(BQ) Part 2 book “Physical chemistry” has contents: Transport processes, reaction kinetics, quantum mechanics, atomic structure, molecular electronic structure, spectroscopy and photochemistry, statistical mechanics,… and other contents.
lev38627_ch15.qxd 3/24/08 6:22 PM Page 474 CHAPTER 15 Transport Processes CHAPTER OUTLINE 15.1 Kinetics 15.2 Thermal Conductivity 15.3 Viscosity 15.4 Diffusion and Sedimentation 15.5 Electrical Conductivity 15.6 Electrical Conductivity of Electrolyte Solutions 15.7 Summary 15.1 KINETICS So far, we have discussed only equilibrium properties of systems Processes in systems in equilibrium are reversible and are comparatively easy to treat This chapter and the next deal with nonequilibrium processes, which are irreversible and hard to treat The rate of a reversible process is infinitesimal Irreversible processes occur at nonzero rates The study of rate processes is called kinetics or dynamics Kinetics is one of the four branches of physical chemistry (Fig 1.1) A system may be out of equilibrium because matter or energy or both are being transported between the system and its surroundings or between one part of the system and another Such processes are transport processes, and the branch of kinetics that studies the rates and mechanisms of transport processes is physical kinetics Even though neither matter nor energy is being transported through space, a system may be out of equilibrium because certain chemical species in the system are reacting to produce other species The branch of kinetics that studies the rates and mechanisms of chemical reactions is chemical kinetics or reaction kinetics Physical kinetics is discussed in Chapter 15 and chemical kinetics in Chapter 16 There are several kinds of transport processes If temperature differences exist between the system and surroundings or within the system, it is not in thermal equilibrium and heat energy flows Thermal conduction is studied in Sec 15.2 If unbalanced forces exist in the system, it is not in mechanical equilibrium and parts of the system move The flow of fluids is the subject of fluid dynamics (or fluid mechanics) Some aspects of fluid dynamics are treated in Sec 15.3 on viscosity If differences in concentrations of substances exist between different regions of a solution, the system is not in material equilibrium and matter flows until the concentrations and the chemical potentials have been equalized This flow differs from the bulk flow that arises from pressure differences and is called diffusion (Sec 15.4) When an electric field is applied to a system, electrically charged particles (electrons and ions) experience a force and may move through the system, producing an electric current Electrical conduction is studied in Secs 15.5 and 15.6 We shall see that the laws describing thermal conduction, fluid flow, diffusion, and electrical conduction all have the same form, namely, that the rate of transport is proportional to the spatial derivative (gradient) of some property Transport properties are important in determining how fast pollutants spread in the environment (see chap of D G Crosby, Environmental Toxicology and Chemistry, Oxford, 1998) Biological examples of transport phenomena include the flow of blood, the diffusion of solute molecules in cells and through cell membranes, and the diffusion of neurotransmitters between nerve cells Transport phenomena such as migration of charged species in an electric field are used to separate biomolecules and played a key role in sequencing the human genome (Sec 15.6) lev38627_ch15.qxd 3/24/08 6:22 PM Page 475 475 T2 Ͼ T1 x Section 15.2 Thermal Conductivity Substance T1 reservoir T2 reservoir Figure 15.1 Conduction of heat through a substance Adiabatic wall 15.2 THERMAL CONDUCTIVITY Figure 15.1 shows a substance in contact with two heat reservoirs at different temperatures A steady state will eventually be reached in which there is a uniform temperature gradient d T/dx in the substance, and the temperature between the reservoirs varies linearly with x from T1 at the left end to T2 at the right end (The gradient of a quantity is its rate of change with respect to a spatial coordinate.) The rate of heat flow dq/dt across any plane perpendicular to the x axis and lying between the reservoirs will also be uniform and will clearly be proportional to Ꮽ, the substance’s cross-sectional area in a plane perpendicular to the x axis Experiment shows that dq/dt is also proportional to the temperature gradient d T/dx Thus dq dT ϭ ϪkᏭ dt dx (15.1) where the proportionality constant k is the substance’s thermal conductivity and dq is the heat energy that in time dt crosses a plane with area Ꮽ and perpendicular to the x axis The minus sign occurs because dT/dx is positive but dq/dt is negative (the heat flows to the left in the figure) Equation (15.1) is Fourier’s law of heat conduction This law also holds when the temperature gradient in the substance is nonuniform; in this case, d T/dx has different values at different places on the x axis, and dq/dt varies from place to place (Fourier, discoverer of the laws of heat conduction, apparently suffered from a thyroid disorder and wore an overcoat in summer.) k is an intensive property whose value depends on T, P, and composition From (15.1), the SI units of k are J KϪ1 mϪ1 sϪ1 ϭ W KϪ1 mϪ1, where watt (W) equals J/s.Values of k for some substances at 25°C and atm are shown in Fig 15.2 Metals are good conductors of heat because of the electrical-conduction electrons, which move relatively freely through the metal Most nonmetals are poor conductors of heat Gases are very poor conductors because of the low density of molecules Diamond has the highest room-temperature thermal conductivity of any substance at 300 K [Some theoretical calculations indicate that carbon nanotubes might have a higher thermal conductivity than diamond, but conflicting results from various calculations and from experiments leave this question in doubt k for an individual single-wall carbon nanotube (like many other intensive properties of nanomaterials) depends on the tube length, which complicates things; see J R Lukes and H Zhong, J Heat Transfer, 129, 705 (2007).] Although the system in Fig 15.1 is not in thermodynamic equilibrium, we assume that any tiny portion of the system can be assigned values of thermodynamic variables such as T, U, S, and P, and that all the usual thermodynamic relations between such variables hold in each tiny subsystem This assumption, called the principle of local state or the hypothesis of local equilibrium, holds well in most (but not all) systems of interest Figure 15.2 Thermal conductivities of substances at 25°C and atm The scale is logarithmic lev38627_ch15.qxd 3/24/08 6:22 PM Page 476 476 Chapter 15 Transport Processes Figure 15.3 Thermal conductivity k of liquid water versus temperature at several pressures The thermal conductivity k is a function of the local thermodynamic state of the system and therefore depends on T and P for a pure substance (Fig 15.3) For solids and liquids, k may either decrease or increase with increasing T For gases, k increases with increasing T (Fig 15.7) The pressure dependence of k for gases is discussed later in this section Thermal conduction is due to molecular collisions Molecules in a highertemperature region have a higher average energy than molecules in an adjacent lowertemperature region In intermolecular collisions, it is very probable for molecules with higher energy to lose energy to lower-energy molecules This results in a flow of molecular energy from high-T to low-T regions In gases, the molecules move relatively freely, and the flow of molecular energy in thermal conduction occurs by an actual transfer of molecules from one region of space to an adjacent region, where they undergo collisions In liquids and solids, the molecules not move freely, and the molecular energy is transferred by successive collisions between molecules in adjacent layers, without substantial transfer of molecules between regions Besides conduction, heat can be transferred by convection and by radiation In convection, heat is transferred by a current of fluid moving between regions that differ in temperature This bulk convective flow arises from differences in pressure or in density in the fluid and should be distinguished from the random molecular motion involved in thermal conduction in gases In radiative transfer of heat, a warm body emits electromagnetic waves (Sec 20.1), some of which are absorbed by a cooler body (for example, the sun and the earth) Equation (15.1) assumes the absence of convection and radiation In measuring k for fluids, great care must be taken to avoid convection currents Kinetic Theory of Thermal Conductivity of Gases The kinetic theory of gases yields theoretical expressions for the thermal conductivity and other transport properties of gases, and the results agree reasonably well with experiment The rigorous equations underlying transport processes in gases were worked out in the 1860s and 1870s by Maxwell and by Boltzmann, but it wasn’t until 1917 that Sydney Chapman and David Enskog, working independently, solved these extremely complicated equations (The Chapman–Enskog theory is so severely mathematical that Chapman remarked that reading an exposition of the theory is “like chewing glass.”) Instead of presenting rigorous analyses, this chapter gives very crude treatments based on the assumption of hard-sphere molecules with a mean free path given by Eq (14.67) The mean-free-path method (devised by Maxwell in 