order phase transition, 181 Second-order rate constant: for a reaction of two small radicals, 687 for a reaction of two spherical molecules, 689 Second-order reaction, 646 Sedimentation coefficient, 779 Selection rules, 387, 460, 463 for electronic spectroscopy, 504 for rotational spectra, 467 Semiconductor quantum well, 834 Semipermeable membranes, 200 Separability, of a Hamiltonian, 320 Series, 885 Shell, 376 Shielding constant, 547 Shock tube, 644 SI system of units, 863 Sigma bond framework, 420 Single-molecule partition function, 572 for an ideal gas, 574 Singlet lifetime, 708 Slater determinant, 372 Smectic liquid crystals, 829 Solid solutions, 205 Solid, one-dimensional, 432 Solid-state chemistry, 803 Solubility, in ideal solutions, 304 Solution, of simultaneous linear equations, 893 Solvents, “good” and “poor”, 765 Space group, 808 Space lattices, 807 Special point groups, 446 Specific rotation, 531 Specific viscosity, 776 Spectral radiant energy density, 460 Spectrometer, UV-visible, 512 Spectroscopic dissociation energy, 479 Spectroscopy: basic ideas, 459 photoelectron, 527 Raman, 491 rotational, 458 ultraviolet photoelectron (UPS), 527 vibrational, 458 X-ray photoelectron (XPS), 527 Spectrum: of hydrogen atoms, 353 vibrational, 475 Specular reflection, 623 Speed of light, 297 in a medium, 793 Speed, 614 mean, 620 most probable, 620 of sound, 621 root-mean-square, 620 Spherical coordinates, 331, 887 Spherical harmonics, 331 Spherical top, 470 Spin angular momentum, 365, 382 Spin multiplicity, 384 Spin quantum number, 365 of a solid, 798 of molecular oxygen, 799 Spin, electron, 362, 364 Spin-echo techniques, 559 Spin-lattice relaxation, 557 Spin-orbit coupling, 387 Spin-spin coupling: direct, 550 indirect, 550 of protons, 551 Spin-spin splitting, 552 Splitting of energy levels, 364 Spontaneous processes, 75 in an isolated system, 78 Standard deviation, 300, 311, 729 for a particle in a box, 315 Standard electrode potentials, 239 table, 240 Standard electromotive force of a cell, 226 Standard enthalpy of an ionic reaction: as a function of ionic strength, 238 as a function of temperature, 62 Standard enthalpy of formation, 60 as a function of ionic strength, 238 of an isomer group, 162 Standard enthalpy of reaction, 61 as a function of temperature, 61 calculation of, 146, 147 Standard Gibbs energy of an ionic reaction,, 238 Standard Gibbs energy of formation, 143 of an ion as a function of ionic strength, 238 of an isomer group, 162 Standard molar entropies of ions, 237 Standard reaction Gibbs energy, 135 Standard states, 58 Standard thermodynamic properties: of ions, 234 of reactions of ions, 235 of a species, 271 Standard transformed enthalpy, of a pseudoisomer group, 272 Standard transformed Gibbs energies: of formation at various pHs, 274 of hydrolysis, 277 of pseudoisomer group, 272 of species, as a function of ionic strength, 271 Standard transformed properties, calculation of, 271 Stark effect, 474 State functions, 76 State of a system, extensive, 156 intensive, 156 State of an ideal gas, reversible changes, 76 State variables, State: macroscopic, microscopic, Stationary state wavefunction, 303 Statistical effects, in polyprotic acids, 284 Statistical mechanics, 568 Statistical probability, and entropy, 88 Steady-state approximation, 652 Steady-state method, 660 