(BQ) Part 2 book Elements of physical chemistry has contents: Quantum theory, quantum chemistry atomic structure, quantum chemistry The chemical bond, materials macromolecules and aggregates, solid surfaces,...and other contents.
Chapter 12 Quantum theory Three crucial experiments 12.1 Atomic and molecular spectra 12.2 The photoelectric effect 12.3 Electron diffraction The dynamics of microscopic systems 12.4 The Schrödinger equation 12.5 The Born interpretation 12.6 The uncertainty principle Applications of quantum mechanics 12.7 Translational motion (a) Motion in one dimension (b) Tunnelling (c) Motion in two dimensions 12.8 Rotational motion (a) Rotation in two dimensions (b) Rotation in three dimensions 12.9 Vibrational motion CHECKLIST OF KEY IDEAS TABLE OF KEY EQUATIONS QUESTIONS AND EXERCISES The phenomena of chemistry cannot be understood thoroughly without a firm understanding of the principal concepts of quantum mechanics, the most fundamental description of matter that we currently possess The same is true of virtually all the spectroscopic techniques that are now so central to investigations of composition and structure Present-day techniques for studying chemical reactions have progressed to the point where the information is so detailed that quantum mechanics has to be used in its interpretation And, of course, the very currency of chemistry—the electronic structures of atoms and molecules—cannot be discussed without making use of quantum-mechanical concepts The role—indeed, the existence—of quantum mechanics was appreciated only during the twentieth century Until then it was thought that the motion of atomic and subatomic particles could be expressed in terms of the laws of classical mechanics introduced in the seventeenth century by Isaac Newton (see Appendix 3), as these laws were very successful at explaining the motion of planets and everyday objects such as pendulums and projectiles However, towards the end of the nineteenth century, experimental evidence accumulated showing that classical mechanics failed when it was applied to very small particles, such as individual atoms, nuclei, and electrons, and when the transfers of energy were very small It took until 1926 to identify the appropriate concepts and equations for describing them Three crucial experiments Quantum theory emerged from a series of observations made during the late nineteenth century As far as we are concerned, there are three crucially important experiments One shows—contrary to what had been supposed for two centuries—that energy can be THREE CRUCIAL EXPERIMENTS 12.1 Atomic and molecular spectra: discrete energies A spectrum is a display of the frequencies or wavelengths (which are related by λ = c/k) of electromagnetic radiation that are absorbed or emitted by an atom or molecule Figure 12.1 shows a typical atomic emission spectrum and Fig 12.2 shows a typical molecular absorption spectrum The obvious hν = E3 – E2 E2 hν = E2 – E1 hν = E3 – E1 E1 Fig 12.3 Spectral lines can be accounted for if we assume that a molecule emits a photon as it changes between discrete energy levels High-frequency radiation is emitted when the two states involved in the transition are widely separated in energy; low-frequency radiation is emitted when the two states are close in energy feature of both is that radiation is absorbed or emitted at a series of discrete frequencies The emission of light at discrete frequencies can be understood if we suppose that Emission intensity • The energy of the atoms or molecules is confined to discrete values, as then energy can be discarded or absorbed only in packets as the atom or molecule jumps between its allowed states (Fig 12.3) • The frequency of the radiation is related to the energy difference between the initial and final states The simplest assumption is the Bohr frequency relation, that the frequency k (nu) is directly proportional to the difference in energy ΔE, and that we can write 415 420 Wavelength, λ /nm ΔE = hk Absorption intensity Fig 12.1 A region of the spectrum of radiation emitted by excited iron atoms consists of radiation at a series of discrete wavelengths (or frequencies) 200 E3 Energy, E transferred between systems only in discrete amounts Another showed that electromagnetic radiation (light), which had long been considered to be a wave, in fact behaved like a stream of particles A third showed that electrons, which since their discovery in 1897 had been supposed to be particles, in fact behaved like waves In this section we review these three experiments and establish the properties that a valid system of mechanics must accommodate (12.1) where h is the constant of proportionality The additional evidence that we describe below confirms this simple relation and gives the value h = 6.626 × 10−34 J s This constant is now known as Planck’s constant, for it arose in a context that had been suggested by the German physicist Max Planck A brief illustration The bright yellow light emitted by sodium atoms in some street lamps has wavelength 590 nm Wavelength and frequency are related by V = c /l, so the light is emitted when an atom loses an energy DE = hc/l In this case, DE = 240 280 Wavelength, λ /nm 320 (6.626 × 10−34 J s) × (2.998 × 108 ms−1) 5.9 × 10−7 m = 3.4 × 10 −19 J or 0.34 aJ (corresponding to 2.1 eV) Fig 12.2 When a molecule changes its state, it does so by absorbing radiation at definite frequencies This spectrum is part of that due to sulfur dioxide (SO2) molecules This observation suggests that molecules can possess only discrete energies, not a continuously variable energy Later we shall see that the shape of this curve is due to a combination of electronic and vibrational transitions of the molecule At this point we can conclude that one feature of nature that any system of mechanics must accommodate is that the internal modes of atoms and molecules can possess only certain energies; that is, these modes are quantized 271 272 CHAPTER 12: QUANTUM THEORY Photoelectrons Energy, E Ek(e–) UV radiation hν Metal Fig 12.4 The experimental arrangement to demonstrate the photoelectric effect A beam of ultraviolet radiation is used to irradiate a patch of the surface of a metal, and electrons are ejected from the surface if the frequency of the radiation is above a threshold value that depends on the metal 12.2 The photoelectric effect: light as particles By the middle of the nineteenth century, the generally acceptable view was that electromagnetic radiation is a wave (see Appendix 3) There was a great deal of compelling information that supported this view, specifically that light underwent diffraction, the interference between waves caused by an object in their path, and that results in a series of bright and dark fringes where the waves are detected However, evidence emerged that suggested that radiation can be interpreted as a stream of particles The crucial experimental information came from the photoelectric effect, the ejection of electrons from metals when they are exposed to ultraviolet radiation (Fig 12.4) The characteristics of the photoelectric effect are as follows: No electrons are ejected, regardless of the intensity of the radiation, unless the frequency exceeds a threshold value characteristic of the metal The kinetic energy of the ejected electrons varies linearly with the frequency of the incident radiation but is independent of its intensity Even at low light intensities, electrons are ejected immediately if the frequency is above the threshold value A brief comment We say that y varies linearly with x if the relation between them is y = a + bx; we say that y is proportional to x if the relation is y = bx These observations strongly suggest an interpretation of the photoelectric effect in which an electron is ejected in a collision with a particle-like projectile, provided the projectile carries enough energy to Photoelectron, e– Free, stationary electron Φ Bound electron Fig 12.5 In the photoelectric effect, an incoming photon brings a definite quantity of energy, hV It collides with an electron close to the surface of the metal target, and transfers its energy to it The difference between the work function, F, and the energy hV appears as the kinetic energy of the ejected electron expel the electron from the metal If we suppose that the projectile is a photon of energy hk, where k is the frequency of the radiation, then the conservation of energy requires that the kinetic energy, Ek, of the electron (which is equal to 12 mev2, when the speed of the electron is v) should be equal to the energy supplied by the photon less the energy Φ (uppercase phi) required to remove the electron from the metal (Fig 12.5): Ek = hk − Φ (12.2) The quantity Φ is called the work function of the metal, the analogue of the ionization energy of an atom Self-test 12.1 The work function of rubidium is 2.09 eV (1 eV = 1.60 × 10−19 J) Can blue (470 nm) light eject electrons from the metal? [Answer: yes] When hk < Φ, photoejection (the ejection of electrons by light) cannot occur because the photon supplies insuAcient energy to expel the electron: this conclusion is consistent with observation Equation 12.2 predicts that the kinetic energy of an ejected electron should increase linearly with the frequency, in agreement with observation When a photon collides with an electron, it gives up all its energy, so we should expect electrons to appear as soon as the collisions begin, provided the photons carry suAcient energy: this conclusion agrees with observation Thus, the photoelectric effect is strong evidence for the particle-like nature of light and the existence of photons Moreover, it provides a route to the determination of h, for a plot of Ek against k is a straight line of slope h THREE CRUCIAL EXPERIMENTS Diffracted electrons Electron beam λ Short wavelength, high momentum Metal λ Fig 12.6 In the Davisson–Germer experiment, a beam of electrons was directed on a single crystal of nickel, and the scattered electrons showed a variation in intensity with angle that corresponded to the pattern that would be expected if the electrons had a wave character and were diffracted by the layers of atoms in the solid Long wavelength, low momentum Fig 12.7 According to the de Broglie relation, a particle with low momentum has a long wavelength, whereas a particle with high momentum has a short wavelength A high momentum can result either from a high mass or from a high velocity (because p = mv) Macroscopic objects have such large masses that, even if they are travelling very slowly, their wavelengths are undetectably short 12.3 Electron diffraction: electrons as waves The photoelectric effect shows that light has certain properties of particles Although contrary to the long-established wave theory of light, a similar view had been held before, but discarded No significant scientist, however, had taken the view that matter is wave-like Nevertheless, experiments carried out in the early 1920s forced people to question even that conclusion The crucial experiment was performed by the American physicists Clinton Davisson and Lester Germer, who observed the diffraction of electrons by a crystal (Fig 12.6) There was an understandable confusion—which continues to this day—about how to combine both aspects of matter into a single description Some progress was made by Louis de Broglie when, in 1924, he suggested that any particle travelling with a linear momentum, p = mv, should have (in some sense) a wavelength λ given by what we now call the de Broglie relation: h