1860) gives results that are qualitatively correct but quantitatively wrong We shall assume that the gas pressure is neither very high nor very low Our treatment is based on collisions between two molecules and assumes no intermolecular forces except at the moment of collision At high pressures, intermolecular forces in the intervals between collisions become important, and the mean-free-path formula (14.67) does not apply At very low pressures, the mean free path l becomes comparable to, or larger than, the dimensions of the container, and wall collisions become important Thus our treatment applies only for pressures such that d V l V L, where d is the molecular diameter and L is the smallest dimension of the container In Sec 14.7, we found l to be about 10Ϫ5 cm at atm and room temperature Since l is inversely proportional to pressure, l is 10Ϫ7 cm at 102 atm and is 10Ϫ2 cm at 10Ϫ3 atm Thus our treatment applies to the pressure range from 10 Ϫ2 or 10 Ϫ3 atm to 101 or 102 atm We make the following assumptions: (1) The molecules are rigid, nonattracting spheres of diameter d (2) All molecules in a given region move at the same speed ͗v͘ characteristic of the temperature of that region and travel the same distance l between successive collisions (3) The direction of molecular motion after a collision is completely random (4) Complete adjustment of the molecular energy e occurs at each collision; this means that a gas molecule moving in the x direction and colliding with lev38627_ch15.qxd 3/24/08 6:22 PM Page 477 477 Section 15.2 Thermal Conductivity Figure 15.4 Three planes separated by 23 l, where l is the mean free path in the gas a molecule in a plane located at x ϭ xЈ will take on the average energy eЈ characteristic of molecules in the plane at xЈ Assumptions and are false Assumption is also inaccurate, in that, after a collision, a molecule is somewhat more likely to be moving in or close to its original direction of motion than in other directions Assumption is not bad for translational energy but is very inaccurate for rotational and vibrational energies Let a steady state be established in Fig 15.1 and consider a plane perpendicular to the x axis and located at x ϭ x0 (Fig 15.4) To calculate k, we must find the rate of flow of heat energy through this plane The net heat flow dq through the x0 plane in time dt is dq ϭ eL dNL Ϫ eR dNR (15.2) where dNL is the number of molecules coming from the left that cross the x0 plane in time dt and eL is the average energy (translational, rotational, and vibrational) of each of these molecules; dNR and eR are the corresponding quantities for molecules crossing the x0 plane from the right Since we are assuming no convection, there is no net flow of gas, and dNL ϭ dNR To find dNL, we think of the plane at x0 as an invisible “wall,” and use Eq (14.56), which gives the number of molecules hitting a wall in time dt as dN ϭ 14(N/V)͗v͘Ꮽdt Therefore dNL ϭ dNR ϭ 14 1N>V2 v9 Ꮽ dt (15.3) where N/V is the number of molecules per unit volume at the x0 plane, whose crosssectional area is Ꮽ The molecules coming from the left have traveled an average distance l since their last collision The molecules move into the x0 plane at various angles By averaging over the angles, one finds that the average perpendicular distance from the x0 plane to the point of last collision is 23l (see Kennard, pp 139–140, for the proof) Figure 15.4 shows an “average” molecule moving into the x0 plane from the left Molecules moving into the x0 plane from the left will, by assumption 4, have an average energy that is characteristic of molecules in the plane at x0 Ϫ 23l Thus, eL ϭ e_, where eϪ is the average molecular energy in the plane at x0 Ϫ 23l Similarly, eR ϭ eϩ, where eϩ is the average molecular energy at the x0 ϩ 23l plane Equation (15.2) becomes dq ϭ eϪ dNL Ϫ eϩ dNR, and substitution of (15.3) for dNL and dNR gives dq ϭ 14 1N>V2 v 9Ꮽ1eϪ Ϫ eϩ dt (15.4) The energy difference eϪ Ϫ eϩ is directly related to the temperature difference TϪ Ϫ Tϩ between the x0 Ϫ 32l and x0 ϩ 23l planes Letting de denote this energy difference, we have de de dT dT ϭ dx dT dT dx (15.5) dx ϭ 1x0 Ϫ 23 l Ϫ 1x0 ϩ 23 l ϭ Ϫ43 l (15.6) eϪ Ϫ eϩ ϵ de ϭ where dT ϵ TϪ Ϫ Tϩ and lev38627_ch15.qxd 3/24/08 6:25 PM Page 478 478 Chapter 15 Transport Processes Since we assumed no intermolecular forces except at the instant of collision, the total energy is the sum of the energies of the individual gas molecules, and the local molar thermodynamic internal energy is Um ϭ NAe, where NA is the Avogadro constant Therefore CV,m d1Um>NA de dUm ϭ ϭ ϭ dT dT NA dT NA (15.7) since CV,m ϭ dUm /dT for an ideal gas [Eq (2.68)] Use of (15.7) and (15.6) in (15.5) gives eϪ Ϫ eϩ ϭ Ϫ 4CV,m l dT 3NA dx (15.8) and (15.4) becomes dq ϭ Ϫ dT N v9 ᏭCV,m l dt 3NAV dx We have N/NAV ϭ n/V ϭ m/MV ϭ r/M, where n, m, r, and M are the number of moles of gas, the mass of the gas, the gas density, and the gas molar mass Therefore r8v CV,m l dT dq ϭϪ Ꮽ dt 3M dx Comparison with Fourier’s law dq/dt ϭ ϪkᏭ d T/dx [Eq (15.1)] gives k Ϸ 13 CV, m l v9 r>M hard spheres (15.9) Because of the crudity of assumptions to 4, the numerical coefficient in this equation is wrong A rigorous theoretical treatment (Kennard, pp 165–180) for hardsphere monatomic molecules gives kϭ 25p CV,m l v9 r 64 M hard spheres, monatomic (15.10) The rigorous extension of (15.10) to polyatomic gases is a very difficult problem that has not yet been fully solved Experiments on intermolecular energy transfer show that rotational and vibrational energy is not as easily transferred in collisions as translational energy The heat capacity CV,m is the sum of a translational part and a vibrational and rotational part [see Sec 14.10 and Eq (14.18)]: CV,m ϭ CV,m,tr ϩ CV,m,vibϩrot ϭ 32 R ϩ CV,m,vibϩrot (15.11) Because vibrational and rotational energy is less easily transferred than translational, it contributes less to k Therefore, in the expression for k, the coefficient of CV,m,vibϩrot should be less than the value 25p/64, which is correct for CV,m,tr [Eq (15.10)] Eucken gave nonrigorous arguments for taking the coefficient of CV,m,vibϩrot as two-fifths that of CV,m,tr, and doing so leads to fairly good agreement with experiment Thus, for polyatomic molecules, 25pCV,m /64 in (15.10) is replaced by 25p 25p 25p 3R 3R 5p C ϩ C ϭ ϩ a CV,m Ϫ b 64 V,m,tr 64 V,m,vibϩrot 64 32 ϭ 5p a CV,m ϩ R b 32 The thermal conductivity of a gas of polyatomic (or monatomic) hard-sphere molecules is then predicted to be kϭ l v 9r 5p 9 RT 1>2 aCV,m ϩ Rb ϭ aCV,m ϩ Rb a b 32 M 16 pM NAd hard spheres (15.12) lev38627_ch15.qxd 3/25/08 9:19 AM Page 479 479 where (14.67) and (14.47) for l and ͗v͘ and the ideal-gas law r ϭ PM/RT were used (Other approaches to the calculation of thermal conductivities are considered in Poling, Prausnitz, and O’Connell, chap 10.) Use of (15.12) to calculate k requires knowledge of the molecular diameter d Even a truly spherical molecule like He does not have a well-defined size, so it is hard to say what value of d should be used in (15.12) In the next section, we shall use experimental gas viscosities to get d values appropriate to the hard-sphere model [see (15.25) and (15.26)] Using d values calculated from 0°C viscosities, one finds the following ratios of theoretical gas thermal conductivities predicted by (15.12) to experimental values at 0°C: 1.05 for He, 0.99 for Ar, 0.96 for O2, and 0.97 for C2H6 How does k in (15.12) depend on T and P? The heat capacity CV,m varies slowly with T and very slowly with P Hence (15.12) predicts k ϰ T 1/2P Surprisingly, k is predicted to be independent of pressure As P increases, the number of heat carriers (molecules) per unit volume increases, thereby tending to increase k However, this increase is nullified by the decrease in l in (15.10) with increasing P As l decreases, each molecule goes a shorter average distance between collisions and is therefore less effective in transporting heat Data show that k for gases does increase with increasing T but faster than the T 1/2 behavior predicted by the rather crude hard-sphere model Molecules are actually “soft” rather than hard Moreover, they attract one another over significant distances Use of improved expressions for intermolecular forces gives better agreement with the observed T dependence of k (Kauzmann, pp 218–231) The prediction that k is independent of P holds well, provided P is not too high or too low (Recall the restriction d V l V L.) Values of k versus P are plotted for some gases at 50°C in Fig 15.5 k is nearly constant for pressures up to about 50 atm At very low pressures (below, say, 0.01 torr), the gas molecules in Fig 15.1 travel back and forth between the reservoirs, making very few collisions with one another At pressures low enough to make l substantially larger than the separation between the heat reservoirs, heat is transferred by molecules moving directly from one reservoir to the other, and the rate of heat flow is proportional to the rate of molecular collisions with the reservoir walls Since the rate of wall collisions is proportional to the pressure, dq/dt becomes proportional to P at very low pressures and goes to zero as P goes to zero One finds that Fourier’s law (15.1) does not hold in this very-lowpressure range (see Kauzmann, p 206), and so k is not defined here Between the pressure range where dq/dt is independent of P and the range where it is proportional to P, there is a transition range in which k falls off from its moderate-pressure value The falloff of k begins at 10 to 50 torr, depending on the gas The pressure dependence