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• Front Matter
• Preface
• Distinguishing Features of This Text
• Acknowledgments
• Authors
• Guide for Students
• 1 World of Atoms and Molecules
• 1.1 Introduction to Physical Chemistry
• 1.2 Theory and Experiment in Physical Chemistry
• 1.3 Atomic and Molecular Energies
• FIGURE 1.1 The energy levels of an idealized harmonic oscillator (left), a rotating diatomic molecule (center), and the hydrogen atom (right). Each horizontal line is drawn according to the energy scale given by the vertical lines and following Equations 1.2 through 1.4, respectively. Notice that the oscillator levels are evenly spaced, whereas the rotational levels become increasingly separated with increasing energy. Also, notice the different energy scales. Typically, rotational energy levels are more closely spaced than vibrational energy levels, and vibrational energy levels are more closely spaced than the levels that develop from the electrons’ motions in an atom or molecule.
• 1.4 Configurations, Entropy, and Volume
• 1.5 Energy, Entropy, and Temperature
• Example 1.1: Energy Level Populations
• 1.6 Distribution Law Derivation
• 1.7 Conclusions
• Point of Interest: James Clerk Maxwell
• Exercises
• Bibliography
• 2 Ideal and Real Gases
• 2.1 The Ideal Gas Laws
• FIGURE 2.1 The pressure of a gas sample, Pint, can be measured by balancing the force due to the pressure, Fint, against an external force, F. The point of balance is when there is no contraction or expansion of the sample. The external force is applied to the system via a piston such that the force F is the gravitational force of the mass m loaded onto the piston. The internal force is the pressure times the area of the piston, A.
• FIGURE 2.2 Pressure–volume isotherms for an ideal gas. Pressure is given by the vertical axis, and volume is given by the horizontal axis. Each curve shows the possible pressure and volume as they vary together at one fixed temperature, those temperatures being 10, 30, 50, 70, and 90K.
• FIGURE 2.3 Three-dimensional representation of Equation 2.7. Pressure and volume are the x and y variables, and temperature is the vertical or z direction. The surface depicts how temperature varies with pressure and volume according to Equation 2.7.
• 2.2 Collisions and Pressure
• FIGURE 2.4 For three orthogonal axes, vx, vy, vz, coordinates, s, q, f, are defined such that s is the magnitude of a vector in (vx, vy, vz) space, q is the elevation angle relative to the vx − vy plane (q = 0 corresponds to a vector pointing perpendicular and up from the vx − vy plane), and f is the angle about the vz-axis.
• FIGURE 2.5 The functional form of the distribution of speeds, s (horizontal axis), of ideal gas particles. This is simply a plot of the function f(s) = s2 exp(−αs2) (f on the vertical axis) versus s with an arbitrarily chosen value of α = m/2kT. The three vertical lines mark, from left to right, the s-values of the most probable speed, the average speed, and the root-mean-squared speed. Notice that this function is not symmetrical about its maximum, and as a consequence, the average speed is somewhat greater than the most probable speed.
• Example 2.1: Most Probable Speed of Gas Particles
• FIGURE 2.6 A hypothetical system consisting of a vessel with a wall that can move freely to any position along the x-axis. Inside the vessel is only one moving particle.
• FIGURE 2.7 With a piston area of A, the volume that includes all particles with velocity component vx that will hit the piston in time dt is V′ = Avxδt.
• 2.3 Nonideal Behavior
• TABLE 2.1 Equations of State for Gases
• 2.4 Thermodynamic State Functions
• TABLE 2.2 Coefficients in the van der Waals Equation of State for Selected Real Gases
• 2.5 Energy and Thermodynamic Relations
• Example 2.2: Isothermal Compressibility
• FIGURE 2.8 A graphical device, the “thermodynamic compass,” for identifying the so-called natural choices of dependent variables and for generating thermodynamic relations without derivation. With the rules given in the discussion, this graph encodes all the Maxwell relations of thermodynamics.
• Example 2.3: Thermodynamic Relations
• Example 2.4: Thermodynamic Compass
• 2.6 Conclusions
• Point of Interest: Intermolecular Interactions
• Exercises
• Bibliography
• 3 Changes of State
• 3.1 Pressure–Volume Work
• FIGURE 3.1 A cylindrical piston system. The volume of the gas inside the cylinder is the cross-sectional area of the cylinder, A, times the height, h. The mass, m, is placed on top of the piston in increments, giving rise to a downward force due to gravity, mg. (The piston is attached to the bottom mass and its mass is included in the value ma.)
• Example 3.1: Work in a Stepwise Gas Expansion
• 3.2 Reversibility, Heat, and Work
• FIGURE 3.2 Expansion of an ideal gas at a specific temperature. The curve is the reversible path for the given temperature. Expansion from V1 to V2 follows this path only if equilibrium is maintained throughout. Then, the area under the segment from V1 to V2 is which is the work associated with the reversible process. Several irreversible expansions are indicated. A one-step irreversible process is for the external pressure to go instantaneously from P1 to P2, and then the gas expands from V1 to V2 against a pressure of P2. The amount of work expended by the gas in this irreversible process is the area of the shaded rectangle. It is clear that the area under any of the irreversible paths is less than the area under the path that follows the isotherm where the gas pressure is always equal to the external pressure. The amount of reversible work is approached through a sequence of irreversible steps as the steps are made smaller and smaller.