λ= p (12.3) The wave corresponding to this wavelength, what de Broglie called a ‘matter wave’, has the mathematical form sin(2πx/λ) The de Broglie relation implies that the wavelength of a ‘matter wave’ should decrease as the particle’s speed increases (Fig 12.7) Equation 12.3 was confirmed by the Davisson–Germer experiment, as the wavelength it predicts for the electrons they used in their experiment agrees with the details of the diffraction pattern they observed Example 12.1 Estimating the de Broglie wavelength Estimate the wavelength of electrons that have been accelerated from rest through a potential difference of 1.00 kV Strategy We need to establish a string of relations: from the potential difference we can deduce the kinetic energy acquired by the accelerated electron; then we need to find the electron’s linear momentum from its kinetic energy; finally, we use that linear momentum in the de Broglie relation to calculate the wavelength Solution The kinetic energy acquired by an electron of charge −e accelerated from rest by falling through a potential difference V is Ek = eV Because Ek = 12 mev2 and p = mev the linear momentum is related to the kinetic energy by p = (2meEk)1/2 and therefore p = (2meeV )1/2 This is the expression we use in the de Broglie relation, which becomes l= h (2meeV )1/ At this stage, all we need is to substitute the data and use the relations C V = J and J = kg m2 s−2: l= = 6.626 × 10−34 J s {2 × (9.110 × 10−31 kg) × (1.602 × 10−19 C) × (1.00 × 103 V)}1/2 6.626 × 10−34 −31 {2 × (9.110 × 10 = 3.88 × 10−11 m Js ) × (1.602 × 10−19) × (1.00 × 103)}1/2 (kg C V)1/2 273 274 CHAPTER 12: QUANTUM THEORY Trajectory The wavelength of 38.8 pm is comparable to typical bond lengths in molecules (about 100 pm) Electrons accelerated in this way are used in the technique of electron diffraction, in which the diffraction pattern generated by interference when a beam of electrons passes through a sample is interpreted in terms of the locations of the atoms Self-test 12.2 Calculate the wavelength of an electron in a 10 MeV particle accelerator (1 MeV = 106 eV; eV (electronvolt) = 1.602 × 10−19 J; energy units are described in Appendix 1) [Answer: 0.39 pm] The Davisson–Germer experiment, which has since been repeated with other particles (including molecular hydrogen and C60), shows clearly that ‘particles’ have wave-like properties We have also seen that ‘waves’ have particle-like properties Thus, we are brought to the heart of modern physics When examined on an atomic scale, the concepts of particle and wave melt together, particles taking on the characteristics of waves, and waves the characteristics of particles This joint wave–particle character of matter and radiation is called wave–particle duality It will be central to all that follows The dynamics of microscopic systems How can we accommodate the fact that atoms and molecules exist with only certain energies, waves exhibit the properties of particles, and particles exhibit the properties of waves? We shall take the de Broglie relation as our starting point, and abandon the classical concept of particles moving along ‘trajectories’, precise paths at definite speeds From now on, we adopt the quantummechanical view that a particle is spread through space like a wave To describe this distribution, we introduce the concept of a wavefunction, ψ (psi), in place of the precise path, and then set up a scheme for calculating and interpreting ψ A ‘wavefunction’ is the modern term for de Broglie’s ‘matter wave’ To a very crude first approximation, we can visualize a wavefunction as a blurred version of a path (Fig 12.8); however, we refine this picture considerably in the following sections Wavefunction Fig 12.8 According to classical mechanics, a particle may have a well-defined trajectory, with a precisely specified position and momentum at each instant (as represented by the precise path in the diagram) According to quantum mechanics, a particle cannot have a precise trajectory; instead, there is only a probability that it may be found at a specific location at any instant The wavefunction that determines its probability distribution is a kind of blurred version of the trajectory Here, the wavefunction is represented by areas of shading: the darker the area, the greater the probability of finding the particle there 12.4 The Schrödinger equation In 1926, the Austrian physicist Erwin Schrödinger proposed an equation for calculating wavefunctions The Schrödinger equation, specifically the timeindependent Schrödinger equation, for a single particle of mass m moving with energy E in one dimension is − H2 d2ψ + V(x)ψ = Eψ 2m dx2 (12.4a) In this expression V(x) is the potential energy; H (which is read h-bar) is a convenient modification of Planck’s constant: H= h = 1.055 × 10−34 J s 2π The term proportional to d2ψ /dx2 is closely related to the kinetic energy (so that its sum with V is the total energy, E) Mathematically, it can be interpreted as the way of measuring the curvature of the wavefunction at each point Thus, if the wavefunction is sharply curved, then d2ψ /dx2 is large; if it is only slightly curved, then d2ψ /dx2 is small We shall develop this interpretation later: just keep it in mind for now You will often see eqn 12.4 written in the very compact form Hˆ ψ = Eψ (12.4b) where ‘Hˆ ψ ’ stands for everything on the left of eqn 12.4a The quantity Hˆ is called the hamiltonian of the system after the mathematician William Hamilton who had formulated a version of classical mechanics that used the concept It is written with a ^ to signify that it is an ‘operator’, something that acts in a particular way on ψ rather than just multiplying it (as E THE DYNAMICS OF MICROSCOPIC SYSTEMS A brief illustration Three simple but important cases, λ sin (2πx/λ) multiplies ψ in Eψ); see Derivation 12.1 You should be aware that a lot of quantum mechanics is formulated in terms of various operators, but we shall not encounter them again in this text.1 For a justification of the form of the Schrödinger equation, see Derivation 12.1 The fact that the Schrödinger equation is a ‘differential equation’, an equation in terms of the derivatives of a function, should not cause too much consternation for we shall simply quote the solutions and not go into the details of how they are found The rare cases where we need to see the explicit forms of its solution will involve very simple functions x/λ • The wavefunction for a particle free to oscillate to-and2 fro near a point is e−x , where x is the displacement from the point • The wavefunction for an electron in the lowest energy state of a hydrogen atom is e−r, where r is the distance from the nucleus As can be seen, none of these wavefunctions is particularly complicated mathematically A justification of the Schrödinger equation We can justify the form of the Schrödinger equation to a certain extent by showing that it implies the de Broglie relation for a freely moving particle By free motion we mean motion in a region where the potential energy is zero (V = everywhere) Then, eqn 12.4a simplifies to 22 d2y = Ey 2m dx Fig 12.9 The wavelength of a harmonic wave of the form sin(2px/l) The amplitude of the wave is the maximum height above the centre line Thus: − 22 d2y 22 d2 sin(kx ) =− 2m dx 2m dx =− as may be verified by substitution of the solution into both sides of the equation and using d cos(kx ) = −k sin(kx ) dx d2 sin(kx ) = −k sin(kx ) dx k 222 22 (−k sin(kx )) = y 2m 2m The final term is equal (according to the Schrödinger equation) to Ey, so we can recognize that E = k 22 2/2m and therefore that k = (2mE )1/2/2 The function sin(kx) is a wave of wavelength l = 2p/k, as we can see by comparing sin(kx) with sin(2px/l), the standard form of a harmonic wave with wavelength l (Fig 12.9) Next, we note that the energy of the particle is entirely kinetic (because V = everywhere), so the total energy of the particle is just its kinetic energy: E = Ek = p2 2m p= 2p h h × = l 2p l which is the de Broglie relation We see, in the case of a freely moving particle, that the Schrödinger equation has led to an experimentally verified conclusion 12.5 The Born interpretation (2mE )1/ k= d sin(kx ) = k cos(kx ) dx (12.5a) A solution of this equation is y = sin(kx) Because E is related to k by E = k 22 2/2m, it follows from a comparison of the two equations that p = k2 Therefore, the linear momentum is related to the wavelength of the wavefunction by Derivation 12.1 − –1 but not putting in various constants are as follows: • The wavefunction for a freely moving particle is sin x, exactly as for de Broglie’s matter wave See, for instance, our Physical chemistry (2006) Before going any further, it will be helpful to understand the physical significance of a wavefunction The interpretation that is widely used is based on a suggestion made by the German physicist Max Born He made use of an analogy with the wave theory of light, in which the square of the amplitude of an electromagnetic wave is interpreted as its intensity and therefore (in quantum terms) as the number of photons present He argued that, by analogy, the square of a wavefunction gives an indication of the probability of finding a particle in a particular region of space To be precise, the Born interpretation asserts that: 275 276 CHAPTER 12: QUANTUM THEORY In other words, ψ is a probability density As for other kinds of density, such as mass density (ordinary ‘density’), we get the probability itself by multiplying the probability density ψ by the volume δV of the region of interest A note on good practice The symbol d is used to indicate a small (and, in the limit, infinitesimal) change in a parameter, as in x changing to x + dx The symbol D is used to indicate a finite (measurable) difference between two quantities, as in DX = Xfinal − Xinitial A brief comment We are supposing throughout that y is a real function (that is, one that does not depend on i, the square-root of −1) In general, y is complex (has both real and imaginary components); in such cases y2 is replaced by y*y, where y* is the complex conjugate of y We not consider complex functions in this book.2 For a small ‘inspection volume’ δV of given size, the Born interpretation implies that wherever ψ is large, there is a high probability of finding the particle Wherever ψ is small, there is only a small chance of finding the particle The density of shading in Fig 12.10 represents this probabilistic interpretation, an interpretation that accepts that we can make predictions only about the probability of finding a particle somewhere This interpretation is in contrast to classical physics, which claims to be able to predict precisely that a particle will be at a given point on its path at a given instant Example 12.2 Interpreting a wavefunction The wavefunction of an electron in the lowest energy state of a hydrogen atom is proportional to e−r/a0, with a0 = 52.9 pm and r the distance from the nucleus (Fig 12.11) Calculate the relative probabilities of finding the electron inside a small cubic volume located at (a) the nucleus, (b) a distance a0 from the nucleus Wavefunction, ψ(r)/ψ (0) The probability of finding a particle in a small region of space of volume δV is proportional to ψ 2δV, where ψ is the value of the wavefunction in the region 0.8 0.6 0.4 0.2 0 Radius, r/a0 Fig 12.11 The wavefunction for an electron in the ground state of a hydrogen atom is an exponentially decaying function of the form e−r /a0, where a0 is the Bohr radius Strategy The probability is proportional to