of dq/dt at very low pressures is the basis of the Pirani gauge and the thermocouple gauge used to measure pressure in vacuum systems These gauges have a heated wire sealed into the vacuum system The temperature T and hence the resistance R of this wire vary with P of the surrounding gas, and monitoring T or R of a properly calibrated gauge gives us P A theoretical equation for k of liquids is given in Prob 15.5 15.3 VISCOSITY Viscosity This section deals with the bulk flow of fluids (liquids and gases) under a pressure gradient Some fluids flow more easily than others The property that characterizes a fluid’s resistance to flow is its viscosity h (eta) We shall see that the speed of flow through a tube is inversely proportional to the viscosity To get a precise definition of h, consider a fluid flowing steadily between two large plane parallel plates (Fig 15.6) Experiment shows that the speed vy of the fluid flow is a maximum midway between the plates and decreases to zero at each plate Section 15.3 Viscosity Figure 15.5 Thermal conductivities of some gases versus P at 50°C lev38627_ch15.qxd 3/25/08 9:24 AM Page 480 480 Chapter 15 Transport Processes Figure 15.6 A fluid flowing between two planar plates The arrows in the figure indicate the magnitude of vy as a function of the vertical coordinate x The condition of zero flow speed at the boundary between a solid and a fluid, called the no-slip condition, has been verified in many experiments, but very small amounts of slip have been detected in certain special situations The no-slip condition probably results from attractions of the molecules of the fluid to molecules of the solid and the trapping of fluid in pockets and crevices on the surface of the rough solid [For reviews of the no-slip condition, see E Lauga et al., arxiv.org/abs/ cond-mat/0501557 (2005); C Neto et al., Rep Progr Phys., 68, 2859 (2005) Knowledge of the correct boundary condition for fluid flow is important to understand flow in microfluidic devices that are currently under development.] Adjacent horizontal layers of fluid in Fig 15.6 flow at different speeds and “slide over” one another As two adjacent layers slip past each other, each exerts a frictional resistive force on the other, and this internal friction gives rise to viscosity Consider an imaginary surface of area Ꮽ drawn between and parallel to the plates (Fig 15.6) Whether the fluid is at rest or in motion, the fluid on one side of this surface exerts a force of magnitude PᏭ in the x direction on the fluid on the other side, where P is the local pressure in the fluid Moreover, because of the change in flow speed as x changes, the fluid on one side of the surface exerts a frictional force in the y direction on the fluid on the other side Let Fy be the frictional force exerted by the slower-moving fluid on one side of the surface (side in the figure) on the fastermoving fluid (side 2) Experiments on fluid flow show that Fy is proportional to the surface area of contact and to the gradient dvy /dx of flow speed The proportionality constant is the fluid’s viscosity h (sometimes called the dynamic viscosity): Fy ϭ ϪhᏭ dvy dx (15.13) The minus sign shows that the viscous force on the faster-moving fluid is in the direction opposite its motion By Newton’s third law of motion (action ϭ reaction), the faster-moving fluid exerts a force hᏭ(dvy /dx) in the positive y direction on the slower-moving fluid The viscous force tends to slow down the faster-moving fluid and speed up the slower-moving fluid Equation (15.13) is Newton’s law of viscosity Experiments show it to be well obeyed by gases and by most liquids, provided the flow rate is not too high When Eq (15.13) applies, we have laminar (or streamline) flow At high rates of flow, (15.13) does not hold, and the flow is called turbulent Both laminar flow and turbulent flow are types of bulk (or viscous) flow In contrast, for flow of a gas at very low pressures, the mean free path is long, and the molecules flow independently of one another; this is molecular flow, and it is not a type of bulk flow When flow is turbulent, addition of extremely small amounts of a long-chain polymer solute to the liquid reduces substantially the resistance to flow through pipes, a phenomenon called the Toms effect after its discoverer The cost of pumping oil through the trans-Alaska pipeline is substantially reduced by the addition of a drag-reducing polymer to the crude oil Injection of the nontoxic polymer poly(ethylene oxide) was found to normalize blood flow to the heart in dogs with an artery that had been constricted by a surrounding balloon, and has been suggested as a treatment for persons with coronary artery disease [J J Pacella et al., Eur Heart J., 27, 2362 (2006); S Kaul and A R Jayaweera, ibid., 27, 2272] lev38627_ch15.qxd 3/24/08 6:22 PM Page 481 481 Blood flow is mainly laminar Turbulent flow is noisy and can be detected with a stethoscope Turbulence-produced noises (murmurs and bruits) heard with a stethoscope indicate abnormalities The onset and cessation of noise is used to measure systolic and diastolic blood pressure with a stethoscope and blood-pressure cuff Atherosclerotic plaques “usually form where the arteries branch—presumably because the constant turbulent blood flow at these areas injures the artery’s wall, making these areas more susceptible” to plaque formation (Merck Manual of Medical Information: Second Home Edition, Merck, 2003, chap 32) Section 15.3 Viscosity A Newtonian fluid is one for which h is independent of dvy /dx For a nonNewtonian fluid, h in (15.13) changes as dvy /dx changes Gases and most pure nonpolymeric liquids are Newtonian Polymer solutions, liquid polymers, and colloidal suspensions are often non-Newtonian An increase in flow rate and in dvy /dx may change the shape of flexible polymer molecules, facilitating flow and reducing h From (15.13), the SI units of h are N mϪ2 s ϭ Pa s ϭ kg mϪ1 sϪ1, since N ϭ kg m sϪ2 The cgs units of h are dyn cmϪ2 s ϭ g cmϪ1 sϪ1, and dyn cmϪ2 s is called one poise (P) Since dyn ϭ 10Ϫ5 N (Prob 2.6), we have P ϵ dyn cmϪ2 s ϭ 0.1 N mϪ2 s ϭ 0.1 Pa s (15.14) cP ϭ mPa s Some values of h in centipoises for liquids and gases at 25°C and atm are Substance C6H6 H2O H2SO4 olive oil glycerol O2 CH4 h/cP 0.60 0.89 19 80 954 0.021 0.011 Gases are much less viscous than liquids The viscosity of liquids generally decreases rapidly with increasing temperature (Molasses flows faster at higher temperatures.) The viscosity of liquids increases with increasing pressure The Earth has a solid inner core surrounded by a liquid outer core The outer core is at very high pressure (1 to Mbar) and is barely a liquid; its viscosity ranges from ϫ 103 Pa s at the top of the outer core to ϫ 1011 Pa s at the bottom [D E Smylie and A Palmer, arxiv.org/abs/0709.3333 (2007)] Figure 15.7a plots h versus T for H2O(l) at atm Also plotted are water’s thermal conductivity k (Sec 15.2) and self-diffusion coefficient D (Sec 15.4) Figure 15.7b plots these quantities for Ar(g) at atm Strong intermolecular attractions in a liquid hinder flow and make h large Therefore, liquids of high viscosity have high boiling points and high heats of vaporization Viscosities of liquids decrease as T increases, because the higher translational H2O(l) Ar(g) Figure 15.7 Viscosity h, thermal conductivity k, and self-diffusion coefficient D versus T at atm for (a) H2O(l); (b) Ar(g) lev38627_ch15.qxd 3/25/08 9:40 AM Page 482 482 Chapter 15 Transport Processes Figure 15.8 Viscosity of liquid sulfur versus temperature at atm The vertical scale is logarithmic kinetic energy allows intermolecular attractions to be overcome more easily In gases, intermolecular attractions are much less significant in determining h than in liquids The viscosities of liquids are also influenced by the molecular shape Long-chain liquid polymers are highly viscous, because the chains become tangled with one another, hindering flow The viscosity of liquid sulfur shows an extraordinary tenthousandfold increase with temperature in the range 155°C to 185°C (Fig 15.8) Below 150°C, liquid sulfur consists of S8 rings Near 155°C, the rings begin to break, producing S8 radicals, which polymerize to long-chain molecules containing an average of 105 S8 units Since Fy ϭ may ϭ m(dvy /dt) ϭ d(mvy )/dt ϭ dpy /dt, Newton’s law of viscosity (15.13) can be written as dpy dvy ϭ ϪhᏭ (15.15) dt dx where dpy /dt is the time rate of change in the y component of momentum of a layer on one side of a surface in the fluid due to its interaction with fluid on the other side The molecular explanation of viscosity is that it is due to a transport of momentum across planes perpendicular to the x axis in Fig 15.6 Molecules in adjacent layers of the fluid have different average values of py, since adjacent layers are moving at different speeds In gases, the random molecular motion brings some molecules from the faster-moving layer into the slower-moving layer, where they collide with slowermoving molecules and impart extra momentum to them, thereby tending to speed up the slower layer Similarly, slower-moving molecules moving into the faster layer tend to slow down this layer In liquids, the momentum transfer between layers occurs mainly by collisions between molecules in adjacent layers, without actual transfer of molecules between layers Flow Rate of Fluids Newton’s viscosity law (15.13) allows the rate of flow of a fluid through a tube to be determined Figure 15.9 shows a fluid flowing in a cylindrical tube The pressure P1 at the left end of the tube is greater than the