Step-growth polymerization, 772 Stern-Volmer equation, 710 Sticking coefficient, 843 Stirling’s approximation, 573 Stoichiometric number matrix, 164 Stoichiometric number of the electron, 226 Stoichiometric number, 57, 642 Stokes lines, 491 Structural pattern, 804 Structure factor, 820 Structure, of liquids, 828 Subshell, 376 Superconductivity, 830 Supermolecule, 690 Superposition, 309, 340 linear, 309 state, 310 Supersonic molecular beam, 703 Surface area, determination of, 845 Surface coverage, 843 Surface dynamics, 840 Index Surface reactions, theory, 852 Surface reconstruction, 856 Surface tension, 44 effect on vapor pressure, 205 of some liquids, 206 Surface work, 45 Surface, of a solid, 857 Surroundings, in thermodynamics, Susceptibility, electric, 788 Svedberg unit, 779 Symbols, for physical quantities, 899 Symmetric top, 470, 472 Symmetric wave functions, 371 Symmetry element, 438 Symmetry number, 584 Symmetry of crystals, 805 Symmetry operations, 438 applied to wavefunctions, 453 multiplication, 450 Symmetry plane, 438, 440 Symmetry, 437 System: closed, isolated, open, T Taylor series, 475 Temperature scale: ideal gas, Kelvin, Temperature, theta, 766 Temperature-jump apparatus, 738 Tensor, 493 Term symbol, of a molecule, 411 Term values, 465 Termination reactions, 677 Tesla, 795 Test for exactness, 37 Theoretical plates, 192 Thermal conduction, in gases, 633 Thermal equilibrium, Thermal neutrons, 813 Thermal wavelength, 576 Thermochemistry, 56 Thermodynamic equilibrium constant, 137 Thermodynamic potentials, 105 Thermodynamic properties: acid dissociation, 256 at several temperatures, table, 874 from statistical mechanics, 588 of a monatomic ideal gas, 52 of species in biochemical reactions, 266 table, 870 Thermodynamic standard states, 58 Thermodynamic temperature, Thermodynamic variables, Thermodynamics of isomer groups, 161 Theta temperature, 766 Third law of thermodynamics, 74, 92 and the calculation of equilibrium constants, 94 apparent deviations, 93 Third virial coefficient, 13 Tie line, 16, 187 Time domain, 545 Time-dependent Schrodinger equation, 337 Titration curve, of a weak acid, 259 Total angular momentum, 333 of an atom, 383 Total osmotic pressure, 767 Total spin angular momentum, 538 Trajectory of a molecular collision, 694 Transformed chemical potential, 268 in a phase with an electric potential, 221 Transformed enthalpy, 269 Transformed entropy, 269 Transformed Gibbs energy, 267 as a criterion for equilibrium, 267 fundamental equation, 268, 270 in a phase with an electric potential, 221 Transistor, 833 Transition dipole moment, 462 Transition moment, 464, 508 Transition state theory, 697 Transition state, 691 Transition-state spectroscopy, femtosecond, 714 Translational contributions to the properties of an ideal gas, 576 Translational partition function, 576 for a hydrogen atom, 576 Transmittance, 511 Transport coefficients, in gases, 635 Transport phenomena in gases, 632 Transpose of a matrix, 166 Trial wave function, 368 Trimolecular reaction, 659, 670 rate constants, 672 Triple point, 15 of water, Triplet state, 374, 521, 563 Tunneling, 326, 338 Turbidity, 781 Turnover number, 747 943 Two-dimensional NMR, 560 U Ultracentrifuge, 779 Ultraviolet photoelectron spectroscopy (UPS), 527, 850 Unbranched chain reactions, 674 Uncertainty principle, 299 Unimolecular