CONCISE PHYSICAL CHEMISTRY
CONTENTS IDEAL GAS LAWS EMPIRICAL GAS LAWS, THE COMBINED GAS LAW, UNITS, THE MOLE, EQUATIONS OF STATE, DALTON’S LAW, PARTIAL PRESSURES, THE MOLE FRACTION, EXTENSIVE AND INTENSIVE VARIABLES, GRAHAM’S LAW OF EFFUSION, MOLECULAR WEIGHT DETERMINATION, THE MAXWELL–BOLTZMANN DISTRIBUTION, FIGURE 1 1 THE PROBABILITY DENSITY FOR VELOCITIES OF IDEAL GAS PARTICLES AT T ?= 0 , BOLTZMANN’S CONSTANT, FIGURE 1 2 A MAXWELL–BOLTZMANN DISTRIBUTION OVER DISCONTINUOUS ENERGY LEVELS , A DIGRESSION ON “SPACE”, FIGURE 1 3 THE GAUSSIAN PROBABILITY DENSITY DISTRIBUTION IN 3-SPACE , THE GAUSSIAN DISTRIBUTION IN 2- AND 3- AND 4-SPACE, THE SUM-OVER-STATES OR PARTITION FUNCTION, FIGURE 1 4 THE PROBABILITY DENSITY OF MOLECULAR VELOCITIES IN A SPHERICAL VELOCITY SPACE , PROBLEMS AND EXERCISES, EXERCISE 1 1, EXERCISE 1 2, COMPUTER EXERCISE 1 14, REAL GASES: EMPIRICAL EQUATIONS THE VAN DER WAALS EQUATION, THE VIRIAL EQUATION: A PARAMETRIC CURVE FIT, THE COMPRESSIBILITY FACTOR, FIGURE 2 1 A QUADRATIC LEAST-SQUARES FIT TO AN EXPERIMENTAL DATA SET FOR THE COMPRESSIBILITY FACTOR OF NITROGEN AT 300 K AND LOW PRESSURES (SIGMAPLOT 11 0 C?) , FILE 2 1 PARTIAL OUTPUT FROM A QUADRATIC LEAST-SQUARES CURVE FIT TO THE COMPRESSIBILITY FACTOR OF NITROGEN AT 300 K (SIGMAPLOT 11 0 C?) , FIGURE 2 2 THE SECOND VIRIAL COEF?CIENT OF THREE GASES AS A FUNCTION OF TEMPERATURE , CORRESPONDING STATES, FIGURE 2 3 THE Z = F (P) CURVE FOR TWO DIFFERENT GASES OR FOR THE SAME GAS AT TWO DIFFERENT TEMPERATURES , THE CRITICAL TEMPERATURE, FIGURE 2 4 THREE ISOTHERMS OF A VAN DER WAALS GAS , FIGURE 2 5 CONVERSION OF A LIQUID TO ITS VAPOR WITHOUT BOILING (1–4) , SUBCRITICAL FLUIDS, THE CRITICAL DENSITY, FIGURE 2 6 DENSITY ? CURVES FOR LIQUID AND GASEOUS OXYGEN , REDUCED VARIABLES, THE LAW OF CORRESPONDING STATES, ANOTHER VIEW, FIGURE 2 7 COMPRESSIBILITY FACTORS CALCULATED FROM THE VAN DER WAALS CONSTANTS , DETERMINING THE MOLAR MASS OF A NONIDEAL GAS, EXERCISE 2 1, FIGURE 2 8 BOYLE’S LAW PLOT FOR AN IDEAL GAS (LOWER CURVE) AND FOR NITROGEN (UPPER CURVE) , EXERCISE 2 2, TABLE 2 1 OBSERVED REAL GAS BEHAVIOR FROM 10 TO 100 BAR EXPRESSED AS (P, PVM) , FIGURE 2 9 EXPERIMENTAL VALUES OF PVM = Z(P) VS P FOR A REAL GAS , TABLE 2 2 OBSERVED REAL GAS BEHAVIOR EXPRESSED AS (P, PVM) , FIGURE 2 10 QUADRATIC REAL GAS BEHAVIOR , FIGURE 2 11 CUBIC REAL GAS BEHAVIOR , THE THERMODYNAMICS OF SIMPLE SYSTEMS CONSERVATION LAWS AND EXACT