pressure P2 at the right end, and the pressure drops continually along the tube The flow speed vy is zero at the walls (the no-slip condition) and increases toward the center of the pipe By the symmetry of the tube, vy can depend only on the distance s from the tube’s center (and not on the angle of rotation about the tube’s axis); thus vy is a function of s only; vy ϭ vy(s) (see also Prob 15.13) The liquid flows in infinitesimally thin cylindrical layers, a layer with radius s flowing with speed vy (s) Using Newton’s viscosity law, one finds (see Prob 15.12a for the derivation) that vy (s) for laminar flow of a fluid in a cylindrical tube of radius r is vy ϭ dP 1r Ϫ s2 aϪ b 4h dy laminar flow (15.16) where dP/dy (which is negative) is the pressure gradient Equation (15.16) shows that vy (s) is a parabolic function for laminar flow in a pipe; see Fig 15.10a (For turbulent Figure 15.9 A fluid flowing in a cylindrical tube The shaded portion of fluid is used in the derivation of Poiseuille’s law (Prob 15.12) s lev38627_ch15.qxd 3/25/08 9:40 AM Page 483 483 flow, there are random fluctuations of velocity with time, and portions of the fluid move perpendicularly to the pipe axis as well as in the axial direction The timeaverage velocity profile for turbulent flow looks like Fig 15.10b.) Application of (15.16) to a liquid shows that (see Prob 15.12b) for laminar (nonturbulent) flow of a liquid in a tube of radius r, the flow rate is pr P1 Ϫ P2 V ϭ t 8h y2 Ϫ y1 laminar flow of liquid Section 15.3 Viscosity (15.17) where V is the volume of liquid that passes a cross section of the tube in time t and (P2 Ϫ P1)/(y2 Ϫ y1) is the pressure gradient along the tube (Fig 15.9) Equation (15.17) is Poiseuille’s law [The French physician Poiseuille (1799–1869) was interested in blood flow in capillaries and measured flow rates of liquids in narrow glass tubes Blood flow is a complex process that is not fully described by Poiseuille’s law For the biophysics of blood flow, see G J Hademenos, American Scientist, 85, 226 (1997).] Note the very strong dependence of flow rate on tube radius and the inverse dependence on fluid viscosity h (A vasodilator drug such as nitroglycerin increases the radius of blood vessels, thereby reducing the resistance to flow and the load on the heart This relieves the pain of angina pectoris.) For a gas (assumed ideal), Poiseuille’s law is modified to (see Prob 15.12c) dn pr4 P12 Ϫ P 22 Ϸ dt 16hRT y2 Ϫ y1 laminar isothermal flow of ideal gas (15.18) where dn/dt is the flow rate in moles per unit time and P1 and P2 are the inlet and outlet pressures at y1 and y2 Equation (15.18) is accurate only if P1 and P2 don’t differ greatly from each other (see Prob 15.13) Measurement of Viscosity Figure 15.10 Measurement of the flow rate through a capillary tube of known radius allows h of a liquid or gas to be found from (15.17) or (15.18) A convenient way to determine the viscosity of a liquid is to use an Ostwald viscometer (Fig 15.11) Here, one measures the time t it takes for the liquid level to fall from the mark at A to the mark at B as the liquid flows through the capillary tube One then refills the viscometer with a liquid of known viscosity using the same liquid volume as before, and again measures t The pressure driving the liquid through the tube is rgh (where r is the liquid density, g the gravitational acceleration, and h the difference in liquid levels between the two arms of the viscometer), and rgh replaces P1 Ϫ P2 in Poiseuille’s law (15.17) Since h varies during the experiment, the flow rate varies From (15.17), the time t needed for a given volume to flow is directly proportional to h and inversely proportional to ⌬P Since ⌬P ϰ r, we have t ϰ h/r, where the proportionality constant depends on the geometry of the viscometer Hence rt/h is a constant for all runs For two different liquids a and b, we thus have ta /ha ϭ rb tb /hb and rb tb hb ϭ ta (15.19) where , , and ta and hb , rb , and tb are the viscosities, densities, and flow times for liquids a and b If , , and rb are known, one can find hb Another way to find h of a liquid is to measure the rate of fall of a spherical solid through the liquid The layer of fluid in contact with the ball moves along with it (no-slip condition), and a gradient of speed develops in the fluid surrounding the sphere This gradient generates a viscous force Ffr resisting the sphere’s motion This viscous force Ffr is found to be proportional to the moving body’s speed v (provided v is not too high) Ffr ϭ fv (15.20) Velocity profiles for fluid flow in a cylindrical pipe: (a) laminar flow; (b) turbulent flow s ϭ corresponds to the center of the pipe lev38627_index.qxd 4/10/08 5:04 PM Page 980 980 Molecular weight—Cont viscosity average, 487 weight average, 487 Molecular-weight determination: from freezing-point depression, 354–355 for gases, 15 from mass spectrometry, 356 from osmotic pressure, 359–360 for polymers, 356, 359–360, 486–487, 493 from sedimentation speed, 493 Molecularity, 531 Møller–Plesset perturbation theory, 709 Moment(s) of inertia, 622–623, 745, 758–759 Momentum: angular See Angular momentum linear, 445, 647 Monochromatic radiation, 742 Monte Carlo calculations, 950–951 MOPAC2007, 721 Motif, 922 Moving boundary method, 500 MP perturbation theory, 709 MP2, 709 Multiplication table of a group, 803 Multiplicity, spin, 657 N Nagle electronegativity scale, 676 Nanometer (nm), 736 Nanoparticles, 227 Nanoscopic systems, 2–3, 8, 102 Nanotubes, 427, 475, 915–916 Natural logarithm, 29–30 Nernst–Einstein equation, 491 Nernst equation, 414–417, 421, 429 Nernst–Simon statement of third law, 156–157 Nerve cells, 490 Nerve impulse, 436 Neutron diffraction, 937 Newton, Isaac, 37 Newton (unit), 38 Newtonian fluid, 481 Newton’s law of viscosity, 480, 482, 485, 494 Newton’s second law, 38, 103, 599, 601 NIST-JANAF tables, 164, 174 NMR spectroscopy, 779–793 of biological molecules, 791 carbon-13, 788, 789 chemical shifts in, 784–785 classical description of, 792–793 of dimethylformamide, 790 double resonance in, 789 dynamic, 792 of ethanol, 785, 787–788 experimental setup in, 783–784, 788–789 first-order analysis in, 786–788 Fourier-transform, 788–789 and imaging, 792 intensities in, 784, 790 and protein structure, 791 proton, 784–788 and rate constants, 790 selection rule in, 782 of solids, 791 spin–spin coupling in, 785–788 two-dimensional, 791 No-slip condition, 480 Node, 609, 620, 683, 685, 692 Nonactivated adsorption, 578 Nondissociative adsorption, 571 Nonideal gases: chemical potential of, 321 enthalpy of, 149–150, 257 entropy of, 158, 257 equation of state of, 23, 245–247, 250–251 internal energy of, 875 internal pressure of, 119, 120 mixtures of, 232–233, 321–322 reaction equilibrium for, 340 thermodynamic properties of, 149–150, 158, 256–257 Nonideal solution(s), 294–321 activities in, 294–297 activity coefficients in See Activity coefficients chemical potentials in, 294–297 liquid–vapor equilibrium for, 368–369 reaction equilibrium in, 330–339, 340–343 reaction rates in, 547–548, 907–908 standard states for components of, 295–297 vapor pressure of, 298–302, 368–369 Noninteracting particles in quantum mechanics, 618–619 Non-Newtonian fluid, 481 Nonpolar molecule, 432 Normal boiling point, 210 Normal distribution, 99, 455 Normal melting point, 211 Normal modes, 763–765, 768 symmetry species of, 810 Normalization, 602–603, 605, 607–608, 624, 650 Normalization constant, 603, 608 nth-order reaction, 523 Nuclear g factor(s), 781 table of See inside back cover Nuclear magnetic dipole moment, 781–782 Nuclear-magnetic-resonance spectroscopy See NMR spectroscopy Nuclear magneton, 782 Nuclear Overhauser effect (NOE), 790 Nuclear spin(s), 749–750, 781–783 energy levels of, 782 and entropy, 859 and rotational levels, 749–750, 760 table of See inside back cover Nuclear wave function, 679, 744–746 Nucleus, 638 Number average molecular weight, 360, 486–487 Numerical integration, 539–541 O Octanol/water partition coefficient, 372–373 Odd function, 684, 693 Ohm, 495 Ohm’s law, 495 Onsager conductivity equation, 507–509 Open system, 4, 47 Operator(s), 613–618 commutator of, 614 Coulomb, 664 electric dipole moment, 738 equality of, 613 exchange, 664 Hamiltonian, 615 See also Hamiltonian operator Hartree–Fock, 664, 692, 714 Hermitian, 627–630 Kohn–Sham, 712, 714 Laplacian, 615 linear, 616 momentum, 614 position, 614 product of, 613 square of, 613 sum of, 613 Oppenheimer, J R., 677 Optical activity, 794 lev38627_index.qxd 4/10/08 9:54 AM Page 981 981 Optical pyrometer, Optical rotatory dispersion, 795 Optimization, 724–725 Orbital(s), 651–652 antibonding, 684 atomic, 643–645, 651, 659, 663–665, 683 canonical, 696 crystal, 940–941 Hartree–Fock, 663–665, 687, 692–693 Kohn–Sham, 712, 713–714, 716 localized See Localized MOs molecular See Molecular orbital(s) pi, 684–685, 692, 693, 717 shape of, 643–644 sigma, 684–685, 692, 693, 717 Slater-type, 665, 708 virtual, 708 Orbital angular momentum, 647–649, 656 Orbital energy, 661–662, 664 Orbital exponent, 654, 665 ORCA, 721 Order–disorder transitions, 226 Order of a group, 801 Order of a matrix, 804 Order of a reaction, 517 determination of, 526–529 half-integral, 538 pseudo, 519 with respect to solvent, 519, 537 Ortho H2, 881 Orthogonal functions, 610, 628–629 Orthonormal set, 629 Oscillating reactions, 565–566 Osmosis, 360–361 Osmotic coefficient, practical, 311 Osmotic pressure, 356–361 definition of, 357, 358 of ideally dilute solution, 357 of nonideally dilute solutions, 358 Ostwald, Wilhelm, Ostwald viscometer, 483 Overall mole fraction, 362 Overall order, 