• Example 3.2: ΔU for an Adiabatic Expansion
• FIGURE 3.3 A hypothetical four-step process for an ideal gas known as a Carnot cycle. Step 1 is a reversible isothermal expansion that follows the T1 isotherm from (P1, V1) to (P2, V2). Step 2 is a reversible adiabatic expansion from (P2, V2) to (P3, V3) which results in a drop in temperature. This is a move to a different isotherm. Step 3 is a reversible isothermal compression from (P3, V3) to (P4, V4), and step 4 is a reversible adiabatic compression from (P4, V4) to (P1, V1), leading to an increase in temperature. In this cycle, the gas does mechanical work on the surroundings using heat transferred from the surroundings. The differential of reversible work is −PdV, and so the area under the line corresponding to any given step is the work associated with that step. The entire process yields a net amount of work corresponding to the enclosed area.
• 3.3 Entropy
• Example 3.3: ΔS for an Adiabatic Expansion
• 3.4 The Laws of Thermodynamics
• Example 3.4: ΔS of an Engine Cycle
• 3.5 Heat Capacities
• TABLE 3.1 Heat Capacity Coefficients of Equation 3.48 for 1mol of Selected Gases
• 3.6 Joule–Thomson Expansion
• FIGURE 3.4 An apparatus for the Joule–Thomson expansion of a gas. The apparatus consists of a cylinder with pistons on both ends. Between the pistons is a porous plug which allows gas molecules to pass slowly from one side to the other. By external mechanical means, the pressure on the left is maintained at some value P1, and the pressure on the right is P2, with P1 > P2. The whole apparatus is insulated and does not exchange heat with the surroundings.
• 3.7 Conclusions
• Point of Interest: Heat Capacities of Solids
• Exercises
• Bibliography
• 4 Phases and Multicomponent Systems
• 4.1 Phases and Phase Diagrams
• FIGURE 4.1 The phase diagram of water drawn on two different scales covering different temperature–pressure ranges. The triple point is shown on the expanded scale on the bottom drawing.
• FIGURE 4.2 P–V isotherms (T7 > T6 > … > T2 > T1) of a hypothetical pure substance that can exist as a gas and as a liquid. The isotherms for the gas phase are similar to those of an ideal gas. Deviations from ideal behavior become apparent at lower temperatures. The one isotherm (at T1) shown for the liquid state has the typical pressure–volume behavior of a liquid, a small change in volume corresponding to a large change in pressure. To change smoothly from one type of behavior to the other, there must be a P–V isotherm (T2) with an inflection (critical) point.
• FIGURE 4.3 The phase diagram of water showing the different solid phases designated ice I, ice II, and so on.
• 4.2 The Chemical Potential
• 4.3 Clapeyron Equation
• FIGURE 4.4 The phase diagram of carbon dioxide. Notice that the slopes of the phase equilibrium curves are positive.
• Example 4.1: The Solid–Liquid Phase Boundary
• TABLE 4.1 Molar Transition Enthalpies and Transition Temperaturesa
• 4.4 First- and Second-Order Phase Transitions
• FIGURE 4.5 Typical behavior of heat capacity, volume, and entropy of a pure substance as a function of temperature given that the system undergoes a first-order phase transition (left) versus a second-order phase transition (right) at the temperature T′.
• 4.5 Conclusions
• Point of Interest: Josiah Willard Gibbs
• Exercises
• Bibliography
• 5 Activity and Equilibrium of Gases and Solutions
• 5.1 Activities and Fugacities of Gases
• Example 5.1: Fugacity and Activity of a Real Gas
• 5.2 Activities of Solutions
• 5.3 Vapor Pressure Behavior of Solutions
• FIGURE 5.1 Vapor pressure of a hypothetical mixture of two species A and B as a function of the mole fraction at constant temperature. The end points give the vapor pressures of the pure liquids. Straight-line behavior is ideal (dashed line). Real systems tend to deviate to higher pressure or to lower pressure with a curved path between the end points (solid curve).
• FIGURE 5.2 Vapor pressure curves for an A–B mixture. The dashed lines show the behavior expected if Raoult’s law held throughout, and the dotted line shows the behavior if Henry’s law held throughout. The solid lines represent the true behavior.
• FIGURE 5.3 For a hypothetical mixture of two species A and B, the composition of liquid and vapor is related to the temperature on this plot. At some pressure such as 1 bar, TA is the boiling point of pure A and TB is the boiling point of pure B. If the system starts at point 1, heating it to the temperature at point 2 will initiate boiling. Because of different volatilities of A and B, the composition of the vapor differs from that of the liquid at the outset. Of course, if boiling were continued until all the liquid was gone, then the composition would have to be the same as that of the original liquid. However, at the outset, the composition of the vapor corresponds to the intersection of a horizontal line at the given temperature with the upper curve, the curve that gives the temperature at which only vapor will exist. So, if the first vapor from the boiling liquid (point 2) is pumped off to a new chamber at the same temperature, the composition will be that corresponding to point 3. The temperature is then lowered to condense the vapor into liquid. This liquid can then be heated to the boiling point, and the first vapor to come off will have the composition corresponding to point 5. Repeated steps will continue to purify the sample of the most volatile substance.
• 5.4 Equilibrium Constants
• 5.5 Phase Equilibria Involving Solutions
• FIGURE 5.4 Progressive effect on phase equilibria of a solvent from increasing the amount of solute. The heavy lines are those of the pure solvent. As the concentration of the solute increases, phase equilibria curves for the liquid–vapor interface and the solid–vapor interface are displaced further and further from those of the pure solvent. This leads to elevation of the boiling point and lowering of the freezing point.
• FIGURE 5.5 Freezing point of a hypothetical binary mixture as a function of composition.
• FIGURE 5.6 Osmotic pressure develops in the solution on the right because of the reduced chemical potential of the solvent on the right relative to that of the pure solvent on the left. The pressure is proportional to the height to which the liquid rises in the tube.