DIFFERENTIALS, THE RECIPROCITY RELATIONSHIP, THERMODYNAMIC CYCLES, FIGURE 3 1 DIFFERENT PATH TRANSFORMATIONS FROM A TO B , HEY, LET’S MAKE A PERPETUAL MOTION MACHINE!, LINE INTEGRALS IN GENERAL, FIGURE 3 2 DIFFERENT SEGMENTS OF A CURVED ROD , MATHEMATICAL INTERLUDE: THE LENGTH OF AN ARC, FIGURE 3 3 PYTHAGOREAN APPROXIMATION TO THE SHORT ARC OF A CURVE , BACK TO LINE INTEGRALS, THERMODYNAMIC STATES AND SYSTEMS, STATE FUNCTIONS, REVERSIBLE PROCESSES AND PATH INDEPENDENCE, FIGURE 3 4 THE ENERGY CHANGE FOR REVERSIBLE EXPANSION OF AN IDEAL GAS , HEAT CAPACITY, ENERGY AND ENTHALPY, THE JOULE AND JOULE–THOMSON EXPERIMENTS, FIGURE 3 5 INVERSION TEMPERATURE TI AS A FUNCTION OF PRESSURE , THE HEAT CAPACITY OF AN IDEAL GAS, TABLE 3 1 HEAT CAPACITIES AND ? FOR SELECTED GASES , FIGURE 3 6 TYPICAL HEAT CAPACITY AS A FUNCTION OF TEMPERATURE FOR A SIMPLE ORGANIC MOLECULE , ADIABATIC WORK, FIGURE 3 7 TWO EXPANSIONS OF AN IDEAL GAS , PROBLEMS AND EXAMPLE, EXAMPLE 3 1, FIGURE 3 8 C = DIAGONAL ALONG X = 1 TO Y = 1 , FIGURE 3 9 C = QUARTER-CIRCULAR ARC , THERMOCHEMISTRY CALORIMETRY, ENERGIES AND ENTHALPIES OF FORMATION, STANDARD STATES, MOLECULAR ENTHALPIES OF FORMATION, FIGURE 4 1 COMBUSTION OF C(GR) AND CO(G) , FIGURE 4 2 A THERMOCHEMICAL CYCLE FOR DETERMINING F H 298(METHANE) , ENTHALPIES OF REACTION, GROUP ADDITIVITY, F H 298(G) FROM CLASSICAL MECHANICS, THE SCHR¨ODINGER EQUATION, VARIATION OF H WITH T, DIFFERENTIAL SCANNING CALORIMETRY, FIGURE 4 3 SCHEMATIC DIAGRAM OF THE THERMAL DENATURATION OF A WATER-SOLUBLE PROTEIN , EXAMPLE 4 1, ENTROPY AND THE SECOND LAW ENTROPY, FIGURE 5 1 AN ENGINE , HEAT DEATH AND TIME’S ARROW, THE REACTION COORDINATE, DISORDER, ENTROPY CHANGES, HEATING, EXPANSION, HEATING AND EXPANSION, SPONTANEOUS PROCESSES, MIXING, HEAT TRANSFER, CHEMICAL REACTIONS, THE THIRD LAW, CHEMICAL REACTIONS (AGAIN), EXAMPLE 5 1, FIGURE 5 2 C P/T VS T FOR METALLIC SILVER AG(S) , THE GIBBS FREE ENERGY COMBINING ENTHALPY AND ENTROPY, FREE ENERGIES OF FORMATION, SOME FUNDAMENTAL THERMODYNAMIC IDENTITIES, THE FREE ENERGY OF REACTION, PRESSURE DEPENDENCE OF THE CHEMICAL POTENTIAL, FIGURE 6 1 A REACTION DIAGRAM FOR G4 , THE EQUILIBRIUM CONSTANT AS A QUOTIENT OF QUOTIENTS, THE TEMPERATURE DEPENDENCE OF THE FREE ENERGY, EXAMPLE 6 1, EQUILIBRIUM THE EQUILIBRIUM CONSTANT, GENERAL FORMULATION, THE EXTENT OF REACTION, FUGACITY AND ACTIVITY, VARIATION OF THE EQUILIBRIUM CONSTANT WITH TEMPERATURE, THE