y2dV evaluated at the specified location The volume of interest is so small (even on the scale of the atom) that we can ignore the variation of y within it and write Probability ∝ y2dV Wavefunction (ψ ) and probability density (ψ 2) with y evaluated at the point in question ψ2 ψ Node Fig 12.10 (a) A wavefunction does not have a direct physical interpretation However, (b) its square tells us the probability of finding a particle at each point The probability density implied by the wavefunction shown here is depicted by the density of shading in (c) Solution (a) At the nucleus, r = 0, so there y2 ∝ 1.0 (because e0 = 1) and the probability is proportional to 1.0 × dV (b) At a distance r = a0 in an arbitrary direction, y2 ∝ e−2 × dV = 0.14 × dV Therefore, the ratio of probabilities is 1.0/0.14 = 7.1 It is more probable (by a factor of 7.1) that the electron will be found at the nucleus than in the same tiny volume located at a distance a0 from the nucleus Self-test 12.3 The wavefunction for the lowest energy state in the ion He+ is proportional to e−2r /a0 Repeat the calculation for this ion Any comment? [Answer: 55; a more compact wavefunction on account of the higher nuclear charge] For the role, properties, and interpretation of complex wavefunctions, see our Physical chemistry (2006) THE DYNAMICS OF MICROSCOPIC SYSTEMS ∞ ∞ (b) Region contributes low kinetic energy Position, x Fig 12.12 The observed kinetic energy of a particle is the average of contributions from the entire space covered by the wavefunction Sharply curved regions contribute a high kinetic energy to the average; slightly curved regions contribute only a small kinetic energy There is more information embedded in ψ than the probability that a particle will be found at a location We saw a hint of that in the discussion of eqn 12.4 when we identified the first term as an indication of the relation between the kinetic energy of the particle and the curvature of the wavefunction: if the wavefunction is sharply curved, then the particle it describes has a high kinetic energy; if the wavefunction has only a low curvature, then the particle has only a low kinetic energy This interpretation is consistent with the de Broglie relation, as a short wavelength corresponds to both a sharply curved wavefunction and a high linear momentum and therefore a high kinetic energy (Fig 12.12) For more complicated wavefunctions, the curvature changes from point to point, and the total contribution to the kinetic energy is an average over the entire region of space The central point to remember is that the wavefunction contains all the dynamical information about the particle it describes By ‘dynamical’ we mean all aspects of the particle’s motion Its amplitude at any point tells us the probability density of the particle at that point and other details of its shape tells us all that it is possible to know about other aspects of its motion, such as its momentum and its kinetic energy The Born interpretation has a further important implication: it helps us identify the conditions that a wavefunction must satisfy for it to be acceptable: It must be single valued (that is, have only a single value at each point): there cannot be more than one probability density at each point It cannot become infinite over a finite region of space: the total probability of finding a particle in a region cannot exceed Wavefunction, ψ Wavefunction, ψ Region contributes high kinetic energy (c) (d) (a) Location, x Fig 12.13 These wavefunctions are unacceptable because (a) it is not single-valued, (b) it is infinite over a finite range, (c) it is not continuous, (d) its slope is not continuous These conditions turn out to be satisfied if the wavefunction takes on particular values at various points, such as at a nucleus, at the edge of a region, or at infinity That is, the wavefunction must satisfy certain boundary conditions, values that the wavefunction must adopt at certain positions We shall see plenty of examples later Two further conditions stem from the Schrödinger equation itself, which could not be written unless: The wavefunction is continuous everywhere It has a continuous slope everywhere These last two conditions mean that the ‘curvature’ term, the first term in eqn 12.4, is well defined everywhere All four conditions are summarized in Fig 12.13 These requirements have a profound implication One feature of the solution of any given Schrödinger equation, a feature common to all differential equations, is that an infinite number of possible solutions are allowed mathematically For instance, if sin x is a solution of the equation, then so too is a sin(bx), where a and b are arbitrary constants, with each solution corresponding to a particular value of E However, it turns out that only some of these solutions fulfill the requirements stated above Suddenly, we are at the heart of quantum mechanics: the fact that only some solutions are acceptable, together with the fact that each solution corresponds to a characteristic value of E, implies that only certain values of the energy are acceptable That is, when the Schrödinger equation is solved subject to the boundary conditions that the solutions must satisfy, we find that the energy of the system is quantized (Fig 12.14) 277 278 CHAPTER 12: QUANTUM THEORY Probability density, ψ Acceptable Unacceptable Fig 12.14 Although an infinite number of solutions of the Schrödinger equation exist, not all of them are physically acceptable In the example shown here, where the particle is confined between two impenetrable walls, the only acceptable wavefunctions are those that fit between the walls (like the vibrations of a stretched string) Because each wavefunction corresponds to a characteristic energy, and the boundary conditions rule out many solutions, only certain energies are permissible Position of particle Position, x Fig 12.15 The wavefunction for a particle with a welldefined position is a sharply spiked function that has zero amplitude everywhere except at the particle’s position 21 We have seen that, according to the de Broglie relation, a wave of constant wavelength, the wavefunction sin(2πx/λ), corresponds to a particle with a definite linear momentum p = h/λ However, a wave does not have a definite location at a single point in space, so we cannot speak of the precise position of the particle if it has a definite momentum Indeed, because a sine wave spreads throughout the whole of space we cannot say anything about the location of the particle: because the wave spreads everywhere, the particle may be found anywhere in the whole of space This statement is one half of the uncertainty principle proposed by Werner Heisenberg in 1927, in one of the most celebrated results of quantum mechanics: It is impossible to specify simultaneously, with arbitrary precision, both the momentum and the position of a particle More precisely, this is the position–momentum uncertainty principle: there are many other pairs of observables with simultaneous values that are restricted in a similar way; we meet some later Before discussing the principle further, we must establish the other half: that if we know the position of a particle exactly, then we can say nothing about its momentum If the particle is at a definite location, then its wavefunction must be nonzero there and zero everywhere else (Fig 12.15) We can simulate such a wavefunction by forming a superposition of many wavefunctions; that is, by adding together the amplitudes of a large number of sine functions (Fig 12.16) This procedure is successful because the Wavefunction, ψ 12.6 The uncertainty principle Position, x 21 Fig 12.16 The wavefunction for a particle with an ill-defined location can be regarded as the sum (superposition) of several wavefunctions of different wavelengths that interfere constructively in one place but destructively elsewhere As more waves are used in the superposition, the location becomes more precise at the expense of greater uncertainty in the particle’s momentum An infinite number of waves are needed to construct the wavefunction of a perfectly localized particle The numbers against each curve are the number of sine waves used in the superpositions amplitudes of the waves add together at one location to give a nonzero total amplitude, but cancel everywhere else In other words, we can create a sharply localized wavefunction by adding together wavefunctions corresponding to many different wavelengths, and therefore, by the de Broglie relation, of many different linear momenta The superposition of a few sine functions gives a broad, ill-defined wavefunction As the number of functions increases, the wavefunction becomes sharper THE DYNAMICS OF MICROSCOPIC SYSTEMS Solution From DpDx ≥ 12 2, the uncertainty in position is Dx ≥ (a) 1.054 × 10−34 J s = 2Dp × (1.0 × 10−3 kg) × (1.0 × 10−6 m s−1) = 5.3 × 10−26 m (b) Fig 12.17 A representation of the content of the uncertainty principle The range of locations of a particle is shown by the circles, and the range of momenta by the arrows In (a), the position is quite uncertain, and the range of momenta is small In (b), the location is much better defined, and now the momentum of the particle is quite uncertain because of the more complete interference between the positive and negative regions of the components When an infinite number of components are used, the wavefunction is a sharp, infinitely narrow spike like that in Fig 12.15, which corresponds to perfect localization of the particle Now the particle is perfectly localized, but at the expense of discarding all information about its momentum The quantitative version of the position–momentum uncertainty relation is ΔpΔx ≥ 12 H (12.6) The quantity Δp is the ‘uncertainty’ in the linear momentum and Δx is the uncertainty in position (which is proportional to the width of the peak in Fig 12.16) Equation 12.6 expresses quantitatively the fact that the more closely the location of a particle is specified (the smaller the value of Δx), then the greater the uncertainty in its momentum (the larger the value of Δp) parallel to that coordinate, and vice versa (Fig 12.17) The position–momentum uncertainty principle applies to location and momentum along the same axis It does not limit our ability to specify location on one axis and momentum along a perpendicular axis Example 12.3 Using the uncertainty principle The speed of a certain projectile of mass 1.0 g is known to within 1.0 mm s−1 What is the minimum uncertainty in its position along its line of flight? Strategy Estimate Dp from mDv, where Dv is the uncertainty in the speed; then use eqn 12.6 to estimate the minimum uncertainty in position, Dx, where x is the direction in which the projectile is travelling This degree of uncertainty is completely negligible for all practical purposes, which is why the need for quantum mechanics was not recognized for over 200 years after Newton had proposed his system of mechanics and why in daily life we are completely unaware of the restrictions it implies However, when the mass is that of an electron, the same uncertainty in speed implies an uncertainty in position far larger than the diameter of an atom, so the concept of a trajectory—the simultaneous possession of a precise position and momentum—is untenable Self-test 12.4 Estimate the minimum uncertainty in the speed of an electron in a hydrogen atom (taking its diameter as 100 pm) [Answer: 580 km s−1] The uncertainty principle captures one of the principal differences between classical and quantum mechanics