517 Overlap integral, 682 Overtone bands, 753, 765 Oxidation, 407 Oxygen molecule, 689, 750, 845 Ozone in stratosphere, 566–567 P P branch, 754–755 Pair-correlation function, 947–948 Paired electrons, 660 Para H2, 881 Parallel spins, 657 Paramagnetism, 663 Parameter, 28, 677–678 Partial derivative(s), 20–22 Partial miscibility: between gases, 371 between liquids, 370–373 Partial molar enthalpy(ies), 268, 273–274 in ideal solution, 291 in ideally dilute solution, 292 Partial molar entropy(ies), 268, 274 in ideal solution, 291 in ideally dilute solution, 292 Partial molar Gibbs energy, 268 See also Chemical potentials Partial molar internal energy, 268 Partial molar quantities, 264–270 in ideal solution, 281, 291 in ideally dilute solution, 281, 292 measurement of, 267, 272–274 in nonideal solution, 292 relations between, 269–270 Partial molar volume(s), 265–268 in ideal gas mixture, 266, 270 in ideal solution, 291 in ideally dilute solution, 292 infinite-dilution, 268 measurement of, 267–268, 272–273 of pure substance, 266 Partial orders, 517 Partial pressure, 16, 186–187, 321 Partial specific volume, 492 Particle in a box: one-dimensional case, 606–610 selection rule for, 738–739 three-dimensional case, 610–612 two-dimensional case, 612 Partition coefficient, 372–373 Partition function See Canonical partition function; Electronic partition function; Molecular partition function; Rotational partition function; Semiclassical partition function; Translational partition function; Vibrational partition function Pascal (unit), 11, 12 Path of a process, 44, 62 Pauli, Wolfgang, 653, 659 Pauli exclusion principle, 659 Pauli repulsion, 689, 864, 881 Pauling, Linus, 675 Pauling electronegativity, 676 Peng–Robinson equation, 251, 254, 260 Perfect gas, 58–62, 69, 122 See also Ideal gases Period, 619 Periodic table, 660–662 Peritectic phase transition, 378 Peritectic point, 379 Peritectic temperature, 377–378 Permeability of vacuum, 780 Permeable wall, Permittivity, 434 relative, 433–434 of vacuum, 395, 637, 735 Perturbation theory, 626–627 of liquids, 948 Møller–Plesset, 709 pH, 425–426, 428 Phase, 5–6, 205 Phase change: continuous, 225–227 discontinuous, 225 enthalpy change for, 63, 213 entropy change for, 87–88, 90, 157, 213–214 first-order, 225, 227 Gibbs energy change for, 113 higher-order, 225–227 lambda, 226–227 order–disorder, 226–227 in polymers, 914 second-order, 225–226 work in, 63 Phase diagram(s): of carbon, 225 for CO2, 212 computer calculation of, 383–385 liquid–liquid, 370–373 liquid–solid, 373–380 liquid–vapor, 362–370 one-component, 210–212 structure of, 381 for sulfur, 221–222 three-component, 385–387 two-component, 361–385 for water, 211–212, 222, 223 Phase equilibrium, 109, 129–131, 205, 214, 361, 402 in electrochemical systems, 402–403 multicomponent, 361–387 one-component, 210–227 solid–solid, 221–225 lev38627_index.qxd 4/10/08 9:54 AM Page 982 982 Phase equilibrium—Cont three-component, 385–387 two-component, 361–385 Phase rule, 205–210 See also Degrees of freedom Phase transition See Phase change Phase-transition loop, 377, 381 Phosphorescence, 797 Photochemical kinetics, 798–800 Photochemical reactions, 796–797, 799 Photochemistry, 796–800 Photoconductivity, 941 Photoelectric effect, 593–594 Photon, 593–594, 595, 597, 650, 737, 796 Photophysical processes, 797 Photosensitization, 798 Photostationary state, 800 Photosynthesis, 796, 798 Physical adsorption, 570–571, 574, 575 Physical chemistry, 1–2 Physical constants, table of See inside back cover Physical kinetics, 474 Physisorption, 570–571, 574 Pi bond, 699–700 Pi orbitals, 705, 747, 855 Pitzer equations, 315, 317–318 pK, 337 Planck, Max, 592–593 Planck’s constant, 592, 600 Plane defects, 947 Plane-polarized wave, 735 Plane of symmetry, 756, 802 Plasma, 561 PM3 method, 719 PM5 method, 719 PM6 method, 719 Point defects, 946–947 Point groups, 802–803 Poise, 481 Poiseuille’s law, 483 Poison, 568, 576, 578 Polar molecule, 432 Polarizability, electric, 432, 434, 773, 862–863 Polarization, 432–434 Polarized wave, 735, 795 Polyatomic molecules: MO theory of, 693–702 rotation of, 758–763 statistical mechanics of ideal gas of, 851–854 vibration of, 763–770 Polyelectrolyte, 306, 503 Polyethylene, 936 Polymer(s), 914 formation of, 554–556 molecular weights of See Polymer molecular-weight determination solid, 913, 914, 916, 936 solutions of See Polymer solutions Polymer molecular-weight determination: from mass spectrometry, 356 from osmotic pressure, 359–360 from sedimentation speed, 493 from solution viscosity, 486–487 Polymer solutions, 235 sedimentation in, 492–493 viscosity of, 481, 486–487 Polymerization kinetics, 554–556 Polymorphism, 221 Pople, John, 708, 721, 727 Population inversion, 741–742 Population of vibrational and rotational levels, 749, 755 Potential difference See Electric potential difference(s) Potential energy, 40–41, 600 electrostatic, 637 gravitational, 40–41 intermolecular, 861–866 Potential-energy surfaces for reactions, 880–887 calculation of rate constants from, 887–889 for F ϩ H2, 885 for H ϩ H2, 881–885 semiempirical calculation of, 886–887 for unimolecular reactions, 904–905 Potentiometer, 408–409 Potentiometric titrations, 426 Powder x-ray diffraction method, 936–937 Powell plot, 528–529 Power, 39 Power plants, 84–85 Practical osmotic coefficient, 311 Precession, 792 Pre-exponential factor, 541–544 collision-theory expression for, 878–879 TST expression for, 900, 909 Pressure, difference of across curved interface, 232 of earth’s atmosphere, 466 effect of surface tension on, 230 and equilibrium constant, 342 high, 223–224 in interphase region, 230 kinetic-theory derivation of, 443–446 negative, 251–252 osmotic, 357–359 partial, 16, 186–187, 321 standard-state, 140, 163 statistical-mechanical expression for, 826–827, 846, 853, 869 units of, 11–12 Pressure equilibrium constant, 179 Pressure-jump method, 558 Primary kinetic salt effect, 908 Primary quantum yield, 800 Primitive unit cell, 924, 928 Principal axes of inertia, 758–759 Principal diagonal, 804–805 Principal moments of inertia, 758–759, 761, 851 Principal quantum number, 640 Printed electrodes, 412 Prions, 795 Probability(ies) See also Probability density in canonical ensemble, 823–825, 828 of chemical reaction, 889–890 and entropy, 98–101 of independent events, 99, 449 in quantum mechanics, 601–603 Probability density, 448, 601, 602, 605 calculation of, 707 in DFT, 711–712, 714 in H atom, 645–646 in H ϩ2 , 683 in solids, 934–935, 936 Problem solving, 70–72 Process, 62 adiabatic, 61, 62, 64 cyclic, 49, 62, 66 irreversible, 44, 45–46, 87, 89–90, 93–95 isobaric, 50, 62, 63 isochoric, 52, 53, 62 isothermal, 23, 60, 62, 88 reversible, 43, 62, 87, 93 Product notation, 178 Products of inertia, 758 Propagation steps, 552, 554 Protein Data Bank, 935 Proteins, 914 antifreeze, 355 lev38627_index.qxd 4/10/08 9:54 AM Page 983 983 charge on, 515 denaturation of, 342–343 electronic absorption spectra of, 777 electrophoresis of, 503, 504 IR bands of, 769 MD simulation of folding of, 950 molecular-weight, determination of, 356, 493 NMR spectra of, 791 ORD and CD spectra of, 795 structures of, 791, 914, 935 T-jump kinetics of folding of, 558, 769–770 Pseudo order, 519 Pulse oximeter, 741 Pulsed-field electrophoresis, 504 Pure-rotation spectra See Rotational spectra Q Q branch, 765 Quadratic formula, 188 Quadratic integrability, 605 Quadruple bond, 705 Quantization of energy, 592–596, 605, 607, 608 Quantum, 593 Quantum chemistry, 1, 590 Quantum chemistry calculations, performing, 720–723 Quantum mechanics, 590–632 Quantum number, 609, 612, 640 for spin, 649–650, 656–657 Quantum yield, 799–800 Quasicrystal, 913, 925 R R branch, 754–755 Radial distribution function: in H atom, 646 in liquids, 947–948 Radiation: electromagnetic, 591, 734–737 of heat, 47, 476 Radiationless deactivation, 797 Radical(s) See Free radical Radioactive decay, 521, 582 Raman-active vibration, 768, 773 Raman effect, 772 Raman shift, 772 Raman spectroscopy, 771–774 Random coil, 498 Raoult’s law, 280–281, 285, 286, 298, 299, 362 deviations from, 285–286, 299, 301 Rate coefficients See Rate constant(s) Rate constant(s), 517 and activity coefficients, 548, 561, 907–908 apparent, 547–548 calculated from potential-energy surface, 887–889 calculated from trajectory calculations, 887–889 collision-theory expression for, 878 determination of, 530 diffusion-controlled, 562–564 and equilibrium constants, 531–532, 546–547 estimation of, 520 for ionic reactions, 908 and isotopic substitution, 901 and NMR spectra, 790 in nonideal systems, 547–548, 907–908 range of values of, 556 solvent effects on, 560 temperature dependence of, 541–546, 900 theories of, 877–910 TST expressions for, 897–898 unimolecular, 548–550 units of, 517 Rate of conversion, 516, 576–577 Rate-determining-step approximation, 532–534, 535, 536–537 Rate law, 516–517 for catalyzed reaction, 565 determination of, 526–530 for elementary reaction, 530–531 integration of, 520–526 in nonideal systems, 547–548, 908 and reaction mechanism, 518, 532–538 Rate-limiting-step approximation, 532–534, 535, 536–537 Rate of reaction, 516–518 See also Rate constant(s); Reaction(s) computer integration for, 539–541 in heterogeneous catalysis, 576–577 isotope effects on, 901 measurement of, 519–520, 556–560, 790 in photochemistry, 799–800 theories of, 877–911 Rayleigh scattering, 772 Reaction(s): autocatalytic, 565 bimolecular, 531 biochemical, 346 catalyzed See Catalysis; Catalyst chain, 551–556 chemically controlled, 564, 906–907 combustion, 554 competing, 525–526 complex, 518 composite, 518 consecutive, 