reaction, 659, 667 Unit cell, 804 body centered, face centered, end centered, 808 of NaCl, 815 primitive, 804 multiple, 804 United atom, 412 Units: atomic, 390 derived, 864 non-SI, 865 of physical quantities, 863 supplementary, 864 V Valence band, 833 Valence bond method, 416 Valence electrons, 385 van der Waals constants, 18 in terms of critical constants, 20 van der Waals equation, 17 van der Waals forces, 428 Van’t Hoff equation, 145 Vapor pressure, 16, 182 effect of radius of curvature, 209 effect on surface tension, 205 from effusion, 625 of a nonideal mixture, 192 of ice and water, 184 Variables: extensive, intensive, state, thermodynamic, Variance, 311 Variational method, 368, 409 Vector force, 31 Vector product, 891 Vectors, 891 Velocity distribution, 615 Velocity gradient, 635 Velocity vector, 614 Vibrational and rotational levels, of diatomic molecules, 484 Vibrational contributions to the properties of an ideal gas, 580 944 Index Vibrational coordinates, 464 Vibrational degrees of freedom, 465 Vibrational quantum number, 325 Vibrational spectra, 458 of diatomic molecules, 475 of HCl, 477 of polyatomic molecules, 486 characteristic, 580 Vibrational term value, 476 Vibration-rotation coupling constant, 485 Vibration-rotation spectra, of diatomic molecules, 483 Virial coefficients: from Lennard-Jones potential, 600 relations between, 13 Virial equation, 11 for a van der Waals gas, 19 Virial expansion, 765 Viscosity, 634 of a liquid, 725 of DNA solution, 778 of water, 726 intrinsic, 776 relative, 776 specific, 776 Viscous flow, in gases, 633 Vision, 718 Volume, of a unit cell, 809 W Water, lone-pair electrons, 418 Watson-Crick base pairs, 283 Wave function, 301 symmetric, 372 Wave number, 353, 459 Wave packet, 299, 340, 715 in a parabolic barrier, 342 Wavefunction: antisymmetric, 372 for a particle in a three-dimensional box, 318 normalized, 303 orthogonal, 303 orthonormal, 303 Wavelength, of a particle, 299 Weak acids: dissociation, 255 practical calculations, 257 Web addresses, 898 Wien displacement law, 340 Work, 31 conjugate pairs, 45 electric, 45 from a reversible isothermal expansion, 43 hydrostatic, 45 of adiabatic expansion, 54 of elongation, 45 of compression, 39 of expansion, 39, 44 SI unit, 31 sign convention, 33 various kinds, 44 X X-ray photoelectron spectroscopy (XPS), 527, 850 Z Zeeman effect, 364 Zero matrix, 164 Zero point energy, 311, 325, 580 Zero-order reaction, 647 Zeroth law of thermodynamics, 3, PHYSICAL CONSTANTSa (ROUNDED TO FOUR SIGNIFICANT FIGURES) Quantity Symbol Value Speed of light in vacuum Permeability of vacuum c 0 Permittivity of vacuum ⑀0 Planck constant h /2 Elementary charge Bohr magneton Nuclear magneton Rydberg constant Bohr radius Hartree energy Electron mass Proton mass Neutron mass Deuteron mass Avogadro constant Atomic mass constant Faraday constant Gas constant h ប e B N Rϱ a0 Eh me mp mn md NA mu F R Boltzmann constant Acceleration due to gravity k g 2.998 ϫ 108 m sϪ1 (exact) 4 ϫ 10Ϫ7 N AϪ2 (exact) Ϸ 12.57 ϫ 10Ϫ7 N AϪ2 1/0 c (exact) Ϸ 8.854 ϫ 10Ϫ12 C2 NϪ1 mϪ2 6.626 ϫ 10Ϫ34 J s 1.055 ϫ 10Ϫ34 J s 1.602 ϫ 10Ϫ19 C 9.274 ϫ 10Ϫ24 J TϪ1 5.051 ϫ 10Ϫ27 J TϪ1 1.097 ϫ 107 mϪ1 5.292 ϫ 10Ϫ11 m 4.360 ϫ 10Ϫ18 J 9.109 ϫ 10Ϫ31 kg 1.673 ϫ 10Ϫ27 kg 1.675 ϫ 10Ϫ27 kg 3.344 ϫ 10Ϫ27 kg 6.022 ϫ 1023 molϪ1 1.661 ϫ 10Ϫ27 kg 9.649 ϫ 104 C molϪ1 8.315 J KϪ1 