Classical mechanics supposed, falsely as we now know, that the position and momentum of a particle can be specified simultaneously with arbitrary precision However, quantum mechanics shows that position and momentum are complementary, that is, not simultaneously specifiable Quantum mechanics requires us to make a choice: we can specify position at the expense of momentum, or momentum at the expense of position As we shall see, there are many other complementary observables, and if any one is known precisely, the other is completely unknown The uncertainty principle has profound implications for the description of electrons in atoms and molecules and therefore for chemistry as a whole When the nuclear model of the atom was first proposed it was supposed that the motion of an electron around the nucleus could be described by classical mechanics and that it would move in some kind of orbit But to specify an orbit, we need to specify the position and momentum of the electron at each point of its path The possibility of doing so is ruled out by the uncertainty principle The properties of electrons in atoms, and therefore the foundations of chemistry, have had to be formulated (as we shall see) in a completely different way 279 564 DATA SECTION Table D1.2 (continued) M/(g mol −1) DfH /(kJ mol −1) DfG /(kJ mol −1) S m /(J K −1 mol −1) Cp,m /(J K −1 mol −1) Silicon Si(s) Si(g) SiO2(s,a) 28.09 28.09 60.09 +455.6 −910.93 +411.3 −856.64 18.83 167.97 41.84 20.00 22.25 44.43 107.87 107.87 107.87 187.78 143.32 231.74 169.88 +284.55 +105.58 −100.37 −127.07 −31.05 −124.39 +245.65 +77.11 −96.90 −109.79 −11.20 −33.41 42.55 173.00 +72.68 107.1 96.2 121.3 140.92 25.351 20.79 +21.8 52.38 50.79 65.86 93.05 22.99 22.99 22.99 40.00 58.44 102.90 149.89 +107.32 −240.12 −425.61 −411.15 −361.06 −287.78 +76.76 −261.91 −379.49 −384.14 −348.98 −286.06 51.21 153.71 +59.0 64.46 72.13 86.82 98.53 28.24 20.79 +46.4 59.54 50.50 51.38 52.09 32.06 32.06 32.06 64.13 32.06 64.06 80.06 98.08 98.08 96.06 97.07 34.08 34.08 33.072 146.05 +0.33 +278.81 +128.37 +33.1 −296.83 −395.72 −813.99 −909.27 −909.27 −887.34 −20.63 −39.7 −17.6 −1209 +0.1 +238.25 +79.30 +85.8 −300.19 −371.06 −690.00 −744.53 −744.53 −755.91 −33.56 −27.83 +12.08 −1105.3 31.80 32.6 167.82 228.18 −14.6 248.22 256.76 156.90 20.1 +20.1 +131.8 205.79 121 +62.08 291.82 118.69 118.69 118.69 134.69 150.69 +302.1 −8.8 −285.8 −580.7 +267.3 −27.2 −256.8 +519.6 51.55 168.49 −17 56.5 52.3 26.99 20.26 20.786 Silver Ag(s) Ag(g) Ag+(aq) AgBr(s) AgCl(s) Ag2O(s) AgNO3(s) Sodium Na(s) Na(g) Na+(aq) NaOH(s) NaCI(s) NaBr(s) NaI(s) Sulfur S(s, a) (rhombic) S(s, b) (monoclinic) S(g) S2(g) S2−(aq) SO2(g) SO3(g) H2SO4(l) H2SO4(aq) SO 42−(aq) HSO 4− (aq) H2S(g) H2S(aq) HS−(aq) SF6(g) 22.64 23.6 23.673 32.47 39.87 50.67 138.9 −293 −293 −84 34.23 97.28 Tin Sn(s,b) Sn(g) Sn2+(aq) SnO(s) SnO2(s) 44.31 52.59 Xenon Xe(g) 131.30 0 169.68 65.37 65.37 65.37 81.37 +130.73 −153.89 −348.28 +95.14 −147.06 −318.30 41.63 160.98 −112.1 43.64 Zinc Zn(s) Zn(g) Zn2+(aq) ZnO(s) * Entropies and heat capacities of ions are relative to H+(aq) and are given with a sign 25.40 20.79 +46 40.25 DATA SECTION Standard potentials Table D2.1a Standard potentials at 298.15 K in electrochemical order Reduction half-reaction E /V Reduction half-reaction E /V +3.0 +2.87 +2.07 +2.05 +1.98 +1.81 +1.78 +1.69 +1.67 +1.63 +1.61 +1.60 +1.51 +1.51 +1.40 +1.36 +1.33 +1.24 +1.23 +1.23 +1.23 +1.09 +0.97 +0.96 +0.92 +0.89 +0.86 +0.80 +0.80 +0.79 +0.77 −0.37 −0.40 −0.40 −0.41 −0.44 −0.44 −0.48 −0.49 −0.61 −0.74 −0.76 −0.81 −0.83 −0.91 −1.18 −1.19 BrO− + H2O + 2e− → Br − + 2OH− Hg2SO4 + 2e− → 2Hg + SO 2− − − MnO 2− + 2H2O + 2e → MnO2 + 4OH − − 2− MnO + e → MnO I2 + 2e− → 2I− Cu+ + e− → Cu I −3 + 2e− → 3I− NiOOH + H2O + e− → Ni(OH)2 + OH− Ag2CrO4 + 2e− → 2Ag + CrO 2− O2 + 2H2O + 4e− → 4OH− ClO −4 + H2O + 2e− → ClO −3 + 2OH− [Fe(CN)6]3− + e− → [Fe(CN)6]4− Cu2+ + 2e− → Cu Hg2Cl2 + 2e− → 2Hg + 2Cl− AgCl + e− → Ag + CI− Bi3+ + 3e− → Bi Cu2+ + e− → Cu+ Sn4+ + 2e− → Sn2+ AgBr + e− → Ag + Br − Ti4+ + e− → Ti3+ 2H+ + 2e− → H Fe3+ + 3e− → Fe O2 + H2O + 2e− → HO 2− + OH− Pb2+ + 2e− → Pb In+ + e− → In Sn2+ + 2e− → Sn AgI + e− → Ag + I− Ni2+ + 2e− → Ni Co2+ + 2e− → Co In3+ + 3e− → In Tl+ + e− → Tl PbSO4 + 2e− → Pb + SO 2− Ti2+ + 2e− → Ti Al3+ + 3e− → Al U3+ + 3e− → U Mg2+ + 2e− → Mg Ce3+ + 3e− → Ce La3+ + 3e− → La Na+ + e− → Na Ca2+ + 2e− → Ca Sr2+ + 2e− → Sr Ba2+ + 2e− → Ba Ra2+ + 2e− → Ra Cs+ + e− → Cs Rb+ + e− → Rb K+ + e− → K Li+ + e− → Li +0.76 +0.62 +0.60 +0.56 +0.54 +0.52 +0.53 +0.49 +0.45 +0.40 +0.36 +0.36 +0.34 +0.27 +0.22 +0.20 +0.16 +0.15 +0.07 0.00 0, by definition −0.04 −0.08 −0.13 −0.14 −0.14 −0.15 −0.23 −0.28 −0.34 −0.34 −0.36 −1.63 −1.66 −1.79 −2.36 −2.48 −2.52 −2.71 −2.87 −2.89 −2.91 −2.92 −2.92 −2.93 −2.93 −3.05 Strongly oxidizing H4XeO6 + 2H+ + 2e− → XeO3 + 3H2O F2 + 2e− → 2F− O3 + 2H+ + 2e− → O2 + H2O − 2− S2O 2− + 2e → 2SO Ag2+ + e− → Ag+ Co3+ + e− → Co2+ HO2 + 2H+ + 2e− → 2H2O Au+ + e− → Au Pb4+ + 2e− → Pb2+ 2HClO + 2H+ + 2e− → Cl2 + 2H2O Ce4+ + e− → Ce3+ 2HBrO + 2H+ + 2e− → Br2 + 2H MnO 4− + 8H+ + 5e− → Mn2+ + 4H2O Mn3+ + e− → Mn2+ Au3+ + 3e− → Au Cl2 + 2e− → 2Cl− + − 3+ Cr2O 2− + 7H2O + 14H + 6e → 2Cr − O3 + H2O + 2e → O2 + 2OH− O2 + 4H+ + 4e− → 2H2O ClO −4 + 2H+ + 2e− → ClO −3 + H2O MnO2 + 4H+ + 2e− → Mn2+ + 2H2O Br2 + 2e− → 2Br − Pu4+ + e− → Pu3+ NO −3 + 4H+ + 3e− → NO + 2H2O 2Hg2+ + 2e− → Hg 22+ ClO− + H2O + 2e− → Cl− + 2OH− Hg2+ + 2e− → Hg NO −3 + 2H+ + e− → NO2 + H2O Ag+ + e− → Ag − Hg 2+ + 2e → 2Hg 3+ − Fe + e → Fe2+ Ti3+ + e− → Ti2+ Cd2+ + 2e− → Cd In2+ + e− → In+ Cr3+ + e− → Cr2+ Fe2+ + 2e− → Fe In3+ + 2e− → In+ S + 2e− → S2− In3+ + e− → In2+ U4+ + e− → U3+ Cr3+ + 3e− → Cr Zn2+ + 2e− → Zn Cd(OH)2 + 2e− → Cd + 2OH− 2H2O + 2e− → H2 + 2OH− Cr2+ + 2e− → Cr Mn2+ + 2e− → Mn V2+ + 2e− → V 565 566 DATA SECTION Table D2.1b Standard potentials at 298.15 K in alphabetical order Reduction half-reaction E /V Reduction half-reaction E /V +0.80 +1.98 +0.0713 +0.22 +0.45 +0.78 −0.15 −1.66 +1.69 +1.40 −2.91 −1.85 +0.20 +1.09 +0.76 −2.87 −0.81 −0.40 −2.48 +1.61 +1.36 +0.89 +1.23 +0.36 +0.92 +0.62 +0.54 +0.53 −0.14 −0.40 −0.44 −0.34 −0.49 −2.93 −2.52 −3.05 −2.36 −1.18 +1.51 +1.23 +1.51 +0.56 +0.60 −2.71 −0.23 +0.49 +0.80 +0.96 +0.10 +0.40 Co2+ + 2e− → Co Co3+ + e− → Co2+ Cr2+ + 2e− → Cr + − 3+ Cr2O 2− + 7H2O + 14H + 6e → 2Cr Cr3+ + 3e− → Cr Cr3+ + e− → Cr2+ Cs+ + e− → Cs Cu+ + e− → Cu Cu2+ + 2e− → Cu Cu2+ + e− → Cu+ F2 + 2e− → 2F− Fe2+ + 2e− → Fe Fe3+ + 3e− → Fe Fe3+ + e− → Fe2+ [Fe(CN)6]3− + e− → [Fe(CN)6] 4− 2H+ + 2e− → H2 2H2O + 2e− → H2 + 2OH− 2HBrO + 2H+ + 2e− → Br2 + 2H2O 2HClO + 2H+ + 2e− → Cl2 + 2H2O H2O2 + 2H+ + 2e− → 2H2O H4XeO6 + 2H+ + 2e− → XeO3 + 3H2O − Hg 2+ + 2e → 2Hg Hg2Cl2 + 2e− → 2Hg + 2Cl− Hg2+ + 2e− → Hg O2 + 4H+ + 4e− → 2H2O O2 + e− → O −2 O2 + H2O + 2e− → HO −2 + OH− O3 + 2H+ + 2e− → O2 + H2O O3 + H2O + 2e− → O2 + 2OH− Pb2+ + 2e− → Pb Pb4+ + 2e− → Pb2+ PbSO4 + 2e− → Pb + SO 2− Pt2+ + 2e− → Pt Pu4+ + e− → Pu3+ Ra2+ + 2e− → Ra Rb+ + e− → Rb S + 2e− → S2− − 2− S2O2− + 2e → 2SO Sn2+ + 2e− → Sn Sn4+ + 2e− → Sn2+ Sr2+ + 2e− → Sr Ti2+ + 2e− → Ti Ti3+ + e− → Ti2+ Ti4+ + e− → Ti3+ Tl+ + e− → Tl U3+ + 3e− → U U4+ + e− → U3+ V 2+ + 2e− → V V 3+ + e− → V2+ Zn2+ + 2e− → Zn −0.28 +1.81 −0.91 +1.33 −0.74 −0.41 −2.92 +0.52 +0.34 +0.16 +2.87 −0.44 −0.04 +0.77 +0.36 0, by definition −0.83 +1.60 +1.63 +1.78 +3.0 +0.79 +0.27 +0.86 +1.23 −0.56 −0.08 +2.07 +1.24 −0.13 +1.67 −0.36 +1.20 +0.97 −2.92 −2.93 −0.48 +2.05 −0.14 +0.15 −2.89 −1.63 −0.37 0.00 −0.34 −1.79 −0.61 −1.19 −0.26 −0.76 Strongly reducing Ag+ + e− → Ag Ag2+ + e− → Ag+ AgBr + e− → Ag + Br − AgCl + e− → Ag + Cl− Ag2CrO4 + 2e− → 2Ag + CrO 2− AgF + e− → Ag + F− AgI + e− → Ag + I− Al3+ + 3e− → Al Au+ + e− → Au Au3+ + 3e− → Au Ba2+ + 2e− → Ba Be2+ + 2e− → Be Bi3+ + 3e− → Bi Br2 + 2e− → 2Br − BrO− + H2O + 2e− → Br − + 2OH− Ca2+ + 2e− → Ca Cd(OH)2 + 2e− → Cd + 2OH− Cd2+ + 2e− → Cd Ce3+ + 3e− → Ce Ce4+ + e− → Ce3+ Cl2 + 2e− → 2Cl− ClO− + H2O + 2e− → Cl− + 2OH− ClO −4 + 2H+ + 2e− → ClO−3 + H2O ClO −4 + H2O + 2e− → ClO−3 + 2OH− 2Hg2+ + 2e− → Hg2+ Hg2SO4 + 2e− → 2Hg + SO 2− I2 + 2e− → 2I− I −3 + 2e− → 3I− In+ + e− → In In2+ + e− → In+ In3+ + 2e− → In+ In3+ + 3e− → In In3+ + e− → In2+ K+ + e− → K La3+ + 3e− → La Li+ + e− → Li Mg2+ + 2e− → Mg Mn2+ + 2e− → Mn Mn3+ + e− → Mn2+ MnO2 + 4H+ + 2e− → Mn2+ + 2H2O MnO −4 + 8H+ + 5e− → Mn2+ + 4H2O MnO −4 + e− → MnO 2− − − MnO 2− + 2H2O + 2e → MnO2 + 4OH + − Na + e → Na Ni2+ + 2e− → Ni NiOOH + H2O + e− → Ni(OH)2 + OH− NO −3 + 2H+ + e− → NO2 + H2O NO −3 + 3H+ + 3e− → NO + 2H2O NO −3 + H2O + 2e− → NO 2− + 2OH− O2 + 2H2O + 4e− → 4OH− Index A—T base pair 375 ab initio method 346 absolute zero 17 absorbance 221, 474 absorption coefficient integrated 476 molar 220, 474 absorption intensity 495 absorption spectroscopy 486 abundant-spin species 512 acceleration 549 acceleration of free fall accommodation (surface) 425 acid 172 acid–base indicator 185 acid–base titration 181 acid catalysis 258 acid ionization constant 173 acidity constant 173 from conductivity 198 acidosis 184 action potential 200 activated complex 237 activated complex theory 237 activation barrier 234 activation-controlled limit 255 activation energy 233, 236 negative 252 activation enthalpy 239 activation entropy 239 activation Gibbs energy 239 active transport 199 activity 134, 194 activity coefficient 134, 194 adenine 375 adenosine diphosphate 164 adenosine triphosphate 164 adiabatic 44 ADP 164 adsorbate 420 adsorbent 420 adsorption 420 dissociative 428 enthalpy 426 extent 424 rate 425 adsorption isotherm 426 AEDANS 493 aerosol 381 AES 421 AFM 423 algebraic equation 543 alkalosis 184 alkylation 434 allosteric effect 166 allotrope 399 allowed transition 305, 453 α electron 304 α-helix 374 AM1 346 amount of substance ampere 551 amphiphilic 382 amphiprotic species 179 analyte 181 angular (shape), 556 angular momentum 285, 303 angular momentum quantum number 287, 298 angular wavefunction 298 anharmonic vibration 459 anharmonicity constant 460 anion, configuration 310 anisotropy 517 anode 203 anti-Stokes line 456 anti-Stokes radiation 456 antibonding orbital 332 anticyclone 20 antiferromagnetic phase 402 antilogarithm 544 antiparallel β-sheet 374 antisymmetric stretch 461 antisymmetric wavefunction 319 approximation Born–Oppenheimer 323 orbital 306 steady-state 251 aquatic life 132 array detector 474 Arrhenius, Svante 232 Arrhenius equation 233 Arrhenius parameters 232 Arrhenius temperature dependence 257 artist’s colour wheel 473 atmosphere 20, 314 temperature profile 488 atmosphere (unit) atmospheric CO2 levels 463 atom, configuration 306 atomic force microscopy 423 atomic orbital 298 atomic radius 311 atomic weight ATP 164 ATP hydrolysis 164 Aufbau principle 308 Auger effect 421 Auger electron spectroscopy 421 Austin Model 346 autoionization 173 autoprotolysis constant 174 autoprotolysis equilibrium 173 Avogadro’s constant Avogadro’s principle 18 AX spectrum 507 AX2 spectrum 508 AX3 spectrum 508 azeotrope 142 azimuth 287 balanced reaction 11 Balmer, Johann 296 Balmer series 296 band gap 393 band structure 465 band theory 392 bar barometer barometric formula 20 base 172 base buffer 183 base catalysis 259 base unit 541 basicity constant 174 Beer–Lambert law 220, 474, 494 bending mode 462 benzene electrostatic potential surface 347 isodensity surface 347 orbitals 342 BET isotherm 429 β electron 304 β sheet 374 bilayer vesicle 384 bimolecular reaction 250 binary mixture 22, 140 binding of O2 165 biochemical cascade 479 bioenergetics 42 biofuel cell 202, 438 biological macromolecule 414 biological membrane 380 biological standard potential 211 biological standard state 164 