524–525 coupled, 345–347 diffusion-controlled, 562–564, 570, 907, 909–911 elementary, 518, 531–532 equilibrium for a See Reaction equilibrium fast, 520, 556–560 first-order, 521 free radicals in, 520, 538, 543 half-life of, 521, 522, 523, 526–527 heterogeneous, 515, 575–578 homogeneous, 515 independent, 194 ionic, 561 mechanisms of See Reaction mechanism(s) molecular dynamics of, 887–892 nth-order, 523 nuclear, 521 order of See Order of a reaction oscillating, 565–566 photochemical, 798–800 polymerization, 554–556 rate of See Rate of reaction recombination, 543–544, 550–551, 563, 906 reversible, 524 second-order, 522–523 simple, 518 in solutions, 560–564, 906–911 slow, 556 third-order, 523 trimolecular, 531, 550–551, 906 unimolecular See Unimolecular reaction(s) Reaction coordinate, 895–896 Reaction dynamics, molecular, 887–892 Reaction equilibrium, 109, 132–134, 174–175, 402, 403 computer programs for, 340–341 in electrochemical systems, 402, 403 in electrolyte solutions, 332–337 at high T, 182, 185 in ideal gases, 174, 177–198 in ideally dilute solutions, 287 lev38627_index.qxd 4/10/08 9:54 AM Page 984 984 Reaction equilibrium—Cont involving pure solids or liquids, 337–339 at low T, 181–182, 185 in nonelectrolyte solutions, 331–332 in nonideal gas mixture, 340 in nonideal solutions, 330–339, 340–343 in nonideal systems, 330–347 qualitative discussion of, 181–182 shifts in, 194–198 Reaction intermediate, 518, 534, 538 Reaction kinetics, 515–581 See also Reaction(s); Rate constant(s) in photochemistry, 798–800 theories of, 877–911 Reaction mechanism(s), 518, 530, 532–539 compilations of, 539 of reverse reaction, 538–539 rules for, 536–538 Reaction order See Order of a reaction Reaction path, 883 Reaction quotient, 187, 195–197, 345, 414 Reaction rate See Rate of reaction Real gas See Nonideal gases Recombination reactions, 543–544, 550–551, 563, 906 Redlich–Kister equation, 326 Redlich–Kwong equation, 245, 246, 251, 252–253, 323 Reduced mass, 621, 622, 639 Reduced pressure, volume, and temperature, 255–256 Reducible representation, 806, 810–811 Reduction, 407 Reference form, 142, 163 Reference length, 724 Reference system in DFT, 711–712 Refractive index, 736 Relative atomic mass, Relative coordinates, 621 Relative molecular mass, See also Molecular weight Relative permittivity, 433–434 Relative viscosity, 486 Relativistic mass, 51–52 Relativity, theory of, 37, 51–52 effect on molecular properties, 679, 708 and spin, 649 Relaxation, 558, 789 Relaxation methods in kinetics, 558–560 Relaxation time, 559, 793 Representations, 805–807, 810–811 Residuals, 220 Resistance, 495 Resistance thermometer, Resistivity, 494 Resolution, 767 Resonance Raman spectroscopy, 774 Resonance structures, 703 Reverse osmosis, 360 Reversible electrodes, 409–411 Reversible process, 43, 62, 87, 93 Reynolds number, 511 Rideal–Eley mechanism, 576, 578 Rigid rotor, two-particle, 622–623, 649, 739–740 selection rule for, 739 Rigid wall, RM1 method, 719 RNA, 568 Room-temperature ionic liquids, 462, 920 Root-mean-square distance in diffusion, 489–490 Root-mean-square speed, 447–448 Rotation, 67 of diatomic molecules, 744–745, 748 of polyatomic molecules, 758–763 Rotational barriers, 705–706, 762 Rotational constants, 745, 752, 759, 760, 761 Rotational energy, 67 classical, 759 of diatomic molecule, 744–745, 748 of linear molecule, 760 and nuclear spin, 750, 760 of polyatomic molecule, 758–760 Rotational partition function, 835, 842–843, 851–852 Rotational spectra: of diatomic molecules, 751–752 of polyatomic molecules, 760–763, 772 Rotor, two-particle rigid, 622–623, 739–740 RRKM theory, 906 Rumford, Count, 48 Runge–Kutta methods, 540–541 Rutherford, Ernest, 638, 670 Rydberg constant, 594, 595, 774 S Saddle point, 884 Salt bridge, 417, 422 Salt effect, 339 kinetic, 908 Sap, 252 Saturated liquid and vapor, 248 Scalar, 38 Scanning tunneling microscope, 938 Scattering of light, 772, 932 SCF wave function 692 See also Hartree–Fock wave function Schrödinger, Erwin, 600, 603 Schrödinger equation: electronic, 677 for nuclear motion, 679, 744–746 time-dependent, 599–601, 604, 615, 737 time-independent, 604–605, 615 Screening constant, 660 in NMR, 784 Second law of thermodynamics, 78–80, 95, 101, 102, 103 and life, 134–135 Second-order phase transition, 225–226 Second-order reaction, integration of rate law of, 522–523 Second virial coefficient, 245–246, 865, 869 Sedimentation, 236 of polymer molecules, 492–493 Sedimentation coefficient, 493 Selection rules, 738–739 for NMR transitions, 782 for particle in a box, 738–739 in Raman spectroscopy, 772–773 for rotational transitions, 739, 751, 760 for spin, 775, 801 for vibration–rotation transitions, 751, 764–765 Self-consistent-field wave function, 692 See also Hartree–Fock wave function Self-diffusion coefficient, 489 Self-interstitial, 946 Semiclassical partition function, 866–868 Semiconductor, 495, 941, 946–947 Semiconductor laser, 742 Semiempirical methods, 717–720, 886 Semipermeable membrane, 91, 356 Sensitivity, 770 Separation of variables, 612, 618 SERS, 774 Shake and Bake, 935 Shielding, 660 Shielding constant, NMR, 784–785 lev38627_index.qxd 4/10/08 9:54 AM Page 985 985 Shifts in reaction equilibrium, 194–198 Shock tube, 559 SI units, 9, 38, 395, 637, 780 Siemens, 495 Sigma bond, 699–701 Sigma orbitals, 684–685, 692, 693, 717 Silver–silver chloride electrode, 410–411 Similarity transformation, 806 Simple cubic lattice, 928 Simple eutectic system, 375 Simple reaction, 518 Simple solution, 326 Simple unit cell, 924, 929 Simultaneous equilibria, 191–194 Single-point calculation, 720 Single-valuedness, 605, 640 Singlet term, 775 Size effects on properties, 2–3, 227 Slater determinant, 658–659 Slater-type orbitals, 665, 708 Sleep, 32 Slope, 17–18 Slope method, 267 Slow reactions, 556 SMILES string, 723 Smog, 544 SN2 reaction, 886 Soave–Redlich–Kwong equation, 251, 254, 260 Sodium chloride: basis in solid, 922, 929 cohesive energy of, 918–920 dipole moment of, 680 dissociation energy of molecule of, 680 structure of solid, 929–930 Sol, 234 Solid(s), 913–947 a and k of, 25 amorphous, 913, 914 band theory of, 939–941 cohesive energies of, 916–921 covalent See Covalent solids crystalline, 913–914 Debye theory of, 943–946 defects in, 946–947 density of, 927 diffusion in, 377, 489, 490 heat capacity of, 158, 594, 943–946 hydrogen-bonded, 916, 918 interatomic distances in, 921–922, 935 ionic See Ionic solids metallic See Metals molecular See Molecular solids NMR of, 791 of polymers, 914, 916 solutions of, 375 statistical mechanics of, 941–946 structures of, 922–937 See also Structures of solids surfaces of, 937–939 thermodynamic properties of, 120–121 van der Waals, 916, 918 vapor pressure of, 211, 461–462 vibrations of, 941–946 Solid–liquid equilibrium, twocomponent, 373–380 compound formation in, 378–379 solid-phase immiscibility in, 373–374 solid-phase miscibility in, 375–376 solid-phase partial miscibility in, 376–378 Solid polymers, 913, 914, 916, 936 Solid–solid phase transitions, 221–225 Solid solutions, 375 Solubility: of gases in liquids, 286–287 of solids in liquids, 381–383 Solubility product, 338–339, 423–424 Solute, 263, 282 Solutions, 263–288 of electrolytes See Electrolyte solutions freezing point of, 352–356, 373–374, 381–383 Gibbs energy of, 269 ideal See Ideal solution(s) ideally dilute See Ideally dilute solution(s) measurement of partial molar quantities in, 267, 272–274 nonideal See Nonideal solution(s) reaction rates in, 560–564, 906–911 simple, 326, 384–385 solid, 375 volume of, 266–266 Solvation, 306, 560 Solvent, 263, 282 effect of on rate constants, 560 order with respect to, 519, 537 Solver in spreadsheet, 191, 193–194, 221, 252–254 Sound, speed of, 458 Source of emf, 404 Space lattice, 922–925 Spacing between molecular energy levels, 748–749 SPARTAN, 721 Specific conductance, 494 Specific enthalpy, 54 Specific heat (capacity), 46, 54 See also Heat capacity Specific internal energy, 124 Specific rotation, 785 Specific volume, 54 Spectroscopy, 737–794 Speed(s), 38, 443 average, 457–458 distribution of, 448–459 of light, 591, 637, 734–735, 736, 780 most probable, 456, 458 root-mean-square, 447–448 of sound, 458 Spherical coordinates, 639, 644 Spherical top, 759, 760 Spherically symmetric function, 642 Spin, 649–650, 652–653, 655, 656–657 half-integral, 652 integral, 652 nuclear, 749–750, 760, 781–784 selection rule for, 775 Spin coordinates, 650 Spin–lattice relaxation time, 793 Spin multiplicity, 657 Spin–orbit interaction, 657 Spin-orbital, 650, 659 Spin–spin coupling, 785–788 Spin–spin relaxation time, 793 Spin–statistics theorem, 652–653, 749–750 Spontaneous emission, 737 Spreadsheets: absolute references in, 155 and equilibrium calculations, 192–194 formulas in, 154 and least-squares fits, 219–221 and liquid–liquid phase diagrams, 384–385 and liquid–vapor equilibrium calculations, 252–254 polynomial fit and, 153–155, 171 relative reference in, 155 reliability of, 155 and simultaneous equilibria, 192–194 Solver in, 191, 193–194, 221, 253–254, 384–385 solving equations with, 191 lev38627_index.qxd 4/10/08 9:54 AM Page 986 986 Square matrix, 804 Stacking error, 947 Standard concentration equilibrium constant, 180, 332, 531–532 Standard deviation, 101 Standard electrode potentials, 417–420 table of, 419 Standard emf, 414, 416, 422–425 and equilibrium constants, 422–424 temperature dependence of, 424 Standard enthalpy change: and cell emf, 424 of combustion, 145 of formation, 142–144 of reaction, 141–149, 151 Standard entropy change, 161 of activation, 903–904, 909 and cell emf, 424–425 Standard equilibrium