• 5.6 Conclusions
• Point of Interest: Gilbert Newton Lewis
• Exercises
• Bibliography
• 6 Chemical Reactions: Kinetics, Dynamics, and Equilibrium
• 6.1 Reaction of Atoms and Molecules
• FIGURE 6.1 Contours in arbitrary energy units of the potential energy surface for the hypothetical colinear reaction A + BC → AB + C. The horizontal axis is the B-to-C separation distance, and the vertical axis is the A-to-B distance. At the saddle point, a special coordinate system is defined by the two axes z1 and z2. Taking the saddle point as the origin of this system (i.e., z1 = 0 and z2 = 0), we see that z1 is the direction in which the energy contours lead downhill away from the saddle point, whereas z2 is a direction in which the energy is greater away from the saddle point.
• FIGURE 6.2 Minimum energy path (heavy line) superimposed on the contours of the potential energy surface for the hypothetical colinear reaction A + BC → AB + C, as in Figure 6.1. The minimum energy path follows the bottom of one trough to the transition state structure and then follows the bottom of the other trough. We can consider the potential energy as a function of the position along the minimum energy path, Figure 6.3, by considering the path to be a (curved) coordinate axis with regular intervals to mark distance along the curve.
• FIGURE 6.3 One-dimensional slice following the z1 coordinate of the potential energy surface in Figure 6.2. The curve gives the energy following the minimum energy path as a coordinate axis. The highest point on this curve is the saddle point, and the energy of that point relative to the energy of the reactants is the energetic barrier that must be surmounted for the reaction to take place.
• FIGURE 6.4 Contours in arbitrary energy units of the potential energy surface for the hypothetical colinear reaction involving a stable triatomic molecule, ABC. The overall process is A + BC → ABC → AB + C. The horizontal axis is the B-to-C separation distance, and the vertical axis is the A-to-B distance. There is a saddle point between the A + BC trough and the ABC triatomic (potential surface minimum) and a saddle point between the minimum and the AB + C trough.
• 6.2 Collisions and Transport
• FIGURE 6.5 A solid ball moving in the direction of the arrow will collide with ball B but will miss A. B extends into the cylinder whose axis is the direction of motion and whose diameter is 2r, where r is the radius of the balls. A is entirely outside the cylinder. The center of a ball just touching the surface of the cylinder will be at a distance R = 2r from the center of the moving ball at the instant of impact.
• FIGURE 6.6 The collision of two hard spheres with relative velocity v can be analyzed at the point of impact in terms of components of the velocity along the line connecting the mass centers and perpendicular to that line. The impact parameter, b, is the projection of the vector between the mass centers onto an axis perpendicular to the velocity vector. On the right is the relative velocity vector in terms of the individual velocity vectors of the colliding particle.
• 6.3 Rate Equations
• FIGURE 6.7 Plot of the form of the curves that show the dependence of the concentration of a reactant [A] on time for reactions that are zero, first, and second order in A.
• 6.4 Rate Laws for Complex Reactions
• Example 6.1: Integrated Rate Expression for an A + B Reaction
• 6.5 Temperature Dependence and Solvent Effects
• FIGURE 6.8 Possible solvent effects on the potential energy surface slice that follows the minimum energy path for a hypothetical bimolecular reaction. The vertical axis is the potential energy, and the horizontal axis is the reaction coordinate. Solvent 1 has an attractive interaction that is essentially constant along the reaction coordinate. Solvent 2 gives rise to a greater energy lowering at the transition state than elsewhere, and so it increases the rate of reaction. Solvent 3 shows the greatest energy lowering for the reactants and thereby increases the activation energy.
• 6.6 Reaction Thermodynamics
• TABLE 6.1 Standard State Thermodynamic Propertiesa
• Example 6.2: Temperature Dependence of a Reaction Enthalpy
• Example 6.3: ΔS and ΔG of a Reaction
• 6.7 Electrochemical Reactions
• FIGURE 6.9 Schematic representation of an electrochemical cell. The electrodes are conducting materials connected by a conductor (e.g., copper wire). A voltage across the electrodes can be measured by a potentiometer inserted into the circuit. It is common to place the anode on the left and the cathode on the right in these types of schematic drawings. Negative charge flows away from the anode (left) into the external part of the circuit. This means that oxidation, a loss of electrons, occurs at or in the vicinity of the anode. The electrons lost from the surrounding material flow into the anode; it is the negative electrode. At the cathode, or the positive electrode, reduction takes place as electrons flow from the external circuit to the substance in the cell. This flow of electrons corresponds to the cell voltage being measured as a positive value (E > 0). When it is negative, the reaction is spontaneous in the reverse direction. As well, introduction of an external voltage source (e.g., a battery) in the circuit may drive the reaction in one direction or the other.
• FIGURE 6.10 Electrochemical cell with an ion-selective membrane separating the two halves. For reactions involving gas phase reactants, a cell can be arranged so that the gases are kept separate as shown. These two, and other approaches, make it possible to physically separate reactions at the anode of a cell from reactions at the cathode.
• Example 6.4: ΔG of an Electrochemical Cell
• TABLE 6.2 Reduction Potentials for Reactions in Water at 298K Relative to a Hydrogen Standard
• Example 6.5: Effect of Temperature on Cell Voltage
• 6.8 Conclusions
• Point of Interest: Galactic Reaction Chemistry
• Exercises
• Bibliography
• 7 Vibrational Mechanics of Particle Systems
• 7.1 Classical Particle Mechanics and Vibration
• FIGURE 7.1 The harmonic oscillator is a special model problem in mechanics. It consists of a particle of mass m attached by a harmonic spring to an unmovable wall. In the one-dimensional harmonic oscillator problem, the particle has only one degree of freedom corresponding to motion to and away from the wall. In the discussion, this is the x-direction.