VAN’T HOFF EQUATION, LE CHATELIER’S PRINCIPLE, ENTROPY FROM THE VAN’T HOFF EQUATION, COMPUTATIONAL THERMOCHEMISTRY, CHEMICAL POTENTIAL: NONIDEAL SYSTEMS, FREE ENERGY AND EQUILIBRIA IN BIOCHEMICAL SYSTEMS, MAKING ATP, THE CELL’S POWER SUPPLY, PROBLEMS AND EXAMPLES, EXAMPLE 7 1, EXAMPLE 7 2, A STATISTICAL APPROACH TO THERMODYNAMICS EQUILIBRIUM, FIGURE 8 1 A TWO-LEVEL EQUILIBRIUM , FIGURE 8 2 A TWO-LEVEL EQUILIBRIUM , DEGENERACY AND EQUILIBRIUM, FIGURE 8 3 A DEGENERATE TWO-LEVEL EQUILIBRIUM , FIGURE 8 4 A DEGENERATE TWO-LEVEL EQUILIBRIUM , FIGURE 8 5 A TWO-LEVEL EQUILIBRIUM WITH MANY A AND MANY B LEVELS , GIBBS FREE ENERGY AND THE PARTITION FUNCTION, ENTROPY AND PROBABILITY, THE THERMODYNAMIC FUNCTIONS, TABLE 8 1 THERMODYNAMIC FUNCTIONS (IRIKURA, 1998) , THE PARTITION FUNCTION OF A SIMPLE SYSTEM, THE PARTITION FUNCTION FOR DIFFERENT MODES OF MOTION, THE EQUILIBRIUM CONSTANT: A STATISTICAL APPROACH, COMPUTATIONAL STATISTICAL THERMODYNAMICS, TABLE 8 2 SOME COMPUTED PARTITION FUNCTIONS FOR MOLECULAR AND ATOMIC SODIUM , EXAMPLE 8 1, EXAMPLE 8 2, THE PHASE RULE COMPONENTS, PHASES, AND DEGREES OF FREEDOM, COEXISTENCE CURVES, FIGURE 9 1 PURE WATER IN ONE PHASE (LEFT) AND TWO PHASES (RIGHT) , FIGURE 9 2 A LIQUID–VAPOR COEXISTENCE CURVE , FIGURE 9 3 A SINGLE-COMPONENT PHASE DIAGRAM , THE CLAUSIUS–CLAPEYRON EQUATION, PARTIAL MOLAR VOLUME, FIGURE 9 4 TOTAL VOLUME OF AN IDEAL BINARY SOLUTION , FIGURE 9 5 VOLUME INCREASE (OR DECREASE) UPON ADDING SMALL AMOUNTS OF SOLUTE N2 TO PURE SOLVENT , GENERALIZATION, FIGURE 9 6 PARTIAL MOLAR VOLUME AS THE SLOPE OF V VS N2 , FIGURE 9 7 VOLUME BEHAVIOR OF A NONIDEAL BINARY SOLUTION , THE GIBBS PHASE RULE, TWO-COMPONENT PHASE DIAGRAMS, TYPE 1, FIGURE 9 8 A TYPE I PHASE DIAGRAM , TYPE II, FIGURE 9 9 A TYPE II PHASE DIAGRAM , TYPE III, FIGURE 9 10 A TYPE III PHASE DIAGRAM , COMPOUND PHASE DIAGRAMS, FIGURE 9 11 A COMPOUND PHASE DIAGRAM WITH A LOW BOILING AZEOTROPE , TERNARY PHASE DIAGRAMS, FIGURE 9 12 A TERNARY PHASE DIAGRAM WITH A TIE LINE , EXAMPLE 9 1, FIGURE 9 13 THE LIQUID–VAPOR COEXISTENCE CURVE OF WATER LEADING TO VAPH(H2O) = 44 90KJMOL?1 , EXAMPLE 9 2, FIGURE 9 14 A TERNARY PHASE DIAGRAM IN WHICH B AND C ARE PARTIALLY MISCIBLE , CHEMICAL KINETICS FIRST-ORDER KINETIC RATE LAWS, FIGURE 10 1 FIRST-ORDER RADIOACTIVE DECAY , FIGURE 10 2 LOGARITHMIC DECAY OF A RADIOACTIVE ELEMENT , SECOND-ORDER REACTIONS, OTHER REACTION ORDERS, MATHEMATICAL INTERLUDE: THE LAPLACE TRANSFORM, BACK TO KINETICS: SEQUENTIAL REACTIONS, REVERSIBLE REACTIONS, EXPERIMENTAL DETERMINATION OF THE RATE EQUATION, REACTION MECHANISMS, THE IN?UENCE OF TEMPERATURE ON RATE, FIGURE 10 3 AN ACTIVATION ENERGY BARRIER BETWEEN AN UNSTABLE POSITION AND A STABLE POSITION , FIGURE 10 4 ENTHALPY LEVEL DIAGRAM FOR AN ACTIVATED COMPLEX [B] , FIGURE 10 5 AN ACTIVATION BARRIER , FIGURE 10 6 A BOLTZMANN DISTRIBUTION OF MOLECULAR SPEEDS , COLLISION THEORY, COMPUTATIONAL KINETICS, EXAMPLE 10 1, EXAMPLE 10 2, FIGURE 10 7 FIRST-ORDER FLUORESCENCE DECLINE FROM ELECTRONICALLY EXCITED IODINE IN MILLISECONDS , FIGURE 10 8 THE NATURAL LOGARITHM OF RELATIVE INTENSITY VS TIME FOR RADIATIVE DECAY , LIQUIDS AND SOLIDS SURFACE TENSION, FIGURE 11 1 INTERMOLECULAR ATTRACTIVE FORCES ACTING UPON MOLECULES AT AN AIR–WATER INTERFACE , FIGURE 11 2 STRETCHING A TWO-DIMENSIONAL MEMBRANE BY MOVING AN EDGE OF LENGTH L , FIGURE 11 3 STRETCHING A TWO-DIMENSIONAL LIQUID BIMEMBRANE , FIGURE 11 4 CAPILLARY RISE IN A TUBE OF RADIUS R , HEAT CAPACITY OF LIQUIDS AND SOLIDS, FIGURE 11 5 HEAT CAPACITY AS A FUNCTION OF TEMPERATURE , VISCOSITY OF LIQUIDS, FIGURE 11 6 APPROXIMATION OF LAMINAR FLOW INSIDE A TUBE , CRYSTALS, FIGURE 11 7 CLOSE PACKING OF MARBLES BETWEEN TWO SHEETS , FIGURE 11 8 A LESS EF?CIENT PACKING OF MARBLES , FIGURE 11 9 BRAGG’S LAW FOR CONSTRUCTIVE RE?ECTION , X-RAY DIFFRACTION: DETERMINATION OF INTERPLANAR DISTANCES, FIGURE 11 10 A FACE-CENTERED CUBIC UNIT CELL , THE PACKING FRACTION, FIGURE 11 11 A TWO-DIMENSIONAL UNIT CELL FOR PACKING OF DISCS , FIGURE 11 12 A SIMPLE CUBIC CELL , BRAVAIS LATTICES, TABLE 11 1 THE BRAVAIS CRYSTAL SYSTEMS AND LATTICES , COVALENT BOND RADII, COMPUTATIONAL GEOMETRIES, LATTICE ENERGIES, PROBLEMS AND EXERCISE, EXERCISE 11 1, FIGURE 11 13 THE BORN–HABER CYCLE FOR NAI , FIGURE 11 14 CLOSE PACKING (LEFT) AND SIMPLE SQUARE UNIT CELLS (RIGHT) , FIGURE 11 15 A BODY-CENTERED PRIMITIVE CUBIC CELL , SOLUTION CHEMISTRY THE IDEAL SOLUTION, FIGURE 12 1 ENTROPY, ENTHALPY, AND GIBBS FREE ENERGY CHANGES FOR IDEAL MIXING AT T > 0 , RAOULT’S LAW, FIGURE 12 2 PARTIAL AND TOTAL PRESSURES FOR A RAOULT’S LAW SOLUTION , A DIGRESSION ON CONCENTRATION UNITS, REAL SOLUTIONS, FIGURE 12 3 CONSISTENT POSITIVE DEVIATIONS FROM RAOULT’S LAW , HENRY’S LAW, FIGURE 12 4 HENRY’S LAW FOR THE PARTIAL PRESSURE OF COMPONENT B AS THE SOLUTE , HENRY’S LAW ACTIVITIES, VAPOR PRESSURE, BOILING