constant, 178, 331 See also Equilibrium constant Standard Gibbs energy change, 161–162, 330, 343–345 of activation, 903–904, 908, 909 and cell emf, 414, 422–423 for formation, 161–162 relation to equilibrium constant, 178, 181–182, 190, 331 Standard potential, 414, 416, 422–425 Standard pressure, 140, 163, 190, 210, 283, 284 Standard pressure equilibrium constant, 178 Standard state(s): Convention I, 295–296, 298–301 Convention II, 296–297, 301–302 for an electrolyte solute, 309 for ideal gas in mixture, 176 for ideal-solution component, 278 for ideally dilute solution components, 283–284 for molality scale, 305 for nonideal gas in mixture, 321 for nonideal-solution components, 295–297 pressure of, 140, 163, 190, 210, 278, 283, 284 for pure gas, 140 for pure solid or liquid, 140 summary of, 343, 344 Standard-state thermodynamic properties, table of, 959–960 Star superscript, 265, 278 Stardust, 237 Stark effect, 762 Stark–Einstein law, 796 State: change of, 62 in classical mechanics, 467, 599 intensive, 206 molecular, 821 in quantum mechanics, 599, 600–601, 603, 613, 821 stationary, 595, 605, 613, 617 steady, 5, 800 thermodynamic, 6, 821 versus energy level, 613, 755, 837 State function (in quantum mechanics), 599–605 State function(s) (in thermodynamics), dependences on T, P, V, 118–121 and line integrals, 65–66 Stationary state, quantum-mechanical, 595, 605, 613, 617 Stationary-state approximation, 534–536, 538 Statistical mechanics, 1, 820–870 of fluids, 866–870 of ideal gases, 834–836, 840–858 of liquids, 948–951 postulates of, 822, 823 of solids, 941–946 Statistical thermodynamics, 820 See also Statistical mechanics Statistical weight, 837 Steady state, 5, 475, 800 Steady-state approximation, 534–536, 538 Steam engine, 84 Steam point, 7, 14 Steric energy, 724–725 Steric factor, 879 Stern model, 444 Sticking coefficient, 578–579 Stimulated emission, 737, 738, 741–742 Stirling’s formula, 834 Stockmayer potential, 865 Stoichiometric coefficients, 132 Stoichiometric molality, 309 Stoichiometric numbers, 132, 518, 547 Stokes–Einstein equation, 491, 563 Stokes’ law, 484, 491, 502 Stokes lines, 772 Stokes shift, 778 Stopped-flow method, 557 Stratosphere, 479, 566–567 Stretching vibration, 764 Structures of molecules See Geometry of molecules Structures of solids, 922–937 for covalent solids, 930–931 determination of, 931–937 examples of, 928–931 general discussion of, 922–929 for ionic solids, 929–930 for metals, 928–929 for molecular solids, 931 Study suggestions, 30–32 Sublimation, 213 Subshell, 660 Substitutional impurity, 946 Substitutional solid solution, 375 Substrate, 568 Sulfur: phase diagram of, 221–222 viscosity of, 482 Sum, 25 replacement of by integral, 841 Superconductivity, 226 Supercooled liquid, 222, 355 Supercritical fluid, 249 Superheated liquid, 222 Supermolecule, 880 Supersaturated vapor, 222 Supersonic jet, 779 Surface chemistry, 227–237, 570–578, 937–939 Surface-enhanced Raman spectroscopy, 774 Surface melting, 222 Surface migration, 579 Surface reconstruction, 938 Surface relaxation, 922, 938 Surface structures of solids, 937–939 Surface tension, 229–233 of liquids, 229–230 measurement of, 232–233 temperature dependence of, 230 Surroundings, Svedberg (unit), 493 Symmetric function, 652, 750 Symmetric top, 759–760, 761 Symmetrical convention, 296 Symmetry elements, 756–757 Symmetry of molecules, 756–757, 801–803 Symmetry number, 842, 851–852, 899 Symmetry operations, 757, 801–802 Symmetry point groups, 801–803 Symmetry species, 809 lev38627_index.qxd 4/10/08 9:54 AM Page 987 987 Synchrotron, 937 System, closed, heterogeneous, homogeneous, isolated, 4, 95, 96 open, 4, 47 in statistical mechanics, 821 Systematic absences, 933–934 T T-jump method, 558 Tables of thermodynamic data, 163–165, 959–960 Taylor series, 257–258 Temperature, 6–8 absolute ideal-gas scale of, 12–14, 97, 446 Celsius, 14 ideal-gas scale of, 12–14, 97, 446 ITS-90, 13–14 low, attainment of, 168–169 measurement of, 7–8, 13 and molecular translational energy, 446–447 thermodynamic scale of, 96–97 Temperature-jump method, 558 Term(s) atomic, 656, 658, 663 molecular, 809, 810 Terminal speed of ions, 499, 501, 503 Terminals, 404, 405 Termination step, 552, 554, 555 Termolecular reactions See Trimolecular reactions Ternary system, 385 Tesla, 780 Tetramethylsilane, 785, 789 Thermal analysis, 380 Thermal conductivity, 475–479 kinetic theory of, 476–479 pressure dependence of, 479 temperature dependence of, 479 Thermal desorption, 579 Thermal equilibrium, 5, 6–7 Thermal expansivity, 24–25, 117 Thermal reaction, 796 Thermistor, Thermochemical calorie, 51, 163 Thermocouple, 8, 400 Thermodynamic control of products, 526 Thermodynamic data tables, 163–165, 959–960 Thermodynamic properties, 5–6 dependence of on T, P, V, 118–121 estimation of, 165–168 of ideal gas, 58–62, 64, 176, 846–849, 853–854 of ideal gas mixture, 176, 289 of ions in solution, 319–321 of nonequilibrium system, 475 of real gases, 149–150, 158, 256–257 of solution components, 318–321 tables of, 162–165, 959–960 Thermodynamic state, 6, 821 local, 475 Thermodynamic temperature scale, 96–97 Thermodynamics, 1, of electrochemical systems, 401–403 equilibrium, first law of, 47–51, 58 of galvanic cells, 412–417 irreversible, and living organisms, 134–135, 347 second law of, 78–80, 95, 101, 102, 103 third law of, 156–157, 168 zeroth law of, 7, Thermometer, 7–8, 13 Third law of thermodynamics, 156–157, 168, 858–860 Third-order reactions, 523 Thomson, George, 596, 638 Thomson, Joseph J., 596, 637–638 Thomson, William, 56, 79, 96 Three-body forces, 861 Three-center bond, 701–702 Three-component phase equilibrium, 385–387 Threshold energy, 877, 878, 889 Threshold frequency, 593 Tie line, 364–365, 369–370 Tilde, 752 Time, direction of, 103–104 TMS, 785, 789 Toms effect, 480 Torr, 11 Torsional vibration, 769 Total differential, 20 Totally symmetric representation, 807, 809 Trace of a matrix, 807 Tracer diffusion coefficient, 489 Trajectory calculations, 887–890 Transference numbers, 504–506, 507 Transition (dipole) moment, 738, 751 Transition elements, 662 Transition state, 884–886, 894, 904–905 Transition-state theory (TST), 892–904 assumptions of, 893, 895, 898 and collision theory, 899–900 for gas-phase reactions, 892–904 for H ϩ H2, 898–899 and isotope effects, 901 rate constant equation in, 897, 903, 907–908 for reactions in solution, 907–909 temperature dependence of rate constant in, 900 tests of, 901–902 thermodynamic formulation of, 902–904 and transport properties, 902 and unimolecular reactions, 904–906 variational, 902 Translation, 67 Translational energy, molecular, 67, 443, 446–447, 622, 744 distribution of, 459 in a fluid, 870 level spacings of, 748 Translational partition function, 835, 840–841, 851 Translational states, number of, 833 Transmembrane potential, 429, 435–436 Transmission coefficient, 897 Transmittance, 741 Transport numbers, 504–506, 507 Transport processes, 474–509 Transverse relaxation time, 793 Triangular coordinates, 385–386 Trimolecular reactions, 531, 550–551, 879–880, 906 Triple bond, 699, 701 Triple point, 13, 210–211, 212, 221 Triplet term, 775 Troposphere, 466 Trouton’s rule, 213 Trouton–Hildebrand–Everett rule, 213 Tunneling, 620, 888, 898, 901 Turbulent flow, 480, 481 Turnover number, 569 Two-component phase equilibrium, 361–385 Two-dimensional NMR, 791 Two-particle rigid rotor See Rigid rotor, two-particle lev38627_index.qxd 4/10/08 9:54 AM Page 988 988 Two-particle system in quantum mechanics, 621–622 Two-photon spectroscopy, 796 U Unattainability of absolute zero, 168 Uncertainty principle, 597–598, 603, 608 Ungerade (u), 684, 693 Unimolecular reaction(s), 531, 548–550 A factor of, 543 activation energy of, 543 falloff of rate constant of, 549–550 Lindemann mechanism of, 548–550 potential-energy surfaces for, 904–905 RRKM theory of, 906 and TST, 904–906 Unit cell, 923–925 body-centered, 924 coordinates of a point in, 925 end-centered, 924 face-centered, 924, 928 number of formula units in, 927 primitive, 924, 928 simple, 924, 928 Unit matrix, 805 Unpolarized light, 735 Unsymmetrical convention, 296 V Vacancy, 946 Vacuum decay, 223 Valence band, 742, 941 Valence-bond method, 702–703 van der Waals equation, 23, 122–123, 245, 246, 250, 255, 257 van der Waals force, 862 van der Waals molecules, 865–866 van der Waals radii, 921–922 van der Waals solid, 916 van der Waals well depth, 883 van’t Hoff, J H., van’t Hoff equation, 183–184 van’t Hoff’s law, 357 Vapor pressure, 210, 211, 216 effect of pressure on, 216 of electrolyte solution, 310 from equation of state, 252–254 of ideal solution, 279–281 of ideally dilute solution, 284–286 lowering of, 351–352 of nonideal solution, 298–302, 368–369 of a small drop, 232, 243 of a solid, 211, 216, 461–462 temperature dependence of, 216–217 Vaporization, heat of, 213, 216, 219 Variance, 206 Variation method, 624–626 Variational integral, 624 Variational transition-state theory, 902 Vector, 38 Velocity(ies), 38, 443 distribution of See Distribution function Velocity space, 449–450 Vibration, 67 of diatomic molecules, 745–749 of polyatomic molecules, 763–770 of solids, 941–946 Vibration–rotation coupling constant, 747–748 Vibration–rotation spectra: of diatomic molecules, 750–755 of polyatomic molecules, 763–770, 772–773 Vibrational energy, 67, 69 of diatomic molecules, 746–747, 748 of polyatomic molecules, 763–770, 773 of solids, 941–946 Vibrational frequencies: bond, 768–769 calculation of, 720, 854 of diatomic molecules, 746 fundamental, 753–754, 765 Vibrational partition function, 835, 843–844, 852 Vibrational relaxation, 778, 797 