• 7.2 Vibration in Several Degrees of Freedom
• FIGURE 7.2 Double harmonic oscillator. This consists of two particles, 1 and 2, that move only in the direction of the x-axis. The masses of 1 and 2 are m1 and m2, respectively, and their positions are x1 and x2. Spring a is connected to an unmovable wall and to particle 1. Spring b connects the two particles.
• FIGURE 7.3 Oscillator system where two particles, each constrained to move along the x-axis, are attached by a harmonic spring with force constant k. The positions of the two particles along the x-axis are designated x1 and x2. The equilibrium length of the spring is the constant x0.
• FIGURE 7.4 Spherical polar coordinates for a two-particle system. r is the distance between m1 and m2. The center of mass is positioned at the origin of the (x,y,z) coordinate system. θ is the angle between the z-axis and the line connecting the two masses. θ ranges from 0 to π. φ measures the rotation about the z-axis of the line connecting the masses. It ranges from 0 to 2π. (Also see Figure 2.4.)
• 7.3 Quantum Phenomena and Wave Character
• 7.4 Quantum Mechanical Harmonic Oscillator
• FIGURE 7.5 The energy levels of the quantum mechanical harmonic oscillator depicted as horizontal lines intersecting the potential energy function for the oscillator.
• FIGURE 7.6 Wavefunctions of low-lying levels of the quantum mechanical harmonic oscillator. The functions are drawn so that the baseline is the energy level line as in Figure 7.5, and that means that the function is zero-valued or has a node at any point where it crosses the energy level line. Since it is the qualitative form that is usually of interest, the vertical axis for the wavefunctions is in arbitrary units. Notice that for the highest levels drawn, the wavefunction has maximum amplitude close to the classical turning points. A truly classical oscillator is moving at its slowest speed as it changes direction at a turning point, and so it is more likely to be found around the turning points than in the middle.
• 7.5 Harmonic Vibration of Many Particles
• FIGURE 7.7 Vibrating system of particles connected by springs. The potential energy of this system is a sum of the potential energies of the springs. Such energies have at least a quadratic dependence on the mass position coordinates, and that is the mathematical basis for the motions of the masses to be coupled.
• FIGURE 7.8 A vibrating system of two particles constrained to move in the x-direction and connected by springs to each other and to two unmovable walls.
• 7.6 Conclusions
• Point of Interest: Molecular Force Fields
• Exercises
• Bibliography
• 8 Molecular Quantum Mechanics
• 8.1 Quantum Mechanical Operators
• Theorem 8.1: Eigenvalues of a Hermitian Operator Are Real Numbers
• Theorem 8.2: Eigenfunctions of a Hermitian Operator with Different Associated Eigenvalues Are Orthogonal Functions
• Corollary 8.2.1
• Theorem 8.3
• Corollary 8.3.1
• 8.2 Information from Wavefunctions
• FIGURE 8.1 Probability density functions of low-lying levels of the quantum mechanical harmonic oscillator. These functions are the squares of the wavefunctions. They are drawn so that the baseline, or point of zero probability density, is the energy level line as in Figure 7.6. Notice that as n increases, the probability density at x = 0 diminishes. Furthermore, at levels such as n = 10 the points of maximum probability density are close to the classical turning points, unlike the situation for the lower states.
• Example 8.1: Position Uncertainty for the Harmonic Oscillator
• 8.3 Multidimensional Problems and Separability
• FIGURE 8.2 Two-dimensional oscillator where a particle of mass m is attached to two harmonic springs with force constants kx and ky. Each spring is connected to an unmovable rod, but the connection points can slide freely along the rods (i.e., along the x-axis or along the y-axis). Thus, the potential energy for the kx spring depends on only the x-coordinate of the particle, while the potential energy of the ky spring depends on only the y-coordinate. The equilibrium lengths of the springs are taken to be zero in the analysis of this example.
• 8.4 Particles with Box and Step Potentials
• FIGURE 8.3 Particle-in-a-box potential, which is an impenetrable potential outside x = 0 and x = l and zero inside, might serve as an approximation of the smoothly changing potential represented by the solid line.
• Example 8.2: Degenerate Energy Levels
• Example 8.3: Particle in a Three-Dimensional Box
• FIGURE 8.4 Hypothetical step potential for a one-dimensional particle. To understand this system, it is helpful to break it into regions. In the first region (I), there is a potential well that is almost like the potential of the particle-in-a-box problem. The second region (II) has a constant but not infinite potential. The third region (III) has a flat potential, and this continues to infinity.
• FIGURE 8.5 Transmission of a beam of particles coming from the right and encountering the potential in Figure 8.4 is represented by three horizontal lines corresponding to three specific particle energies. The width of each line is a representation of the transmission probability as a function of x. In the highest energy case, there is a negligible diminishment at the boundary between regions II and III, but at lower energies, there is a noticeable diminishment.
• 8.5 Rigid Rotator and Angular Momentum
• TABLE 8.1 Associated Legendre Polynomials through l = 4
• FIGURE 8.6 Angular momentum vectors of the Ylm states of the rigid rotator for l = 1 (solid lines), l = 2 (dashed lines), and l = 3 (dotted lines). The arrows represent the orientation with respect to the z-axis for the particular (l,m) state. The orientations with respect to the x-axis and the y-axis are not determined, and so this picture represents a planar slice through three-dimensional space such that the slice includes the z-axis and is perpendicular to the x–y plane. The angle between this planar slice and the x-axis or the y-axis, though, is arbitrary.