POINT ELEVATION, FIGURE 12 5 BOILING OF PURE SOLVENT (LEFT) AND A SOLUTION OF SOLVENT AND NONVOLATILE SOLUTE (RIGHT) , OSMOTIC PRESURE, FIGURE 12 6 OSMOTIC PRESSURE, ? , COLLIGATIVE PROPERTIES, FIGURE 12 7 LOWERING OF THE FREEZING POINT OF WATER BY AMMONIA , PROBLEMS, EXAMPLES, AND EXERCISE, EXAMPLE 12 1, TABLE 12 1 VAPOR PRESSURES OF ACETONE OVER DILUTE BINARY SOLUTIONS OF ACETONE IN DIETHYL ETHER , EXAMPLE 12 2, EXERCISE 12 1, EXERCISE 12 2, COULOMETRY AND CONDUCTIVITY ELECTRICAL POTENTIAL, MEMBRANE POTENTIALS, FIGURE 13 1 THE POTENTIAL DROP BETWEEN CHARGED PLATES IS V = ?(0) ??(L) , FIGURE 13 2 AN ION-PERMEABLE MEMBRANE (SCHEMATIC) , RESISTIVITY, CONDUCTIVITY, AND CONDUCTANCE, MOLAR CONDUCTIVITY, FIGURE 13 3 KOHLRAUSCH’S LAW FOR CONDUCTANCE OF THE STRONG ELECTROLYTES HCL AND NAOAC AND THE WEAK ELECTROLYTE HOAC , PARTIAL IONIZATION: WEAK ELECTROLYTES, ION MOBILITIES, FIGURE 13 4 MOVING BOUNDARY DETERMINATION OF THE MOBILITY OF H+ , FARADAY’S LAWS, MOBILITY AND CONDUCTANCE, THE HITTORF CELL, FIGURE 13 5 A THREE-COMPARTMENT HITTORF CELL , ION ACTIVITIES, EXAMPLE 13 1, EXAMPLE 13 2, EXAMPLE 13 3, ELECTROCHEMICAL CELLS THE DANIELL CELL, HALF-CELLS, FIGURE 14 1 THE HYDROGEN HALF-CELL , HALF-CELL POTENTIALS, TABLE 14 1 A FEW SELECTED REDUCTION POTENTIALS , CELL DIAGRAMS, ELECTRICAL WORK, THE NERNST EQUATION, CONCENTRATION CELLS, FINDING E?, FIGURE 14 2 EXTRAPOLATION TO E?= 0 2223 FOR THE STANDARD HYDROGEN–SILVER–SILVER CHLORIDE CELL , SOLUBILITY AND STABILITY PRODUCTS, MEAN IONIC ACTIVITY COEF?CIENTS, THE CALOMEL ELECTRODE, THE GLASS ELECTRODE, EXAMPLE 14 1, EXAMPLE 14 2, FIGURE 14 3 THE MEAN ACTIVITY COEF?CIENT OF HCL AS A FUNCTION OF M1/2 , EARLY QUANTUM THEORY: A SUMMARY THE HYDROGEN SPECTRUM, FIGURE 15 1 THE HYDROGEN EMISSION SPECTRUM , FIGURE 15 2 THE FIRST SIX SOLUTIONS OF THE H ATOM ENERGY CALCULATED BY BOHR (1913) , EARLY QUANTUM THEORY, SCHR¨ODINGER, HEISENBERG, AND BORN: AN INTRODUCTION, THE HAMILTONIAN OPERATOR, MOLECULAR QUANTUM CHEMISTRY, HEITLER AND LONDON, HARTREE AND FOCK, ANTISYMMETRY AND DETERMINANTAL WAVE FUNCTIONS, THE HARTREE INDEPENDENT ELECTRON METHOD, A DIGRESSION ON ATOMIC UNITS, EXAMPLE 15 1, EXAMPLE 15 2, WAVE MECHANICS OF SIMPLE SYSTEMS WAVE MOTION, FIGURE 16 1 GRAPH OF SIN(X), SIN(2X), AND SIN(3X) SHOWN OVER THE INTERVAL 0, ? , WAVE EQUATIONS, EIGENVALUES AND EIGENVECTORS, QUANTUM MECHANICAL SYSTEMS, IS A VECTOR, THE EIGENFUNCTION POSTULATE, THE PARTICLE