molϪ1 8.315 ϫ 10Ϫ2 L bar KϪ1 molϪ1 1.987 cal KϪ1 molϪ1 8.206 ϫ 10Ϫ2 L atm KϪ1 molϪ1 1.381 ϫ 10Ϫ23 J KϪ1 9.806 65 m sϪ2 a The best values and their uncertainties are given in Appendix B SOME NUMERICAL CONSTANTS AND CONVERSION FACTORS ס3.141 592 65 e ס2.718 281 828 ln x סlog x / log e ס2.302 585 09 log x 101 325 N mϪ2 atmϪ1 105 N mϪ2 barϪ1 1.013 25 bar atmϪ1 2.54 cm inchϪ1 453.6 g lbϪ1 4.184 J calϪ1 (exactly) 1.602 ϫ 10Ϫ19 J eVϪ1 10Ϫ3 m3 LϪ1 133.32 Pa torrϪ1 TABLE OF RELATIVE ATOMIC MASSES 1991a Scaled to the relative atomic mass, Ar (12 C) ס12 Values given here apply to elements as they exist naturally on earth The uncertainties in the values are indicated by the figures given in parentheses, which are applicable to the last digits Relative Atomic Number Name Symbol Atomic Mass Number Name Symbol Atomic Mass 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 Hydrogen Helium Lithium Beryllium Boron Carbon Nitrogen Oxygen Fluorine Neon Sodium Magnesium Aluminum Silicon Phosphorus Sulfur Chlorine Argon Potassium Calcium Scandium Titanium Vanadium Chromium Manganese Iron Cobalt Nickel Copper Zinc Gallium Germanium Arsenic Selenium Bromine Krypton Rubidium Strontium Yttrium Zirconium Niobium Molybdenum Technetium Ruthenium Rhodium Palladium Silver Cadmium Indium Tin Antimony (stibium) Tellurium Iodine Xenon Cesium H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs 1.007 94(7) 4.002 602(2) 6.941(2) 9.012 182(3) 10.811(5) 12.011(1) 14.006 74(7) 15.999 4(3) 18.998 403 2(9) 20.179 7(6) 22.989 768(6) 24.305 0(6) 26.981 539(5) 28.085 5(3) 30.973 762(4) 32.066(6) 35.452 7(9) 39.948(1) 39.098 3(1) 40.078(4) 44.955 910(9) 47.88(3) 50.941 5(1) 51.996 1(6) 54.938 05(1) 55.847(3) 58.933 20(1) 58.693 4(2) 63.546(3) 65.39(2) 69.723(1) 72.61(2) 74.921 59(2) 78.96(3) 79.904(1) 83.80(1) 85.467 8(3) 87.62(1) 88.905 85(2) 91.224(2) 92.906 38(2) 95.94(1) 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 Barium Lanthanum Cerium Praseodymium Neodymium Promethium Samarium Europium Gadolinium Terbium Dysprosium Holmium Erbium Thulium Ytterbium Lutetium Hafnium Tantalum Tungsten Rhenium Osmium Iridium Platinum Gold Mercury Thallium Lead Bismuth Polonium Astatine Radon Francium Radium Actinium Thorium Protactinium Uranium Neptunium Plutonium Americium Curium Berkelium Californium Einsteinium Fermium Mendelevium Nobelium Lawrencium Rutherfordium Dubnium Seaborgium Bohrium Hassium Meitnerium Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr Rf Db Sg Bh Hs Mt 137.327(7) 138.905 5(2) 140.115(4) 140.907 65(3) 144.24(3) Atomic a 101.07(2) 102.905 50(3) 106.42(1) 107.868 2(2) 112.411(8) 114.88(3) 118.710(7) 121.75(3) 127.60(3) 126.904 47(3) 131.29(2) 132.905 43(5) IUPAC Commission on Atomic Weights, Pure Appl Chem 64:1520 (1992) Relative 150.36(3) 151.965(9) 157.25(3) 158.925 34(3) 162.50(3) 164.930 32(3) 167.26(3) 168.934 21(3) 173.04(3) 174.967(1) 178.49(2) 180.947 9(1) 183.84(1) 186.207(1) 190.23(3) 192.22(3) 195.08(3) 196.966 54(3) 200.59(3) 204.383 3(2) 207.2(1) 208.980 37(3) 232.038 1(1) 231.035 88(2) 238.028 9(1) RELATIVE ATOMIC MASSES AND ISOTOPIC ABUNDANCESa,b Z Symbol H He Li Be B C N O 10 F Ne 11 12 Na Mg 13 14 Al Si 14 15 16 Si P S 17 Cl 35 Br 53 I a b A 3* 10 11 12 13 14* 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 35 37 79 81 127 Relative Atomic Mass, ma /u 1.007 825 037(10) 2.014 