• 8.6 Coupling of Angular Momenta
• FIGURE 8.7 For two angular momentum vectors, with associated quantum numbers j1 and j2 both equal to 1, there are (2j1 + 1)(2j2 + 1) = 9 different combinations of their possible orientations with respect to the z-axis. These 9 ways are shown here, and for each a resultant vector from the sum of has been drawn. In each case, the z-component of the resultant vector can be obtained from the sum of the z-component quantum numbers m1 and m2.
• 8.7 Variation Theory
• Example 8.4: Variational Treatment of a Quartic Oscillator
• 8.8 Perturbation Theory
• FIGURE 8.8 Harmonic oscillator potential, a parabola, with a small perturbing potential, a bump in the middle. The composite potential is called a double-minimum potential or a double-well potential. Perturbation theory offers an excellent approach to understanding how the wavefunctions of the unperturbed (harmonic) system differ from the system with a bump in the potential. An example of this type of potential is found for the water molecule. Water is a bent molecule, which means that the minimum of its potential energy as a function of the atomic positions is at a structure that is bent. We can envision a water molecule being straightened into a linear arrangement of the atoms. Of course, that is necessarily a higher-energy arrangement. If the distortion process is continued, the molecule will be bent again but in the opposite sense. Eventually it will look like a mirror image of the original equilibrium structure, and so it will then be at a potential minimum. If the bending angle is taken to be a coordinate, then the potential energy as a function of that angle will have the form of a double-well potential. In the potential curve mentioned earlier, q could be the bending angle. Then, q = 0 would correspond to the linear arrangement of the atoms in the water molecule, and to the left and to the right of q = 0 would be the two potential minima.
• FIGURE 8.9 On the left is a harmonic vibrational potential with the exact energy levels and corresponding wavefunctions shown for the first several states. On the right is the same potential augmented, or perturbed, by a bump in the middle. The solid horizontal lines drawn with this potential are the energy levels obtained from an extensive variational treatment of this problem, whereas the dashed lines are the energy levels obtained from first-order perturbation theory using the system on the left as the zero-order picture and the bump as the perturbation. The wavefunctions on the right for the perturbed system are those obtained from a variational treatment. They demonstrate the detailed changes in the wavefunctions due to the perturbing potential, the most noticeable changes being in the lowest states.
• 8.9 Conclusions
• Point of Interest: The Quantum Revolution
• The Solvay Conference
• Exercises
• FIGURE 8.10 A particle-in-a-box problem, but with a barrier step potential. The potential is taken to be infinite, except in regions I, II, and III. In regions I and III, the potential is zero. In region II, the potential is the constant value V0.
• Bibliography
• Mathematical Background
• Introductory Level
• Intermediate and Advanced Level
• 9 Vibrational–Rotational Spectroscopy
• 9.1 Molecular Spectroscopy and Transitions
• FIGURE 9.1 Dipole in a uniform electric field arising from two oppositely charged parallel plates. A dipole moment, a property of a separated positive and negative charge, interacts differently with the field at the three orientations shown since each end of the dipole is attracted toward the oppositely charged plate.
• 9.2 Vibration and Rotation of a Diatomic Molecule
• FIGURE 9.2 Energy levels of a vibrating–rotating diatomic molecule according to Equation 9.30. Each horizontal line in this figure represents an energy level with energy increasing in the vertical direction. The levels are labeled by the vibrational quantum number, n, and the rotational quantum number, J. The long, thick lines are all J = 0 levels, and the energy spacing between any pair of these lines is The diagram has been terminated at n = 3, but the energy levels continue on infinitely in the same pattern. Levels for which J is not zero have been drawn as short, thin lines to help organize the figure, but there is nothing fundamentally different between these and the J = 0 levels. These lines appear as stacks or manifolds originating from each J = 0 line. The leftmost manifold is for the states for which n = 0, and the rightmost manifold is for all those for which n = 3. The numbers above four of these lines show the sequence of the J quantum numbers. Higher J levels exist but are not shown. The degeneracy of each level is 2J + 1. In order to show qualitative features without congestion, the levels have been drawn assuming a ratio of the vibrational frequency to the rotational constant of about 20. Usually, this ratio is greater, for example, it is about 200 for HF and 1000 for CO. Thus, the spacing between the rotational levels is very much smaller relative to the vibrational spacings depicted here.
• FIGURE 9.3 Plot of the function V(s) = s2/2 and of the function V′(s) = 1 + s + s2/2. Both functions are parabolas with the same curvature; however, the minimum of V′(s) has been shifted by an amount δ = −1, and the value of the potential at that point is V′(δ) = 1/2 rather than zero.
• 9.3 Vibrational Anharmonicity and Spectra
• FIGURE 9.4 Functional form of the Morse potential in Equation 9.36.
• FIGURE 9.5 Representative correlation diagram of several of the low-lying energy levels of a harmonic oscillator with the energy levels of an anharmonic oscillator. The anharmonic energy levels are those given by Equation 9.39 assuming that the potential function is the power series expansion of a Morse potential truncated at the cubic term. The closer spacing of the levels with increasing vibrational quantum number is typical.
• FIGURE 9.6 Dipole moment function of the hydrogen fluoride molecule as a function of the separation distance between H and F in angstroms. This curve is based on spectroscopic data. The equilibrium separation distance is at 0.92Å, and in the vicinity of the equilibrium, the dipole moment curve is very nearly linear.