Vibrational spectra See Vibration– rotation spectra Vinyl cation, 710 Virial coefficients, 244–245, 323, 865, 869 Virial equation, 245–246, 257, 323, 869 Virtual orbitals, 708 Viscosity, 479–489 definition of, 480 and diffusion, 504 intrinsic, 486 kinetic theory of, 484–485 measurement of, 483–484 Newton’s law of, 480, 482, 485, 494 of polymer solutions, 481, 486–487 pressure dependence of, 485 relative, 486 of sulfur, 482 temperature dependence of, 481, 485 Viscosity-average molecular weight, 487 Viscosity ratio, 486 Viscous flow, 461 Volt, 397 Voltage, 495 Voltaic cell See Galvanic cell(s) Volume mean molar, 247 molar, 22 partial molar See Partial molar volume(s) partial specific, 492 of a solution, 266–267 units of, 12 Volume change: for forming an ideal solution, 276, 278 irreversible, 45–46 for mixing, 264–265, 267, 278 reversible, 42–43 Volume element in spherical coordinates, 644 Volumetric equation of state See Equation of state Vonnegut, Kurt, 223 VSEPR method, 673 W Wall, kinds of, Wall collisions, 460–461 Water: cohesive energy of ice, 920 dimer of, 710–711 expansion of, 7, 24 freezing-point depression constant of, 354 heat capacity of, 69 ionization of, 332–333, 342, 556, 559 isotherms of, 23, 247, 255 localized MOs of, 699 Monte Carlo simulation of liquid, 950–951 normal boiling point of, 14 normal modes of, 764 phase diagram of, 210–212, 223 supercritical, 249 symmetry operations of, 802 thermal conductivity of, 476 vibrational bands of, 765 Watt, 39 lev38627_index.qxd 4/10/08 9:54 AM Page 989 989 Wave function(s), 605 See also Harmonic oscillator; Hydrogen atom; etc of a degenerate level, 642–643 electronic, 677–678 nuclear, 679, 744–746 spin, 650, 652, 658 time-dependent, 599–605 time-independent, 604–605 units of, 608 well-behaved, 605 Wave mechanics, 600 Wave–particle duality, 594, 596–597 Wavelength, 591, 735 de Broglie, 596 Wavenumber, 736, 753 Weight, 38 Weight average molecular weight, 487 Weight percent, 264 Well-behaved function, 605 Wheatstone bridge, 496–497 Woodward–Hoffman rules, 718, 887 Work, 39–40, 42, 50–51 calculation of, 59–60, 63–64 irreversible, 45–46 non-P-V, 114–115 P-V, 42–46 surface, 229 Work–energy theorem, 39–40 Work function, 112–114, 594 X X-ray diffraction of solids, 931–937 X-ray emission, 775 Y YouTube, 950 Z Z-matrix, 721–722 Zero-point energy, 67, 608, 620, 705, 747, 764, 901 Zeroth law of thermodynamics, 7, Zinc sulfide structure, 931 lev38627_index.qxd 4/10/08 9:54 AM Page 990 Be 9.012 12 Mg 24.31 20 Ca 40.08 38 Sr 87.62 56 Ba 137.3 88 Ra (226) 11 Na 22.99 19 K 39.10 37 Rb 85.47 55 Cs 132.9 87 Fr (223) 89 Ac (227) 57 La 138.9 39 Y 88.91 104 Rf (267) 72 Hf 178.5 40 Zr 91.22 22 Ti 47.88 4B 106 Sg (271) 59 Pr 140.9 91 Pa 231.0 58 Ce 140.1 90 Th 232.0 74 W 183.8 42 Mo 95.96 24 Cr 52.00 6B 105 Db (268) 73 Ta 180.9 41 Nb 92.91 23 V 50.94 5B 92 U 238.0 60 Nd 144.2 107 Bh (272) 75 Re 186.2 43 Tc (98) 25 Mn 54.94 7B 93 Np (237) 61 Pm (145) 108 Hs (270) 76 Os 190.2 44 Ru 101.1 26 Fe 55.85 94 Pu (244) 62 Sm 150.4 109 Mt (276) 77 Ir 192.2 45 Rh 102.9 27 Co 58.93 8B 95 Am (243) 96 Cm (251) 64 Gd 157.3 97 Bk (247) 65 Tb 158.9 112 111 Rg (280) 110 Ds (281) 63 Eu 152.0 80 Hg 200.6 79 Au 197.0 48 Cd 112.4 30 Zn 65.38 12 2B 78 Pt 195.1 47 Ag 107.9 29 Cu 63.55 28 Ni 58.69 46 Pd 106.4 11 1B 10 98 Cf (251) 66 Dy 162.5 113 81 Tl 204.4 49 In 114.8 31 Ga 69.72 99 Es (252) 67 Ho 164.9 114 82 Pb 207.2 50 Sn 118.7 32 Ge 72.64 14 Si 28.09 13 Al 26.98 100 Fm (257) 68 Er 167.3 115 83 Bi 209.0 51 Sb 121.8 33 As 74.92 15 P 30.97 N 14.01 101 Md (258) 69 Tm 168.9 116 84 Po (209) 52 Te 127.6 34 Se 78.96 16 S 32.07 O 16.00 16 6A 102 No (259) 70 Yb 173.0 (117) 85 At (210) 53 I 126.9 35 Br 79.90 17 Cl 35.45 F 19.00 17 7A 103 Lr (262) 71 Lu 175.0 118 86 Rn (222) 54 Xe 131.3 36 Kr 83.80 18 Ar 39.95 10 Ne 20.18 12:17 PM 21 Sc 44.96 3B C 12.01 B 10.81 15 5A 2/20/08 Li 6.941 14 4A 13 3A He 4.003 H 1.008 2A 18 8A 1A lev38627_es.qxd Page lev38627_es.qxd 2/20/08 12:17 PM Page Atomic Numbers and Atomic Weights a Actinium Aluminum Americium Antimony Argon Arsenic Astatine Barium Berkelium Beryllium Bismuth Boron Bromine Cadmium Calcium Californium Carbon Cerium Cesium Chlorine Chromium Cobalt Copper Curium Dysprosium Einsteinium Erbium Europium Fermium Fluorine Francium Gadolinium Gallium Germanium Gold Hafnium Helium Holmium Hydrogen Indium Iodine Iridium Iron Krypton Lanthanum Lawrencium Lead Lithium Lutetium Magnesium Manganese Mendelevium aFrom Ac Al Am Sb Ar As At Ba Bk Be Bi B Br Cd Ca Cf C Ce Cs Cl Cr Co Cu Cm Dy Es Er Eu Fm F Fr Gd Ga Ge Au Hf He Ho H In I Ir Fe Kr La Lr Pb Li Lu Mg Mn Md 89 13 95 51 18 33 85 56 97 83 35 48 20 98 58 55 17 24 27 29 96 66 99 68 63 100 87 64 31 32 79 72 67 49 53 77 26 36 57 103 82 71 12 25 101 (227) 26.981538 (243) 121.760 39.948 74.92160 (210) 137.327 (247) 9.012182 208.98040 10.811 79.904 112.41 40.078 (251) 12.011 140.116 132.90545 35.453 51.9961 58.93320 63.546 (247) 162.50 (252) 167.26 151.964 (257) 18.998403 (223) 157.25 69.723 72.64 196.96657 178.49 4.002602 164.93032 1.00794 114.818 126.90447 192.22 55.845 83.798 138.9055 (262) 207.2 6.941 174.967 24.3050 54.93805 (258) Mercury Molybdenum Neodymium Neon Neptunium Nickel Niobium Nitrogen Nobelium Osmium Oxygen Palladium Phosphorus Platinum Plutonium Polonium Potassium Praseodymium Promethium Protactinium Radium Radon Rhenium Rhodium Rubidium Ruthenium Rutherfordium Samarium Scandium Selenium Silicon Silver Sodium Strontium Sulfur Tantalum Technetium Tellurium Terbium Thallium Thorium Thulium Tin Titanium Tungsten Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium Hg Mo Nd Ne Np Ni Nb N No Os O Pd P Pt Pu Po K Pr Pm Pa Ra Rn Re Rh Rb Ru Rf Sm Sc Se Si Ag Na Sr S Ta Tc Te Tb Tl Th Tm Sn Ti W U V Xe Yb Y Zn Zr 80 42 60 10 93 28 41 102 76 46 15 78 94 84 19 59 61 91 88 86 75 45 37 44 104 62 21 34 14 47 11 38 16 73 43 52 65 81 90 69 50 22 74 92 23 54 70 39 30 40 200.59 95.96 144.24 20.1797 (237) 58.6934 92.90638 14.00674 (259) 190.23 15.9994 106.42 30.97376 195.08 (244) (209) 39.0983 140.90765 (145) 231.03588 (226) (222) 186.207 102.90550 85.4678 101.07 (267) 150.36 44.95591 78.96 28.0855 107.8682 22.989769 87.62 32.065 180.9479 (98) 127.60 158.92535 204.3833 232.0381 168.93421 118.710 47.867 183.84 238.0289 50.9415 131.29 173.05 88.90585 65.38 91.224 “Atomic Weights of the Elements 2007” (www.chem.qmul.ac.uk/iupac/AtWt/) A value in parentheses is the mass number of the longest-lived isotope lev38627_es.qxd 2/20/08 12:17 PM Page Fundamental Constantsa Constant Symbol Gas constant R 8.3145 J molϪ1 KϪ1 8.3145 m3 Pa molϪ1 KϪ1 Avogadro constant Faraday constant Speed of light in vacuum Planck constant Boltzmann constant Proton charge Electron rest mass Proton rest mass Electric constant NA F 6.022142 ϫ 1023 molϪ1 96485.34 C molϪ1 c h k e me mp e0 4pe0 1/4pe0 m0 2.99792458 ϫ 108 m sϪ1 6.626069 ϫ 10Ϫ34 J s 1.38065 ϫ 10Ϫ23 J KϪ1 1.6021765 ϫ 10Ϫ19 C 9.109382 ϫ 10Ϫ31 kg 1.672622 ϫ 10Ϫ27 kg 8.85418782 ϫ 10Ϫ12 C NϪ1 mϪ2 1.112650056 ϫ 10Ϫ10 C NϪ1 mϪ2 8.98755179 ϫ 109 N m2 CϪ2 4p ϫ 10Ϫ7 N CϪ2 s2 G 6.674 ϫ 10Ϫ11 m3 sϪ2 kgϪ1 Magnetic constant Gravitational constant SI value Non-SI value 8.3145 ϫ 107 erg molϪ1 KϪ1 83.145 cm3 bar molϪ1 KϪ1 82.0575 cm3 atm molϪ1 KϪ1 1.9872 cal molϪ1 KϪ1 aAdapted from P J Mohr, B N Taylor, and D B Newell (2007), “CODATA Recommended Values of the Fundamental Physical Constants: 2006” (physics.nist.gov/constants and arxiv.org/abs/0801.0028) Defined Constants Standard gravitational acceleration gn ϵ 9.80665 m/s2 Zero of the Celsius scale ϵ 273.15 K Greek Alphabet Alpha Beta Gamma Delta Epsilon Zeta Eta Theta 〈 〉 ⌫ ⌬ ⌭ ⌮ ⌯ ⌰ a b g d e z h u Iota Kappa Lambda Mu Nu Xi Omicron Pi ⌱ ⌲ ⌳ ⌴ ⌵ ⌶ ⌷ ⌸ i k l m n j o p Rho Sigma Tau Upsilon Phi Chi Psi Omega ⌹ ⌺ ⌻ ⌼ ⌽ ⌾ ⌿ ⍀ r s t y f x c v lev38627_es.qxd 2/20/08 12:17 PM Page Conversion Factorsa eV ϭ 1.6021765 ϫ 10Ϫ19 J Å ϵ10Ϫ10 m ϭ 10Ϫ8 cm L ϵ 1000 cm3 ϭ1 dm3 D ־3.335641 ϫ 10Ϫ30 C m P ϭ 0.1 N s mϪ2 G ־10Ϫ4 T atm ϵ 101325 Pa 1 torr ϵ 760 atm ϭ133.322 Pa bar ϵ 105 Pa ϭ 0.986923 atm ϭ 750.062 torr dyn ϭ 10Ϫ5 N 1erg ϭ 10Ϫ7 J calth ϵ 4.184 J a The symbol ־means “corresponds to.” SI Prefixes 10Ϫ1 10Ϫ2 10Ϫ3 10Ϫ6 10Ϫ9 10Ϫ12 10Ϫ15 10Ϫ18 10Ϫ21 deci centi milli micro nano pico femto atto zepto d c m n p f a z 10 102 103 106 109 1012 1015 1018 1021 deca hecto kilo mega giga tera peta exa zetta da h k M G T P E Z Properties of Some Isotopesa Isotope 1H 2H 11B 12C 13C 14N 15N 16O 19F 23Na 31P 32S 35Cl 37Cl 39K 79Br 81Br 127I aAbundances Abundance,% Atomic mass I gN 99.988 0.012 80.1 98.9 1.1 99.64 0.36 99.76 100 100 100 95.0 75.8 24.2 93.26 50.7 49.3 100 1.0078250 2.014102 11.009305 12.000 13.003355 14.003074 15.00011 15.994915 18.998403 22.98977 30.97376 31.972071 34.968853 36.965903 38.96371 78.91834 80.91629 126.90447 1/2 3/2 1/2 1/2 1/2 3/2 1/2 3/2 3/2 3/2 3/2 3/2 5/2 5.58569 0.85744 1.7924 — 1.40482 0.40376 Ϫ0.56638 — 5.25774 1.47844 2.2632 — 0.547916 0.456082 0.261005 1.40427 1.51371 1.1253 are for the earth’s crust Atomic masses are the relative masses of the neutral atoms on the 12C scale ... /(cm2 sϪ1) H2 O2 N2 HCl CO2 C2H6 Xe 1.5 0.19 0.15 0. 12 0.10 0.09 0.05 Liquid (25 °C) H2O C6H6 Hg CH3OH C2H5OH n-C3H7OH 105Djj /(cm2 sϪ1) 2. 4 2. 2 1.7 2. 3 1.0 0.6 Diffusion coefficients at atm and 25 °C... Hq2O,C2H5OH ϭ 2. 4 ϫ 10Ϫ5 cm2 sϪ1 at 25 °C and atm Some Dq values at 25 °C and atm for the solvent H2O are: i Ϫ1 Dq i,H2O>1cm s 10 N2 LiBr NaCl n-C4H9OH sucrose hemoglobin 1.6 1.4 2. 2 0.56 0. 52. .. artery disease [J J Pacella et al., Eur Heart J., 27 , 23 62 (20 06); S Kaul and A R Jayaweera, ibid., 27 , 22 72] lev38 627 _ch15.qxd 3 /24 /08 6 :22 PM Page 481 481 Blood flow is mainly laminar Turbulent