biopolymer, melting 379 biradical 339 black-body radiation 463 blood 132 buffer action 184 Blue Mountains 21 body-centred cubic 411 Bohr, Niels 300 Bohr effect 184 Bohr frequency condition 271, 296, 447 Bohr magneton 500 Bohr radius 300 boiling 114 boiling point 114 boiling point elevation 134 boiling temperature 114 Boltzmann, Ludwig 96, 532 Boltzmann distribution 525 and chemical equilibrium 534 Boltzmann formula 96, 532 Boltzmann’s constant 96 bomb calorimeter 53 bond classification 323 covalent 323 high-energy phosphate 164 ionic 323 π 325, 336 polar 339 σ 324, 334 568 INDEX bond angle, and hybridization 328 bond bending 376 bond enthalpy 68 bond formation, Pauli principle and 324 bond length 323 bond order 338 bond stretching 376 bond torsion 376 bonding orbital 332 Born, Max 275 Born–Haber cycle 397 Born interpretation 275 Born–Meyer equation 399 Born–Oppenheimer approximation 323 borneol 170 boson 305, 318, 451 boundary condition 277, 548 cyclic 286 boundary surface 301, 303 Boyle, Robert 16 Boyle’s law 16 Brackett series 296 Bragg, William and Lawrence 407 Bragg’s law 408 branch 465 branching ratio 490 branching step 262 Bravais lattice 404 breathing 132 bremstrahlung 407 broadening 456 Brønsted–Lowry theory 172 Brunauer, Stephen 429 buffer action 183 buffer solution 183 building-up principle 308 Butler–Volmer equation 439 C—G base pair 375 caesium-chloride structure 412 cage effect 254 calculus 546 calorie 549 calorimeter 50, 53 differential scanning 57 calorimeter constant 50 Calvin–Benson cycle 488 capillary action 386 capillary electrophoresis 372 carbohydrate 74, 82 carbon dioxide atmospheric 463 experimental isotherms 30 phase diagram 118 supercritical 116 carbon dioxide laser 483 carbon nanotube 401 carbonic acid 178 Carnot cycle 103 carotene 348 casein 381 catalysis examples 434 heterogeneous 432 homogeneous 258 mechanism 433 catalyst 258 and equilibrium 162 heterogeneous 258 homogeneous 258 catalyst ensemble 433 catalytic constant 260 catalytic efficiency 260 cathode 203 cation, configuration 310 cavity resonant mode 482 CCD 474 cell 200 cell membrane 200 cell notation 206 cell overpotential 443 cell potential 207 see also standard cell potential cell reaction 206 equilibrium constant 208 Celsius scale centigrade scale see Celsius scale centrifugal distortion constant 451 cesium see caesium chain-branching explosion 263 chain carrier 262 chain reaction 262 rate law 262 channel former 200 charge-coupled device 474 charge-transfer transition 478 Charles’s law 17 chemical amount chemical bond 322 chemical equilibrium, statistical basis 534 chemical exchange 513 chemical kinetics 219 chemical potential 125, 194 solute 133 solvent 130 standard 125 variation with concentration 133 variation with partial pressure 125 chemical shift 504 chemisorption 425 chemisorption abilities 435 chemisorption enthalpy 425 chemistry chlorophyll 472 chloroplast 488 cholesteric phase 379 cholesterol 380 CHPs 202, 438 chromatic aberration 478 chromophore 477 chromosphere 314 circular polarization 552 Clapeyron equation 111 classical mechanics 270, 549 failures 270 classical thermodynamics 41 clathrate 361 Clausius–Clapeyron equation 112 Clebsch–Gordan series 315 climate change 463 close-packed structure 410 closed shell 307 closed system 42 cloud point 383 CMC 382 CNDO 346 coadsorption 429 coagulation 384 coefficient activity 134, 194 cooling performance 104 extinction 220 heating performance 104 Hill 171 molar absorption 220, 474 osmotic virial 138 stoichiometric 11 virial 32 viscosity 255 colatitude 287 cold denaturation 247 cold-pack 44 colligative properties 134 collision cross-section 28 collision flux 420 collision frequency 28, 235 collision theory 234 collisional deactivation 456, 492 colloid 379 colour 472 colour wheel 473 combined gas equation 19 combined heat and power system 202, 438 combining standard potentials 212 combustion 71 standard enthalpy 72 common-ion effect 189 common logarithm 544 common unit 542 competitive inhibition 261 complementary 279 complete neglect of differential overlap 346 complete shell 307 components, number 117 composition of vector 545 compression, effect on K 166 compression factor 32 computational chemistry 345 concentration formal 174 measures of 123 two absorbing species 221 concentration polarization 441 condensation 66 condition of stability 105 conduction band 394 conductivity 197 conductivity cell 196 cone (eye) 478 configuration atom 306 cation and anion 310 conformational conversion 512 conformational energy 376 conformational entropy 373 conjugate acid 173 conjugate base 173 consecutive reactions 248 conservation of energy consolute temperature 144 constant acid ionization 173 acidity see acidity constant anharmonicity 460 autoprotolysis 174 Avogadro’s basicity 174 Boltzmann’s 96 calorimeter 50 INDEX catalytic 260 critical 31 cryoscopic 135 dissociation 173 ebullioscopic 135 equilibrium see equilibrium constant Faraday’s 207, 550 force 289, 458 gas 16 Henry’s law 131 hyperfine coupling 518 Madelung 399 Michaelis 259 normalization 280 Planck’s 271 rate see rate constant rotational 448 Rydberg 296 solubility 187 solubility product 187 spin–spin coupling 507 time 231 constant-current mode 423 constant-volume heat capacity 49, 54 constant-z mode 423 contact interaction 510 continuum generation 486 contour length 372 contrast agent 506 convection 20, 27, 256 converting between units cooling curve 144 cooling performance coefficient 104 Cooper pair 395 cooperative binding 166 cooperative transition 247 coordination number 411 cornea 478 corona 314 correlation spectroscopy 515 cosines, law of 545 COSY 515 Coulomb interaction 550 Coulomb potential 550 Coulombic potential energy 4, 194, 296, 324, 549 Coulomb’s inverse-square law of force 550 couple 202 coupled reactions 164 covalent bond 323 covalent bonding 554 covalent solid 392 cracking 434, 436 criteria of spontaneity 157 critical constant 31 critical isotherm 31 critical micelle concentration 382 critical molar volume 31 critical point 31, 115 critical pressure 31, 115 critical solution temperature 144 critical temperature 31, 395 crixivan 367 cross-product 546 cross-section, collision 28 cryoscopic constant 135 crystal diode 448 crystal structure 403 crystal system 403 crystallinity 376 cubic cage 436 cubic close-packed 410 cubic system 403 Curie temperature 402 current, electric 551 current density 202, 439 cyclic boundary condition 286 cyclic voltammetry 441 cytosine 375 d block 309 d-metal complex 478 d orbital 303 occupation 309 Dalton, John 21 Dalton’s law 21 Daniell cell 206 Davisson, Clinton 273 Davisson–Germer experiment 273 de Broglie, Louis 273 de Broglie relation 273 deactivation 490 Debye, Peter 195, 353, 408 debye 353 Debye–Hückel limiting law 195 Debye–Hückel theory 194 Debye T3 law 94 decay 490 exponential 227 fluorescence 220 decomposition temperature 158 defect 420 definite integral 548 degeneracy 283 degenerate 285 degrees of freedom, number 117 dehydration 434 dehydrogenation 434 delocalization energy 345 delocalized 343 δ orbital 338 δ scale 504 denaturant 247 denaturation 375 cold 247 density, kinetic energy 40 density functional theory 346 deoxyribonucleic acid 375 depression of freezing point 135 deprotonation 173 derivative 547 derived unit 541 deshielded 504 desorption 420, 425 activated process 431 desulfurization 434 detergent 382 DFT 346 dialysis 382 diamagnetic 339, 400 diamond 399 diathermic 43 diatomic molecule heteronuclear 339 homonuclear 338 Period 335 properties 461 structure 333 VB theory 324 dielectric constant see relative permittivity differential equation 548 differential overlap 346 differential scanning calorimetry 57 differentiation 546 diffraction 272, 406 electron 273 low-energy electron 422 X-ray 406 diffraction grating 474 diffraction pattern 406 diffractometer 408 four-circle 409 diffuse double layer 438 diffusion 27, 255 Fick’s laws of 256, 265 temperature dependence 257 diffusion coefficient 256 diffusion-controlled limit 255 diffusion equation 256 dilute-spin species 512 diode laser 483 dipole–dipole interaction 356 dipole–induced-dipole interaction 357 dipole interaction 355 dipole moment 353 induced 357 resolution 354 transition 453 disorder 95 disperse phase 369 disperse system 379 dispersion interaction 358, 382 dissociation 68, 479 dissociation constant 173 dissociation limit 479 dissociative adsorption 428 distribution Boltzmann see Boltzmann distribution Maxwell 25, 235, 455 molecular speeds 25 disulfide link 354 DNA 375 domain 402 dopant 394 Doppler-broadened spectral line 455 Doppler effect 455 dot product 546 double helix 375 drift velocity 198 DSC 57 duality 274 Dubosq colorimeter 496 dye laser 484 dynamic equilibrium 111 dynamic light scattering 372 Eadie–Hofstee plot 268 Earth atmosphere 463 surface temperature 463 ebullioscopic constant 135 eddy 20 effect allosteric 166 Auger 421 Bohr 184 cage 254 common-ion 189 Doppler 455 greenhouse 463 569 570 INDEX effect (cont’d) hydrophobic 361 Joule–Thomson 36 Meissner 403 nuclear Overhauser 513 photoelectric 272 effective atomic number 311 effective mass 458 effective nuclear charge 307 effective rate constant 225 effector molecule 200 effusion 27 Einstein, Albert 495 Einstein coefficients 495 Einstein relation 257 Einstein–Smoluchowski equation 257 elastomer 376 electric current 551 electric dipole 352 electric dipole moment 353 electric double layer 384 electric eel 201 electric field 198, 551 electric field jump 220 electrical double layer 437 electrical work electrochemical cell 200 electrochemical series 212 electrochemistry 42 electrode 200 electrode compartment 200 electrode concentration cell 206 electrode process 437 electrode solution interface 437 electrodialysis 382 electrokinetic potential 384 electrolysis 443 electrolyte concentration cell 206 electrolyte solution 123, 194 electrolytic cell 201 electromagnetic field 551 electromagnetic radiation 551 electromagnetic spectrum 271, 552 electromotive force see cell potential electron g-value 500 promoted 326 σ 331 valence 308 electron affinity 313 electron-deficient compound 555 electron diffraction 273 electron gain 68 electron gain enthalpy 68 electron pair arrangement 556 electron paramagnetic resonance 220, 503 electron spin resonance 503 electron spin 304 electron transfer 492 electronegativity 339 electronic conductor 392 electronic partition function 530 electronvolt 472 electro-osmotic drag 202, 438 electrophoresis 372 electrostatic potential surface 347 electrostatics 550 elementary reactions 249 elevation of boiling point 134 Eley–Rideal mechanism 433 elpot surface 347 emf see cell potential emission spectrum 295 Emmett, Paul 429 emulsifying agent 381 emulsion 381 encounter pair 254 end point 186 endergonic compound 159 endergonic reaction 157 endocytosis 380 endothermic 44 endothermic compound 76 energy 3, 42 conformational 376 conservation of delocalization 345 gravitational potential 14 as heat 51 internal see internal energy ionization 298, 312 kinetic 3, 549 potential see potential energy quantization 271 reorganization 492 tendency to become disordered 84 total 4, 549 zero-point 281 energy density 40 energy level harmonic oscillator 289 hydrogen atom 297 particle in a box 281 rotational 448 vibrational 458 energy reserves 73 energy transfer, resonance 492 enthalpy 55 of activation 239 of adsorption 426 chemisorption 425 mixing 127, 130 physisorption 425 reaction see reaction enthalpy standard reaction 72, 213 temperature variation 56 enthalpy density 73 entropy 85 of activation 239 Boltzmann formula 96, 