• FIGURE 9.7 Idealized high-resolution spectrum of a low-density gas phase sample of hydrogen fluoride. The frequencies of the peaks are the transition frequencies, and the relative heights of the peaks correspond to the intensities. The characteristic form of an infrared spectrum of a diatomic molecule is that of two sets of lines, called branches, with diminishing peak height to the left and to the right. The branch on the left, at lower transition frequencies, is the P-branch, whereas the branch on the right is the R-branch. The separation between the two branches is roughly twice the separation between lines within the branches.
• Example 9.1: Diatomic Molecule Vibrational Spectrum
• FIGURE 9.8 High-resolution infrared spectrum of the fundamental band of HCl. The horizontal axis gives the frequency in wavenumbers. The vertical axis is the absorbance in arbitrary units. The P-branch is on the right in this presentation of the spectrum, and the R-branch is on the left. Each line appears to be split in two. In fact, this is a result of the sample used for this spectrum being a mixture of H35Cl and H37Cl at natural abundance of the chlorine isotopes. The mass differences lead to slightly different rotational constants, and so the spectrum is really a superposition of the spectra of H35Cl and H37Cl.
• FIGURE 9.9 High-resolution infrared spectrum of the fundamental band of HBr. The vertical axis is the absorbance in arbitrary units. Notice the difference in the band center of the spectra of HBr and HCl.
• FIGURE 9.10 Plot of the relative transition intensities of the rotational lines in a diatomic molecule’s vibrational absorption band. The curves are plots of the quantity (2J + 1)–exJ(J+1) but with J treated as a continuous variable. The numerical values of x are shown next to each curve. Notice that as the temperature increases, x becomes smaller.
• 9.4 Rotational Spectroscopy
• TABLE 9.1 Spectroscopic Constants (cm−1) of Certain Diatomic Moleculesa
• 9.5 Harmonic Picture of Polyatomic Vibrations
• FIGURE 9.11 Representation of the low-lying vibrational state energy levels of the water molecule in a harmonic picture showing the increasing number of states per energy interval at higher energies.
• TABLE 9.2 Vibrational Frequencies (cm−1) of Selected Small Molecules
• 9.6 Polyatomic Vibrational Spectroscopy
• TABLE 9.3 Characteristic Vibrational Stretching Frequencies (cm-1) of Certain Functional Groupsa
• FIGURE 9.12 The 10 lowest-lying vibrational energy levels of a one-dimensional system experiencing a double-minimum potential. The potential is harmonic away from the middle, and the highest levels hown have an energy spacing characteristic of a harmonic oscillator. However, the energies of the first six states are significantly perturbed by the bump in the potential. The perturbation brings the energies of pairs of states closer together
• 9.7 Conclusions
• Point of Interest: Laser Spectroscopy
• Exercises
• Bibliography
• 10 Electronic Structure
• 10.1 Hydrogen and One-Electron Atoms
• TABLE 10.1 Hydrogen Atom Radial Functionsa
• TABLE 10.2 Energy Levels of the Hydrogen Atoma
• FIGURE 10.1 Radial functions, Rnl(r), of the hydrogen (one-electron) atom. The top three functions are the l = 0 functions on a common vertical axis scale; the first two l = 1 functions are shown below on a different vertical scale. The horizontal scale is in Bohr radii (0.52918Å).
• FIGURE 10.2 Radial probability functions of the hydrogen atom, (The factor r2 is included because the radial volume element is r2dr.) The number of nodes increases with both n and l.
• 10.2 Orbital and Spin Angular Momentum
• FIGURE 10.3 Representation of several of the squared spherical harmonic functions with m = 0. The shapes depict the surface of a contour drawn for some fixed probability density amplitude, and the alternation in shading designates regions of different phases (different signs) of the wavefunction.
• FIGURE 10.4 Probability density representations of the real parts of the angular functions formed from linear combinations of spherical harmonic functions.
• 10.3 Atomic Orbitals and Atomic States
• FIGURE 10.5 Lowest energy levels of the hydrogen atom (horizontal lines) and the allowed transitions. States with the same term symbol have been placed in the same column for clarity.
• 10.4 Molecules and the Born–Oppenheimer Approximation
• 10.5 Antisymmetrization of Electronic Wavefunctions
• 10.6 Molecular Electronic Structure
• TABLE 10.3 Comparison of Calculated and Measured Values of Certain Molecular Properties
• FIGURE 10.6 Orbital energy correlation diagram for the linear combination of atomic orbitals to form molecular orbitals. The vertical axis is an orbital energy scale, and the leftmost and rightmost horizontal lines represent the orbital energies of two noninteracting hydrogen atoms. In the middle of the figure are two horizontal lines representing the energies of mixed functions of the left and right hydrogen 1s orbitals. Their energies are qualitatively deduced from first-order degenerate perturbation theory which requires that the two mixed states be above and below the energies of the unmixed degenerate states by equal amounts. The qualitative form of the mixed orbitals is also shown.
• FIGURE 10.7 Orbital correlation diagram for carbon monoxide. The carbon atomic orbital energies are on the left, and the oxygen atomic orbital energies are on the right. The molecular orbitals that form from mixing of the atomic orbitals are represented by the horizontal lines in the center at their approximate orbital energies in the CO molecule. The vertical lines indicate the orbital occupancy.