101 787(21) 3.016 049 286(32) 3.016 029 297(33) 4.002 603 25(5) 6.015 123 2(8) 7.016 004 5(9) 9.012 182 5(4) 10.012 938 0(5) 11.009 305 3(5) 12 (by definition) 13.003 354 839(17) 14.003 241 993(24) 14.003 074 008(23) 15.000 108 978(38) 15.994 914 64(5) 16.999 130 6(8) 17.999 159 39(32) 18.998 403 25(14) 19.992 439 1(5) 20.993 845 3(12) 21.991 383 7(6) 22.989 769 7(9) 23.985 045 0(8) 24.985 839 2(12) 25.982 595 4(10) 26.981 541 3(7) 27.976 928 4(7) 28.976 496 4(9) 29.973 771 7(10) 30.975 363 8(11) 31.972 071 8(6) 32.971 459 1(8) 33.967 867 74(29) 35.967 079 0(16) 34.968 852 729(68) 36.965 902 62(11) 78.918 336 1(38) 80.916 290(6) 126.904 477(5) Isotopic Abundance, x /% 99.985(1) 0.015(1) 0.000 138(3) 99.999 862(3) 7.5(2) 92.5(2) 100 19.9(2) 80.1(2) 98.90(3) 1.10(3) 99.634(9) 0.366(9) 99.762(15) 0.038(3) 0.200(12) 100 90.51(9) 0.27(2) 9.22(9) 100 78.99(3) 10.00(1) 11.01(2) 100 92.23(1) 4.67(1) 3.10(1) 100 95.02(9) 0.75(1) 4.21(8) 0.02(1) 75.77(5) 24.23(5) 50.69(5) 49.31(5) 100 IUPAC Commission on Atomic Weights, Pure Appl Chem 56:653 (1984) An asterisk denotes an unstable nuclide The standard error in parentheses is applicable to the last digits quoted a 1.00000000E00 3.33564095E-05 5.03411250E04 8.06554093E03 2.19474629E05 8.35934612E01 3.49755041E02 6.95038770E-01 2.99792458E04 1.00000000E00 1.50918896E09 2.41798834E08 6.57968386E09 2.50606892E06 1.04853924E07 2.08367381E04 MHz 1.98644746E-05 6.62607550E-10 1.00000000E00 1.60217733E-01 4.35974820E00 1.66054019E-03 6.94770014E-03 1.38065800E-05 aJ 1.23984245E-04 4.13566924E-09 6.24150636E00 1.00000000E00 2.72113961E01 1.03642721E-02 4.33641146E-02 8.61738569E-05 eV 4.55633530E-06 1.51982986E-10 2.29371045E-01 3.67493088E-02 1.00000000E00 3.80879838E-04 1.59360124E-03 3.16682968E-06 Eh 1.19626582E-02 3.99031324E-07 6.02213670E02 9.64853090E01 2.62549996E03 1.00000000E00 4.18400000E00 8.31451121E-03 kJ/mol 2.85914392E-03 9.53707754E-08 1.43932522E02 2.30605423E01 6.27509552E02 2.39005736E-01 1.00000000E00 1.98721587E-03 kcal/mol 1.43876866E00 4.79921566E-05 7.24292330E04 1.16044475E04 3.15773218E05 1.20271652E02 5.03216592E02 1.00000000E00 K Prefix deci centi milli micro nano pico femto atto zeyto yocto Fraction 10Ϫ1 10Ϫ2 10Ϫ3 10Ϫ6 10Ϫ9 10Ϫ12 10Ϫ15 10Ϫ18 10Ϫ21 10Ϫ24 d c m n p f a z y Symbol 10 102 103 106 109 1012 1015 1018 1021 1024 Multiple PREFIXES deka hecto kilo mega giga tera peta exa zetta yotta Prefix da h k M G T P E Z Y Symbol When a physical quantity is expressed in the units in the left-hand column, the factor is the number to multiply by to convert to the units at the head of the columns to the right cmϪ1 ס MHz ס aJ ס eV ס Eh ס kJ/mol ס kcal/mol ס 1Kס cmϪ1 ENERGY CONVERSION FACTORSa ... 89 136 119. 82 0 7918.1 49 793.3 Te /cm Ϫ1 325 . 321 1854.71 28 58.5 559.7 21 69.814 1518 .2 4401 .21 1358.09 26 48.98 29 90.95 23 09.01 21 4.50 28 1 23 58.57 1733.39 20 47.18 366 1904.04 1904 .20 1580.19 1483.5... 13.34 63.0 2. 67 13 .28 8 19.40 121 .34 20 .888 45 .21 8 52. 819 39.644 0.614 1.30 14. 324 14. 122 28 .445 2. 0 14.100 14.075 11.98 12. 9 10.65 84.811 e xe /cm Ϫ1 0.0 821 07 1.8198 14.457 0 .24 39 1.93 128 1 1.6115... 1.1845 0 .23 328 0.30718 0.1689 1.13 ϫ 10Ϫ4 7.89 ϫ 10Ϫ4 0.017318 0.0179 0.0187 1. 62 ϫ 10Ϫ3 0.01 82 0.0171 0.0159 0.0171 0.0 120 6 0. 724 2 ␣˜ e /cm Ϫ1 120 .7 52 121 .56 160.43 96.966 115.077 22 8.10 124 .25 111.99