532 cell reaction 213 conformational 373 determination 93 experimental determination 90 fusion 90 mixing 127, 130 from partition function 532 perfect gas expansion 87 perfectly ordered crystal 93 phase transition 90 residual 97, 532 standard molar 94 standard reaction 98 surroundings 92 temperature variation 89 Third-Law 94 vaporization 90 enzyme 258 enzyme kinetics 219 epitaxy 401 EPR 220, 503 EPR spectra 517 EPR spectrometer 503 equation algebraic 543 Arrhenius 233 Born–Meyer 399 Butler–Volmer 439 Clapeyron 111 Clausius–Clapeyron 112 diffusion 256 Einstein 257 Einstein–Smoluchowski 257 Eyring 239 Goldman 218 Henderson–Hasselbalch 182 Hückel 344 Karplus 509 Laplace 385 McConnell 520 Nernst 208 quadratic 543 Schrödinger see Schrödinger equation secular 343 Stern–Volmer 491 thermochemical 64 van der Waals 33, 34 van’t Hoff 137, 163 wave 551 equation of state 15 perfect gas 16 van der Waals 33, 34 virial 33 equilibrium autoprotolysis 173 dynamic 111 mechanical proton transfer 172 solubility 187 statistical basis 534 thermal equilibrium bond length 323 equilibrium composition 160 equilibrium constant 156 calculation 535 cell reaction 208 concentration 161 from partition function 538 relation to Gibbs energy 156 relation to rate constants 245 equivalence point 181 ER mechanism 433 ESR 503 essential symmetry 404 ethene, Hückel equations 344 eutectic composition 146 eutectic halt 146 exchange current density 202, 439 excimer laser 484 exciplex 484 exciplex laser 484 exclusion principle 306, 318, 333, 451 exclusion rule 466 exergonic compound 160 exergonic reaction 157, 164 exocytosis 380 exothermic 44 exothermic compound 76 expansion work 45 expectation value 344 INDEX explosion 263 explosion limit 263 explosion region 263 exponential decay 227 exponential function 25, 544 extended Debye–Hückel law 196 extensive property extinction coefficient 220 eye 478 Eyring equation 239 f block 310 factorial 547 Fahrenheit scale 13 Faraday’s constant 207, 550 fast reactions 220 fat 74 FEMO 350, 498 femtochemistry 220, 238 Fermi contact interaction 510 Fermi level 394 fermion 305, 318, 451 ferromagnetism 402 fibre 378 Fick’s first law 256, 265 Fick’s second law 256, 266 field 551 fine structure 507 fingerprint region 462 first ionization energy 312 first ionization enthalpy 67 First Law 53 first-order differential equation 548 flash desorption 425 flash photolysis 220, 221 flocculation 384 flow method 221 fluid, supercritical 31, 116 fluid mosaic model 380 fluorescence 479, 480, 490 quantum yield 491 X-ray 421 fluorescence decay 220 fluorescence lifetime 490 fluorescence microscopy 498 fluorescence quenching 491 fluorescence resonance energy transfer 493 flux 255 Fock, V 310 food 74 forbidden transition 305, 453 force 3, 549 intermolecular 352 force constant 289, 458 formal concentration 174 formation Gibbs energy 158 standard enthalpy 75 Förster, T 492 Förster theory 492 four-circle diffractometer 409 four-level laser 482 Fourier synthesis 409 Fourier transform NMR 503 fraction deprotonated 174 fraction protonated 174 fractional composition 178 fractional coverage 424 fractional saturation 165 fractionating column 141 Franck–Condon principle 476 free-electron molecular orbital (FEMO) theory 350, 498 free expansion 46 freely jointed chain 372 freezing 66 freezing point 115 freezing point depression 135 freezing temperature 115 frequency 448, 551 frequency condition 271, 296, 447 frequency doubling 483 FRET 493 Friedrich, Walter 407 frontier orbital 341 fructose-6-phosphate 154 FT-NMR 503 fuel 73 fuel cell 201, 438 function 543 exponential 25, 544 Gaussian 25 function of a function 547 functional 346 functional MRI 506 fusion 66 entropy 90 standard enthalpy 66 g,u classification 333 g-value 500, 517 gain 482 Galvani potential difference 437 galvanic cell 201 gas kinetic model 23 liquefaction of 35 real 16, 29 gas constant 16 gas electrode 204 gas exchange 132 gas laser 483 Gaussian function 25 Gaussian-type orbital 346 gel 381 gel electrophoresis 372 gerade symmetry 333 Gerlach, Walther 304 Germer, Lester 273 Gibbs, J W 98 Gibbs energy 99 activation 239 equilibrium constant and 156 formation 158 mixing 127, 130, 144 partial molar 125 from partition function 533 perfect gas 108 reaction 154 standard molar 534 standard reaction 155, 158, 213 variation with pressure 106 variation with temperature 108 glacier motion 118 glancing angle 408 glass electrode 211 glass transition temperature 379 global warming 463 globar 460 glucose-6-phosphate 154 Goldman equation 218 Gouy–Chapman model 438 Graham, Thomas 27 Graham’s law of effusion 27 graph 543 graphite 399 gravimetry 425 gravitational potential energy 14 greenhouse effect 463 gross selection rule 453 Grotrian diagram 305 Grotthus mechanism 199 ground state 295 GTO 346 guanine 375 guest 360 Gunn diode 448 Haber, Fritz 165 haemerythin 470 haemoglobin 165, 184, 375 fractional saturation 165 half-life 229 half-reaction 201 Hall–Hérault process 201 Halley’s comet 118 Hanes plot 268 hard-sphere potential 362 harmonic oscillator 289 energy levels 289 wavefunctions 290 harmonic wave 551 Harned cell 213 harpoon mechanism 236 Hartree, D R 310 Hartree–Fock self-consistent field 310 HBr formation 224, 263, 490 head group 382 heat 43 equivalence to work 51 influx during expansion 51 molecular nature 44 heat capacity 48 at constant pressure 49, 58 at constant volume 49, 54 exact relation 62 from partition function 531 relation between 59 temperature dependence 58 heat engine 86 heat pump 87, 104 heating 43 heating performance coefficient 104 Heisenberg, Werner 278 helium, phase diagram 119 helix–coil transition 247, 375 Helmholtz layer model 437 hemoglobin see haemoglobin Henry, William 131 Henry’s law 131 Henry’s law constant 131 hertz 551 Hess’s law 74 heterogeneity index 369 heterogeneous catalysis 432 heterogeneous catalyst 258 heteronuclear diatomic molecule 339 hexagonal system 404 hexagonally close-packed 410 571 572 INDEX HF-SCF procedure 310 high-energy phosphate bond 164 high-temperature superconductor 392, 395 highest occupied molecular orbital 341 Hill coefficient 171 histidine 192 HIV 360, 367 homeostasis 184 HOMO 341 homogeneous catalysis 258 homogeneous catalyst 258 homogeneous mixture 123 homonuclear diatomic molecule 338 Hooke’s law 378 host 360 host–guest complex 360 HTSC 392, 395 Hückel, Erich 195, 343 Hückel equation 344 Hückel method 343 Hull, Albert 408 Humphreys series 319 Hund’s rule 309, 316 hybrid orbital 326 hybridization 326 variation with bond angle 328 hydrodynamic radius 198 hydrogen atom energy levels 297 spectrum 296 hydrogen bond 359, 377 hydrogen burning 314 hydrogen electrode 204, 209 hydrogen molecule 333 hydrogen/oxygen fuel cell 202, 438 hydrogen–oxygen reaction 263 hydrogenation 434, 435 hydrogenic atom 295 hydrolysis of ATP 164 hydronium ion 172 hydrophilic 380 hydrophobic 361, 380 hydrophobic effect 361 hydrostatic pressure hyperbaric oxygen chamber 132 hyperbola 17 hyperfine coupling constant 518 hyperfine structure 518 hypervalent molecule 555 hyperventilation 184 ice residual entropy 532 structure 118 ideal-dilute solution 131 ideal solution 128 IHP 437 incomplete octet 555 indefinite integral 548 indicator 185 INDO 346 induced dipole moment 357 infinitesimal calculus 546 infrared active 459 infrared activity, gross selection rule 462 infrared inactive 459 infrared spectroscopy 458 inhibition step 262 inhibitor 261 initial condition 548 initial rate 225 initiation step 262 inner Helmholtz plane 437 instantaneous rate 222 insulator 392 integral 548 integral protein 380 integrated absorption coefficient 476 integrated rate law 227 integration 547 intensity, nuclear magnetic resonance transition 502 intensive property interaction Coulomb 550 dipole–dipole 356 dipole–induced-dipole 357 dispersion 358, 382 π-stacking 360 potential energy 361 van der Waals 352 intercept 543 interference 406 Intergovernmental Panel on Climate Change 463 intermediate 251 intermediate neglect of differential overlap 346 intermetallic compound 395 internal conversion 479 internal energy 51 as independent of volume 52 from partition function 531 International System of units (SI) 2, 541 intersystem crossing (ISC) 481, 490 inversion symmetry 333 ion channel 199, 380 ion–ion interaction 194 ion pump 199, 380 ionic atmosphere 195 ionic bond 323 ionic conductivity 197 ionic–covalent resonance 329 ionic crystal 412 ionic model 396 ionic radius 413 ionic solid 391 ionic strength 196 ionization energy 298, 312 ionization enthalpy 67 IPCC 463 ISC see intersystem crossing isobar 20 isodensity surface 346 isoelectric point 215, 385 isolated system 42 isolation method 225 isomerization 434 isomorphous replacement 410 isosbestic point 476 isosteric enthalpy of adsorption 428 isotherm 17, 30 adsorption 426 BET 429 critical 31 Langmuir 426, 427 isothermal, reversible expansion 47 isotope separation 485 isotopomer 485 Jablonski diagram 480 Joule, James 3, 51 joule 3, 549 Joule–Thomson effect 36 K and Kc, relation between 162 Kamerlingh Onnes, Heike 395 Karplus equation 509 Kekulé structure 330 Kelvin scale 7, 18 kilogram kinetic control 253 kinetic energy 3, 549 kinetic energy density 40 kinetic model of gases 23 kinetic molecular theory 37 kinetic techniques 220 kinetics 219 Kirchhoff’s law 78 klystron 448 Knipping, Paul 407 Kohlrausch, Friedrich 197 Krafft temperature 382 lamellar micelle 380 Langmuir–Hinshelwood mechanism 433 Langmuir isotherm 426, 427 lanthanide contraction 312 Laplace equation 385 Larmor procession frequency 502 laser 482 applications 484 lattice enthalpy 396 law Beer–Lambert 220, 474, 494 Boyle’s 16 Charles’s 17 conservation of energy of cosines 545 Coulomb’s inverse-square 550 Debye–Hückel 195 Debye T3 94 diffusion 256, 265 effusion 27 Fick’s first 256, 265 Fick’s second 256, 266 First 53 Graham’s 27 Henry’s 131 Hess’s 74 Hooke’s 378 integrated rate 227 Kirchhoff’s 78 limiting 16, 129, 195 Nernst distribution 146 Newton’s second 3, 549 Ohm’s 196, 440, 551 Raoult’s 128 rate 223 Second 85 Stokes’ 198 Third 93 LCAO 330 Le Chatelier’s principle 162 LED 395 LEED 422 Lennard-Jones (12,6)-potential 362 level 316 lever rule 142 Lewis, G N 554 INDEX Lewis theory 554 LH mechanism 433 lifetime 456 lifetime broadening 456 light 472, 552 speed of 551 light-emitting diode 395 light-harvesting complex 488 limit, taking 167 limiting law 16, 129, 195 limiting molar conductivity 197 Linde refrigerator 36 Lindemann, Frederick 253 Lindemann mechanism 253 linear (shape), 556 linear combination 326 of atomic orbitals 330 linear graph 543 linear momentum 273, 549 linear rotor 448 linear-sweep voltammetry 441 Lineweaver–Burk plot 260 linewidth 455 Maxwell distribution 455 lipid raft model 380 liquefaction of gas 35 liquid molecular structure 119 liquid crystal 379 liquid junction 200 liquid junction potential 206 liquid–liquid phase diagram 142 liquid–solid phase diagram 144 liquid surface 385 liquidus 144 local contribution 505 logarithm 193, 543 London force see dispersion interaction London formula 358 long period 309 long-range dispersion attraction 382 long-range order 119 loop, van der Waals 34 low-energy electron diffraction 422 lower consolute temperature 144 lower critical solution temperature 144 lower explosion limit 263 lowest unoccupied molecular orbital 341 lumiflavin 390 LUMO 341 Lyman series 296 lyophilic 381 lysine 192 macular pigment 478 Madelung constant 399 magic-angle spinning 517 magnetic field 551 magnetic properties 400 magnetic quantum number 298 magnetic resonance 499 magnetic resonance imaging 506 magnetic susceptibility 400 magnetization 