• 10.7 Visible–Ultraviolet Spectra of Molecules
• FIGURE 10.8 Potential energy curves for two electronic states, the ground state and an excited state of the same spin, of a hypothetical diatomic molecule. Each potential can be analyzed independently to yield vibrational–rotational states and energy levels. The lowest four vibrational levels are shown. In an absorption spectrum, transitions can originate from the vibrational–rotational levels of the ground electronic state and end in the vibrational–rotational levels of the excited state following the appropriate selection rules. An arrow from the n′ = 0 to the n″ = 1 level depicts the initial and final states of one possible transition. The energies of the transitions are sums of the energy difference between the bottoms of the potential wells, designated Te, and the difference between the vibrational–rotational state energies within their respective potentials.
• FIGURE 10.9 Vibrational wavefunctions and overlaps for two identical harmonic oscillator wavefunctions with offset minima.
• Example 10.1: Diatomic Molecule Electronic Absorption Bands
• 10.8 Properties and Electronic Structure
• FIGURE 10.10 Influence of an external electric field in the z-direction on a hydrogen atom. We can think of the field as arising from two oppositely charged parallel plates. We can think of the electron of the hydrogen atom as being attracted toward the positively charged plate, while the proton is attracted toward the negatively charged plate. This represents a separation of the centers of negative and positive charge in the hydrogen atom, as shown schematically. The result is a nonzero dipole moment. The charge distribution of the electron is roughly that of the superposition of a 1s orbital with a 2p orbital along the z-axis. The 2p orbital adds to the wavefunction amplitude on one side of the nucleus and diminishes the amplitude on the other side where it is of opposite phase.
• 10.9 Conclusions
• Point of Interest: John Clarke Slater
• Exercises
• Bibliography
• Advanced Texts and Monographs
• 11 Statistical Mechanics
• 11.1 Probability
• 11.1.1 Classical Behavior
• 11.2 Ensembles and Arrangements
• 11.3 Distributions and the Chemical Potential
• 11.3.1 High-Temperature Behavior
• 11.3.2 Low-Temperature Behavior
• 11.3.3 Dilute Behavior
• 11.4 Molecular Partition Functions
• Example 11.1: Products of Partition Functions of Independent Systems
• 11.5 Thermodynamic Functions
• Example 11.2: Internal Energy of an Ideal Diatomic Gas
• 11.6 Heat Capacities
• FIGURE 11.1 Plot of the function f(α) = e−1/α/α2(1−e−1/α)2 versus α. In the expression for constant-volume heat capacity, Equation (11.63), this function is the vibrational contribution to CV, with α being proportional to temperature.
• 11.7 Conclusions
• Point of Interest: Lars Onsager
• Exercises
• Bibliography
• 12 Magnetic Resonance Spectroscopy
• 12.1 Nuclear Spin States
• FIGURE 12.1 Nuclear spin energy levels of a molecule with two noninteracting protons in different chemical environments. The levels separate in energy with increasing strength of the external field, according to Equation 12.6. The levels are labeled by the mI quantum numbers of the two protons. The vertical arrows indicate the allowed NMR transitions. Clearly, the energy of these transitions depends on the strength of the applied field.
• FIGURE 12.2 Energy level diagram for the four spin states of H19F in an external magnetic field of fixed strength. The allowed transitions are represented by the vertical arrows. Because of the sizable difference in g values, the transition energies for the 19F spin flip are much different than the transition energies of the proton. In practice, two different instrumental setups are required to observe the two transitions.
• 12.2 Nuclear Spin–Spin Coupling
• FIGURE 12.3 Energy levels for a hypothetical molecule with two protons that are nearby but in different chemical environments. On the left are the energy levels found in the absence of spin–spin interaction. On the right are the energy levels with spin–spin interaction treated via first-order perturbation theory. The vertical arrows show the allowed transitions, and the lengths of these arrows are proportional to the transition energies. Thus, the effect of the spin–spin interaction is seen to split the transitions, that is, to take each pair of transitions that would occur at the same frequency and shift one to a higher frequency and one to a lower frequency.
• FIGURE 12.4 Representation of the NMR spectra for a molecule with the energy levels and transitions given in Figure 12.3. The top spectrum corresponds to the energy levels in the absence of spin–spin interaction, whereas the bottom spectrum includes the effect.
• FIGURE 12.5 Energy level diagram for the nuclear spin states of a hypothetical molecule with three interacting protons in different chemical environments. The energy levels on the left neglect spin–spin interaction. The allowed transitions give rise to a spectrum of three lines, shown as a stick representation at the bottom. These three lines correspond to a spin “flip” of each of the three different nuclei, and their relative transition frequencies give the chemical shift of each. The energy levels on the right have been obtained by first including the 1–2 coupling, then the 1–3 coupling, and finally the 2–3 coupling, assuming J12 > J13 > J23. Transition lines are drawn for the final set of levels, and the resulting stick spectrum is shown at the bottom. The values of the three coupling constants can be obtained directly from the spectrum, as shown.
• FIGURE 12.6 Nuclear spin energy levels of the HD molecule in a magnetic field. The allowed transitions, represented by vertical arrows, are those for which the mI quantum number of the deuterium changes by 1, or the mI quantum number of the proton changes by 1. The deuterium and proton stick spectra are shown at the bottom.
• FIGURE 12.7 Energy levels and transitions for the proton spin states of formaldehyde. On the left are the levels obtained with neglect of spin–spin interaction, and they are labeled by the mI quantum numbers of the two protons. The middle level is doubly degenerate. If spin–spin interaction is treated with first-order degenerate perturbation theory, the levels on the right result, with an assumed size of the coupling constant, J. A transition from the I = 0 state to an I = 1 state would measure J, but that is a forbidden transition. No splitting of the line in the spectrum occurs.
• FIGURE 12.8 Representation of the NMR spectrum of a hypothetical molecule with two equivalent protons and a third, nearby proton in a different chemical environment. This spectrum may be understood as a combination of the spectrum of a system with an I = 1 particle interacting with the third proton and the spectrum from an I = 0 particle interacting with the third proton. These two pseudoparticles correspond to the possible couplings of the spins of the two equivalent nuclei.
• Example 12.1: NMR Energy Levels of Methane
• 12.3 Electron Spin Resonance Spectra
• FIGURE 12.9 High-field energy levels for a system with electron spin S = 1/2 and with a magnetic nucleus with I = 1/2. The levels on the left are the zero-order energies, and the levels on the right include the first-order corrections due to electron spin–nuclear spin interaction. The two vertical arrows show the allowed transitions that would be observed in an ESR experiment. Not shown are the allowed transitions for which ΔMS = 0 since these are the NMR transitions and occur at very different energies.
• FIGURE 12.10 High-field energy levels for a system with electron spin S = 1/2 and magnetic nuclei with I = 1/2 and I = 1. On the left are the levels at zero order where only the interaction with the external field is included. On the right are the energy levels according to Equation 12.22, where the spin–spin interactions have been included by first-order perturbation theory. The nuclear spin–spin interaction has been exaggerated relative to typical values in order to show the effect. The vertical lines correspond to the allowed ESR transitions, and below them is a stick representation of the spectrum that corresponds to these transitions.
• 12.4 Extensions of Magnetic Resonance
• FIGURE 12.11 Two sample tubes embedded in a larger sample tube. Spectra are taken in the vicinity of the transition seen for a type of proton center in the substance in the small tubes. The field strength varies in the direction shown by the arrow. As the small tubes are rotated with respect to the applied field, the signal changes from a single peak to two peaks and back again thereby encoding spatial information in the data.
• FIGURE 12.12 An MRI image of an axial section of a human brain. The bright region in the right cerebral hemisphere is an indication of a tumor. The MRI used to obtain these images operates with a 1.5 Tesla magnet, and the image is based on signals collected for a region that was 5mm thick.
• FIGURE 12.13 Two-dimensional 1H NMR spectrum of guanosine triphosphate. The small circles are contours of the spectrum’s “spikes,” the equivalent of peaks in a 1D spectrum. (They appear as spikes when the intensities are expressed on a scale in the direction perpendicular to the plane of the paper.) Greater intensity is reflected in more contours being shown-more closely spaced contour circles.
• 12.5 Conclusions
• Point of Interest: The NMR Revolution
• Exercises
• Bibliography
• 13 Introduction to Surface Chemistry
• 13.1 Interfacial Layer and Surface Tension
• FIGURE 13.1 Diagram of an apparatus used to determine the surface tension of a liquid.
• 13.2 Adsorption and Desorption
• FIGURE 13.2 An idealized adsorption isotherm for a typical gas surface interaction.
• FIGURE 13.3 Comparison of the chemisorption and physisorption processes.
• 13.3 Langmuir Theory of Adsorption
• 13.4 Temperature and Pressure Effects on Surfaces
• FIGURE 13.4 Typical relationship between the surface coverage and the pressure of an adsorbed gas.
• 13.5 Surface Characterization Techniques
• FIGURE 13.5 A plot of the probability of escape from a surface as a function of the surface depth.
• 13.6 Conclusions
• Point of Interest: Irving Langmuir
• Exercises
• Bibliography
• Back Matter
• Appendix A: Mathematical Background
• A.1 Power Series
• FIGURE A.1 The tangent to some function f(x) at the point x = c shows how f changes linearly with c. The slope of the tangent line times some incremental change in x is an estimate of the value of f at the new point. As the increment, d, increases in size, the tangent line tends to be less accurate as an estimator of the value for the function displayed.
• A.2 Curve Fitting
• A.3 Multivariable Partial Differentiation
• A.4 Matrix Algebra
• A.5 Matrix Eigenvalue Methods
• A.5.1 Simultaneous Diagonalization
• Appendix B: Molecular Symmetry
• B.1 Symmetry Operations
• B.2 Molecular Point Groups
• B.3 Symmetry Representations
• FIGURE B.1 Application of water’s C2 symmetry operator to individual hydrogen 1s orbitals rotates them into each other. Two specific linear combinations of the orbitals prove to be eigenfunctions of this symmetry operator.
• FIGURE B.2 Linear combinations formed from 2pz atomic orbitals on the hydrogen atoms in the water molecule and the effect of applying symmetry operators to these functions.
• B.4 Symmetry-Adapted Functions
• B.5 Character Tables
• B.6 Application of Group Theory
• TABLE B.1 Character Tables of Common Molecular Point Groupsa
• Appendix C: Special Quantum Mechanical Approaches
• C.1 Angular Momentum Raising and Lowering Operators
• C.2 Matrix Methods in Variation and Perturbation Theory
• Appendix D: Table of Integrals
• Appendix E: Table of Atomic Masses and Nuclear Spins
• Appendix F: Fundamental Constants and Conversion of Units
• F.1 Systems of Units
• TABLE F.1 Conversion Factors
• F.2 Fundamental Constants
• TABLE F.2 Values of Constantsa
• F.3 Atomic Units
• TABLE F.3 Energy Conversion Factors
• F.4 Electromagnetic Units and Constants
• F.5 Frequencies
• TABLE F.4 The Electromagnetic Spectrum
• F.6 The Electromagnetic Spectrum
• Appendix G: List of Tables
• Appendix H: Points of Interest
• Appendix I: Atomic Masses and Percent Natural Abundance of Light Elements
• Appendix J: Values of Constants
• Appendix K: The Greek Alphabet
• Answers to Selected Exercises
• Index