400 magnetogyric ratio 500 magneton Bohr 500 nuclear 500 MALDI 370 MALDI-TOF spectrum 371 malleable 411 many-electron atom 295, 305 Marcus, R A 492 Marcus theory 492 Mars van Krevelen mechanism 436 MAS 517 mass matrix-assisted laser desorption/ionization 370 matter states of tendency to become disordered 84 matter wave 273 maximum population 453 maximum velocity 259 Maxwell distribution of speeds 25, 235, 455 linewidth 455 MBE 401 McConnell equation 520 mean activity coefficient 194 mean bond enthalpy 70 mean free path 28 mean speed 24 and temperature 24 mechanical equilibrium mechanism Eley–Rideal 433 heterogeneous catalysis 433 Langmuir–Hinshelwood 433 Mars van Krevelen 436 Michaelis–Menten 259 mechanism of reaction 219, 249 bimolecular 250 unimolecular 249 Meissner effect 403 melting, biopolymer 379 melting point 115 melting temperature 115, 378 membrane cell 200 semipermeable 137 mercury cadmium telluride (MCT) detector 460 mesophase 379 metabolic acidosis 184 metabolic alkalosis 184 metal crystal 410 metal–insoluble-salt electrode 205 metallic conductor 392 metallic solid 391 meteorology 20 methane, atmospheric 463 micelle 382 Michaelis constant 259 Michaelis–Menten mechanism 259 Michelson interferometer 460 microporous material 436 microwave spectroscopy 448 migration of ions 196 Miller indices 405 millimetre of mercury (mmHg) 13 MINDO 346 mixing enthalpy 127, 130 entropy 127, 130 Gibbs energy 127, 130, 144 mixture binary 22, 140 homogeneous 123 volatile liquids 140 mmHg 13 MO theory 322, 330 mobility 198 model 2, 23 modified neglect of differential overlap 346 molality 10, 123 molar absorption coefficient 220, 474 molar concentration 9, 123 standard 133 molar conductivity 197 molar enthalpy 55 molar heat capacity 49 molar internal energy 51 molar mass molar partition function 534 molar quantity molar solubility 187 molar volume 18 molarity 10, 123 mole mole fraction 10, 22, 129 molecular beam epitaxy 401 molecular crystal 413 molecular dynamics simulation 363 molecular interaction 29 molecular mechanics simulation 377 molecular modelling 77 molecular orbital 330 molecular orbital (MO) theory 322, 330 molecular partition function 530 molecular potential energy curve 323, 457 molecular recognition 360 molecular solid 392, 413 molecular weight molecularity 249 moment of inertia 285, 448, 468 momentum angular 285, 303 linear 273, 549 monochromator 474 monoclinic system 403 monodisperse 369 monolayer 384 monomer 368 Monte Carlo method 363 MRI 506 Mulliken, Robert 340 multiphoton process 485 multiplicity 315 multiwalled nanotube 401 MWNT 401 myoglobin 165 fractional saturation 165 n-to-π* transition 477 n-type semiconductivity 394 NAD 203 nanodevice 400 nanometre-scale structures 280 nanotechnology 400 nanotube 401 nanowire 401 natural linewidth 456 natural logarithm 543 Nd-YAG laser 483 Néel temperature 403 neighbouring group contribution 505 nematic phase 379 573 574 INDEX neodymium laser 483 Nernst distribution law 146 Nernst equation 208 Nernst filament 460 network solid 392 Newton, Isaac 3, 270 newton 3, 550 Newton’s second law of motion 3, 549 nicad cell 201 nickel–cadmium cell 201 nicotinamide adenine dinucleotide 203 nicotine 177 nitric oxide 250 nitrogen, fixation 339 NMR 220, 502 solid-state 516 two-dimensional 515 NMR spectrometer 503 nodal plane 303, 332 node 281, 302 NOE 513 noncompetitive inhibition 261 nondegenerate 286 nonelectrolyte solution 123 nonexpansion work 53, 99 maximum 99 nonlinear optical phenomena 483 nonpolar molecule 353 nonpolarizable 441 nonspontaneous change 83 normal boiling point 114 normal freezing point 115 normal melting point 115 normal mode 462 normalization constant 280 notation for cells 206 nuclear g-factor 500 nuclear magnetic resonance 220, 502 solid-state 516 two-dimensional 515 nuclear magnetogyric ratio 500 nuclear magneton 500 nuclear model 296 nuclear Overhauser effect 513 nuclear spin quantum number 500 nuclear statistics 451, 457 number-average molar mass 369 number of components 117 nylon-66 378 observed fluorescence lifetime 490 occupation of d orbitals 309 octahedral 556 octet expansion 555 ocular fluid 478 ohm 551 Ohm’s law 196, 440, 551 OHP 437 open system 42 optical density 221 orbital antibonding 332 atomic 298 bonding 332 δ 338 frontier 341 hybrid 326 molecular 330 π 336 σ 330 orbital angular momentum quantum number 287, 298 orbital approximation 306 order 224, 250 ordinary differential equation 548 ortho-hydrogen 452 orthorhombic system 404 osmometry 138 osmosis 137 reverse 140 osmotic pressure 137 osmotic virial coefficient 138 outer Helmholtz plane 437 overall order 224 overall quantum yield 489 overlap, symmetry and 337 overlap integral 331, 343 overpotential 439 oxidation 434 gas-phase 250 propene 436 oxidation number 554 oxygen binding 165 electron configuration 339 paramagnetic 339 reaction with hydrogen 263 oxygen chamber 132 p band 393 P branch 465 p–n junction 395 p orbital 299, 303 p-type semiconductivity 394 packing fraction 411 paired spins 306 pairing, reason for 334 para-hydrogen 452 parabolic potential energy 289 parallel band 462 parallel β-sheet 374 paramagnetic 339, 400 parameters Arrhenius 232 van der Waals 34 parcel (of air) 20 partial charge 352 partial molar Gibbs energy 125 partial molar property 124 partial molar volume 124 partial negative charge 340 partial positive charge 340 partial pressure 22 partial vapour pressure 128 partially miscible liquids 142 particle in a box 280 energy levels 281 wavefunctions 281 particle on a ring 285 particle on a sphere 287 partition function 525 electronic 530 interpretation 527 molar 534 molecular 530 rotational 529, 538 translational 529, 537 vibrational 528 pascal Pascal’s triangle 508 Paschen series 296 passive transport 199 patch clamp technique 200 Pauli, Wolfgang 307 Pauli exclusion principle 306, 318, 333, 451 Pauli principle 306, 318, 451 and bond formation 324 Pauling, Linus 340 penetration 307 pentagonal bipyramidal 556 peptide link 247 peptizing agent 381 perfect elastomer 376 perfect gas 16 chemical potential 126 condition for 29 entropy 87 equation of state 16 expansion 47 internal energy 52 relation between heat capacities 59 periodic trends 310 peripheral protein 380 permittivity 352 relative 352 vacuum 4, 296, 550 perpendicular band 462 Pfund series 296 pH 173 pH curve 181 phase 64, 552 phase boundary 109 phase diagram 109, 381 carbon dioxide 118 helium 119 liquid–liquid 142 liquid–solid 144 temperature–composition 140 water 117 phase problem 409 phase rule 117 phase transition 64, 105 entropy 90 phosphatidyl choline 380 phosphine decomposition 433 phosphorescence 220, 479, 481 time constant 490 photobleaching 498 photochemical reaction 487 photochemistry 487 photodiode 474 photoejection 486 photoelectric effect 272 photoelectron spectrometer 487 photoelectron spectroscopy 486 photoelectron spectrum 486 photoemission spectroscopy 421 photoisomerization 478 photolysis flash 220, 221 of HI 489 photon 272, 552 photosphere 314 photosynthesis 488 photosystems I and II 488 photovoltaic device 460 physical chemistry physical quantity 541 physical state INDEX physisorption 425 enthalpy 425 π bond 325, 336 π-electron binding energy 344 π orbital 336 π-stacking interaction 360 π-to-π* transition 477 Planck, Max 271 Planck’s constant 271 plane polarized 552 planes, separation of 405 plastic 378 plastic crystal 516 plot Lineweaver–Burk 260 Ramachandran 376 Stern–Volmer 491 Tafel 440 polar bond 339 polar molecule 353 polarimetry 220 polarizability 357, 457 polarizability volume 357 polarizable 357, 441 polarization mechanism 510 polyatomic molecule structure 341 VB theory 326 vibrations 460 polychromator 474 polydisperse 369 polydispersity index 369 polyelectron atom see many-electron atom polymer 368 polymerization 434 polymorph 118, 399 polynucleotide 375 polypeptide 247 partial charge 352 structure 373 polyprotic acid 177 population 500 maximum 453 population inversion 482 potential, variation with pH 210 potential difference 551 potential energy 4, 549 Coulombic 4, 194, 296, 324, 549 interaction 361 parabolic 289 potential energy curve 323, 457 powder diffractometer 408 power 549, 551 power series 547 precession 502 precursor state 431 pre-exponential factor 233, 236 pressure critical 31, 115 hydrostatic osmotic 137 partial 22 standard 5, 19 units see also vapour pressure pressure broadening 456 pressure jump 220, 246 primary quantum yield 489 primary structure, polypeptide 373 principal quantum number 297, 298 principle Aufbau 308 Avogadro’s 18 building-up 308 exclusion 306, 318, 333, 451 Franck–Condon 476 Le Chatelier’s 162 Pauli see Pauli principle uncertainty 278 probabilistic interpretation 276 probability density 276 probe 486 projection reconstruction 506 promoted electron 326 promotion 326 propene oxidation 436 property colligative 134 extensive intensive protein 74 integral 380 peripheral 380 protein structure prediction 376 protein unfolding 247 proton decoupling 512 proton magnetic resonance 503 proton transfer 172 protonation 173 pseudofirst-order rate law 225 pseudosecond-order rate law 225 pulse radiolysis 222 pump (laser) 482 pump (spectroscopic) 486 Q-band 503 Q branch 465 QCM 425 quadratic equation 543 quantization of energy 271 quantum dot 401 quantum number 281 magnetic 298 nuclear spin 500 orbital angular momentum 287, 298 principal 297, 298 rotational 448 spin 304 spin angular momentum 315 spin magnetic 304 total angular momentum 315 total orbital angular momentum 315 vibrational 289 quantum yield 487 fluorescence 491 quartz crystal microbalance 425 quaternary structure, polypeptide 375 quenching 491 quenching method 222 quinoline 176 R branch 465 radial distribution function 301 radial node 302 radial velocity 320 radial wavefunction 298, 302 radiation, black-body 463 radiative decay 479 radical chain reaction 262 radius of gyration 373 radius ratio 412 radius-ratio rule 412 radius of shear 384 Ramachandran plot 376 Raman gross selection rule 460 Raman microscopy 485 Raman spectra rotational 456 vibrational 460, 465 Raman spectroscopy 456, 460, 466 random coil 372 random walk 255 Rankine scale 13 Raoult, François 128 Raoult’s law 128 rate adsorption 425 definition 222 formation of HBr 263, 490 initial 225 instantaneous 222 law 223 surface process 431 temperature dependence 232 rate constant 223 combination 252 effective 225 relation to equilibrium constant 245 rate-determining step 252 rate law formation 250 Rayleigh radiation 456 reaction bimolecular 250 endergonic 157 exergonic 157, 164 Gibbs energy 154 hydrogen and oxygen 263 in solution 254 unimolecular 249, 253 reaction centre 488 reaction coordinate 237 reaction enthalpy 72 variation with temperature 78 reaction entropy, from cell potential 213 reaction mechanism see mechanism of reaction reaction profile 234 reaction quotient 156 real gas 16, 29 real solution 134 real-time analysis 221 rearranging 543 redox couple 202 redox electrode 205 redox reaction 193 reduced mass 297 reference state 75 reforming 436 refrigerator 87, 104 Linde 36 relative atomic mass relative molar mass relative permittivity 352 relaxation 246, 511 spin 511 relaxation time 248, 511 reorganization energy 492 residual entropy 97, 532 resistance 551 resistivity 197 575 576 INDEX resonance 329, 499 ionic–covalent 329 resonance condition 501, 518 resonance energy transfer 492 resonance hybrid 329, 555 resonance Raman spectroscopy 466 resonance stabilization 330 resonant mode 482 respiration 184 respiratory acidosis 184 respiratory alkalosis 184 resting potential 200 resultant vector 545 retardation step 262 retina 478 retinal 349, 478 reverse osmosis 140 reversible expansion 47 reversible process 46 rhodopsin 478 rhombohedral system 404 ribonucleic acid 375 ridge 20 rigid rotor 448 ring current 505 RNA 375 rock-salt structure 412 rod (eye) 478 root 543 root mean square distance 257 root mean square separation 373 root-mean-square speed 23 rotation 285 rotational constant 448 rotational energy level 448, 468 rotational partition function 529, 538 rotational quantum number 448 rotational Raman spectra 456 rotational spectrum 453 rotational state population 451 rotational transition 453 rotationally active 453 rotationally Raman active 457 rotationally Raman inactive 457 rule Hund’s 309, 316 lever 142 phase 117 radius-ratio 412 selection see selection rule Trouton’s 91 Russell–Saunders coupling 314 Rydberg constant 296 s band 393 s electron 299 s orbital 299 salt, effect on solubility 189 salt bridge 200 salts in water 180 SAM see scanning Auger electron microscopy; self-assembled monolayer SATP 19 saturated solution 187 saturation 511 fractional 165 scalar product 546 scanning Auger electron microscopy 422 scanning tunnelling microscopy 423 scCO2 116 SCF see self-consistent field; supercritical fluid Scherrer, Paul 408 Schrödinger, Erwin 274 Schrödinger equation 274 justification 275 scuba diving 132 second derivative 547 second-harmonic generation 483 second ionization energy 312 second ionization enthalpy 67 Second Law 85 second-order differential equation 548 secondary structure, polypeptide 373 secular equation 343 sedimentation 371 see-saw shape 556 selection rule 306, 317, 453 Raman 460 vibrational 459 self-assembled monolayer 426 self-assembly 368 self-consistent field 310, 346 semiconductor 392 semiempirical method 346 semipermeable membrane 137 separation of planes 405 separation of variables procedure 283 series, spectroscopic line 296 SFC 116 SHE 209 shell 299 closed 307 complete 307 shielded 504 shielded nuclear charge 307 shielding 307 shielding constant 504 short-range order 119 SI 2, 541 SI prefix 541 σ bond 324, 334 σ electron 331 σ orbital 330 single-walled nanotube 401 singlet state 481 slice selection 506 slip plane 411 slope 543 slower-growing faces 420 smectic phase 379 smog 21 soap 382 sodalite cage 436 sol 381 solar energy 463 solid band theory 392 molecular 392, 413 solid-state NMR 516 solidus 144 solubility 187 solubility constant 187 solubility equilibrium 187 solubility product 187 solubility product constant 187 solute 9, 123 chemical potential 133 solution electrolyte 123, 194 ideal-dilute 131 ideal 128 nonelectrolyte 123 real 134 saturated 187 solvent 123 chemical potential 130 and local magnetic field 507 solvent contribution 505 sp hybridization 327 sp2 hybrid orbital 327 sp3 hybrid orbital 327 sparingly soluble compound 187 speciation 192 specific enthalpy 73 specific heat capacity 49 specific selection rule 453, 459 spectrometer 460 spectrophotometry 220 spectroscopic line 296 spectroscopy, general features 447 spectrum atomic hydrogen 296 complex atom 314 electromagnetic 271, 552 rotational 453 speed 549 mean 24 root-mean-square 23 speed of light 551 spherical harmonics 287 spherical micelle 383 spherical rotor 450 spherically symmetrical 300 spin, electron 304 spin-1 particle 305 spin-1/2 particle 304 spin angular momentum quantum number 315 spin correlation 309 spin density 520 spin–lattice relaxation time 511 spin magnetic quantum number 304 spin–orbit coupling 316 spin quantum number 304 spin relaxation 511 spin–spin coupling constant 507 spin–spin relaxation time 511 spontaneity 98, 109 criteria 157 spontaneous change 83 spontaneous emission 456, 479, 495 SPR 425 square planar 556 square pyramidal 556 stability condition 105 stable, thermodynamically 160 standard ambient temperature and pressure 19 standard boiling point 115 standard cell potential 208, 213 Gibbs energy from 213 standard chemical potential 125 standard electrode potential 209 standard enthalpy combustion 72 electron gain 68 formation 75 fusion 66 ionization 67 INDEX reaction 72, 213 sublimation 66 vaporization 64 standard Gibbs energy of formation 158 standard hydrogen electrode 209 standard molar concentration 133 standard molar Gibbs energy 534 standard potential 209 biological 211 combining 212 standard pressure 5, 19 standard reaction enthalpy 72, 213 standard reaction entropy 98 standard reaction Gibbs energy 155, 158, 213 standard reduction potential 209 standard state 63, 134 standard temperature and pressure 19 star hydrogen burning 314 spectroscopy 314 state biological standard 164 equation of see equation of state ground 295 physical precursor 431 reference 75 standard 63, 134 thermally accessible 527 transition 237 state function 52 states of matter statistical thermodynamics 41, 524 steady-state approximation 251 step defect 420 steric factor 236 Stern, Otto 304 Stern–Gerlach experiment 304 Stern–Volmer equation 491 Stern–Volmer plot 491 sticking probability 431 stimulated absorption 495 stimulated emission 482, 495 STM 423 stoichiometric coefficient 11 stoichiometric point 181 Stokes’ law 198 Stokes line 456 Stokes radiation 456 stopped flow 220 stopped-flow technique 221 STP 19 strong acid 173 strong base 174 structure factor 409 sublimation 66, 109 standard enthalpy 66 sublimation vapour pressure 110 subshell 299 substrate 420 Sun 40, 314 supercage 436 supercoiled DNA 248 superconductor 392 supercritical carbon dioxide 116 supercritical fluid 31, 116 supercritical fluid chromatography 116 superfluid 119 water 119 superimposition wavefunction 324 superpair 557 superposition 278 surface excess 386 surface plasmon resonance 425 surface process rate 431 surface structure 423 surface tension 385 surfactant 382, 386 surroundings 42 entropy changes 92 susceptibility 400 SWNT 401 symmetric rotor 450 symmetric stretch 461 symmetry, and overlap 337 symmetry number 530 synchrotron radiation 407 system 42 Système International (SI) 2, 541 T-weighted image 506 Tafel plot 440 Taylor expansion 547 Taylor series 547 TDS 432 Teller, Edward 429 temperature 7, 43, 115 boiling 114 consolute 144 critical 31, 395 critical solution 144 Curie 402 decomposition 158 effect of 163 freezing 115 glass transition 379 Krafft 382 mean speed and 24 melting 115, 378 Néel 403 transition see transition temperature temperature–composition diagram 140 temperature dependence diffusion 257 heat capacity 58 reaction rate 232 vapour pressure 113 temperature inversion 21 temperature jump 220, 246 temperature profile, atmosphere 488 temperature programmed desorption 432 temperature variation of enthalpy 56 term 314 term symbol 314 termination step 262 terrace defect 420 tertiary structure, polypeptide 374 tesla 500 tetragonal system 404 tetrahedral 556 theory activated complex 237 band 392 Brønsted–Lowry 172 collision 234 covalent bonding 554 Debye–Hückel 194 density functional 346 Förster 492 free electron molecular orbital 350, 498 Lewis 554 Marcus 492 molecular orbital (MO) 322, 330 transition state 237 valence 322 valence bond (VB) 322, 323 VSEPR 323, 555 thermal analysis 110 thermal desorption spectroscopy 432 thermal equilibrium thermal explosion 263 thermally accessible state 527 thermochemical equation 64 thermochemistry 41 thermodynamically stable 160 thermodynamically unstable 159 thermodynamics 41 classical 41 First Law 53 Second Law 85 statistical 41, 524 Third Law 93 thermogram 57 third body 263 Third law 93 Third-Law entropy 94 thymine 375 time constant 231 phosphorescence 490 time of flight 28 time-resolved spectroscopy 485 titrant 181, 183 titration 181 torr total angular momentum quantum number 315 total energy 4, 549 total orbital angular momentum 315 TPD 432 trajectory 274 transfer coefficient 439 transition 296, 305 cooperative 247 helix–coil 247, 375 rotational 453 transition dipole moment 453 transition state 237 transition state theory 237 transition temperature 109 modification 136 translation 280 translational partition function 529, 537 transmission coefficient 239 transmittance 220, 474 trial wavefunction 329 triclinic system 404 trigonal bipyramidal 556 trigonal planar 556 trigonal pyramidal 556 triple point 115 triplet state 481 trough 20 Trouton’s rule 91 tungsten–iodine lamp 474 tunnelling 283 turning point 477 turnover frequency 260 two-dimensional NMR 515 577 578 INDEX UHV 421 ultra-high vacuum 421 ultracentrifugation 371 ultrapurity 147 ultrasonic absorption 220 uncertainty broadening 456 uncertainty principle 278 ungerade symmetry 333 unimolecular reaction 249, 253 unit cell 403 units 541 converting between pressure unstable, thermodynamically 159 upper consolute temperature 144 upper critical solution temperature 144 upper explosion limit 263 UPS 421 uracil 375 vacuum permittivity 4, 296, 550 valence band 394 valence bond (VB) theory 322, 323 valence electron 308 valence-shell electron pair repulsion model 323, 555 valence theory 322 van der Waals, Johannes 33 van der Waals equation of state 33, 34 van der Waals interaction 352 van der Waals loop 34 van der Waals molecule 238 van der Waals parameters 34 van’t Hoff equation 137, 163 vaporization entropy 90 standard enthalpy 64 vapour deposition 66 vapour diffusion 414 vapour pressure 110 partial 128 sublimation 110 temperature dependence 113 variation theorem 329 VB theory 322, 323 vector 545 composition 545 vector model 288 velocity 549 maximum 259 vertical transition 477, 480 vibration 288, 457 vibration–rotation spectra 465 vibrational energy level 458 vibrational modes, number 460 vibrational partition function 528 vibrational quantum number 289 vibrational Raman spectra 460, 465 vibrational Raman spectroscopy 460 vibrational selection rule 459 vibrational spectra 457 vibrational structure 476 vibrational transitions 458 virial coefficient 32 osmotic 138 virial equation of state 33 viscosity 198, 255 water 258 vision 478 volcano curve 434 volt 551 voltaic cell 201 voltammetry 441 volume molar 18 partial molar 124 volume magnetic susceptibility 400 von Laue, Max 407 VSEPR 323, 555 water phase diagram 117 superfluid phase 119 VB description 326 viscosity 258 watt 549, 551 wave 551 wave equation 551 wave–particle duality 274 wavefunction 274 angular 298 antisymmetric 319 harmonic oscillator 290 particle in a box 281 radial 298, 302 superimposition 324 trial 329 wavelength 551 wavenumber 447, 464, 551 weak acid 173 weak base 174 weather 20 weather map 21 weight, configuration 532 weight-average molar mass 369 work 3, 42, 550 electrical equivalence to heat 51 expansion 45 maximum 99 molecular nature 44 nonexpansion see nonexpansion work reversible isothermal expansion of a perfect gas 47 work function 272 wrinkle, Nature abhors 257 X-band 503 X-ray 407 X-ray crystallography 414 X-ray diffraction 406 X-ray diffractometer 408 X-ray fluorescence 421 xanthophyll 478 XPS 421 Z-average molar mass 370 zeolite 436 zero-current cell potential 207 zero-point energy 281 zeta potential 384 zone levelling 147 zone refining 147 ... EnY For details of the calculation, see our Physical chemistry (20 06) ( 12. 12a) = 2 nX h n2 h2 ⎛ n2 n2 ⎞ h2 + Y = ⎜⎜ X2 + Y2 ⎟⎟ 8mLX 8mLY ⎝ LX LY ⎠ 8m ( 12. 12b) 28 3 28 4 CHAPTER 12: QUANTUM THEORY... of frequency 2 V = (n final − ninitial ) The energy difference between adjacent levels is = ( 122 − 1 12) × n + 2n + Δ E = En +1 − En = (n + 1 )2 = (2n + 1) h2 8m mL2 h2 8mL2 h2 h2 − n2 8mL 8mL2... 12. 4a simplifies to 22 d2y = Ey 2m dx Fig 12. 9 The wavelength of a harmonic wave of the form sin(2px/l) The amplitude of the wave is the maximum height above the centre line Thus: − 22 d2y 22