IN A ONE-DIMENSIONAL BOX, FIGURE 16 2 WAVE FORMS FOR THE FIRST THREE WAVE FUNCTIONS OF THE PARTICLE IN A BOX , FUNDAMENTALS AND OVERTONES, FIGURE 16 3 A MATHCAD C?SKETCH OF THE BORN PROBABILITY DENSITIES AT THE FIRST THREE LEVELS OF THE PARTICLE IN A BOX , THE PARTICLE IN A CUBIC BOX, SEPARABLE EQUATIONS, ORBITALS, FIGURE 16 4 THE GROUND STATE ORBITAL OF A PARTICLE CON?NED TO A CUBIC BOX , DEGENERACY, FIGURE 16 5 THE FIRST EXCITED STATE OF A PARTICLE CON?NED TO A CUBIC BOX , NORMALIZATION, FIGURE 16 6 THE DEGENERATE ENERGY LEVELS FOR THE HYDROGEN ATOM , THE HYDROGEN ATOM, THE RADIAL EQUATION AND PROBABILITY “SHELLS”, BREAKING DEGENERACY, FIGURE 16 7 REDUCED DEGENERACY IN ENERGY LEVELS FOR HYDROGEN-LIKE ATOMS , HIGHER EXACT SOLUTIONS FOR THE HYDROGEN ATOM, TABLE 16 1 THE FIRST SIX WAVE FUNCTIONS FOR HYDROGEN , TABLE 16 2 THE FIRST THREE S WAVE FUNCTIONS FOR HYDROGEN (SIMPLI?ED FORM) , FIGURE 16 8 ROOTS OF THE RADIAL 3S WAVE FUNCTION OF ATOMIC HYDROGEN AS A FUNCTION OF DISTANCE R , FIGURE 16 9 THE RADIAL PROBABILITY DENSITY FOR AN ELECTRON IN THE 3S ORBITAL OF HYDROGEN , ORTHOGONALITY AND OVERLAP, FIGURE 16 10 THE RADIAL NODE OF THE 2P ATOMIC ORBITAL , FIGURE 16 11 FAVORABLE SPX AND UNFAVORABLE SPZ OVERLAP OF ORBITALS DEPENDING UPON ORBITAL SYMMETRY , MANY-ELECTRON ATOMIC SYSTEMS, THE HARTREE METHOD, THE VARIATIONAL METHOD: ATOMS MORE ON THE VARIATIONAL METHOD, THE SECULAR DETERMINANT, A VARIATIONAL TREATMENT FOR THE HYDROGEN ATOM: THE ENERGY SPECTRUM, OPTIMIZING THE GAUSSIAN FUNCTION, SIMULTANEOUS MINIMA, THE EXACT WAVE FUNCTION, THE GAUSSIAN APPROXIMATION, A GAUSSIAN C?HF CALCULATION OF EATOM: COMPUTER FILES, FILE 17 1 GAUSSIAN GEN INPUT FOR THE HYDROGEN ATOM , FILE 17 2 ENERGIES DRAWN FROM THE GAUSSIAN GEN OUTPUT FILE FOR THE HYDROGEN ATOM , HELIUM, AN SCF VARIATIONAL IONIZATION POTENTIAL FOR HELIUM, SPIN, BOSONS AND FERMIONS, SLATER DETERMINANTS, THE AUFBAU PRINCIPLE, THE SCF ENERGIES OF FIRST-ROW ATOMS AND IONS, FIGURE 17 1 CALCULATED IP1 FOR ELEMENTS 1–10 , SLATER-TYPE ORBITALS (STO), TABLE 17 1 SLATER’S RULES , SPIN–ORBIT COUPLING, FIGURE 17 2 LINEAR AND ANGULAR MOMENTUM VECTORS , EXAMPLE 17 1, FILE 17 3 MATHCAD C?CALCULATION OF THE IONIZATION POTENTIAL OF HELIUM , EXAMPLE 17 2, EXPERIMENTAL DETERMINATION OF MOLECULAR STRUCTURE THE HARMONIC OSCILLATOR, FIGURE 18 1 A CLASSICAL HARMONIC OSCILLATOR , THE HOOKE’S LAW POTENTIAL WELL, FIGURE 18 2 PARABOLIC POTENTIAL WELLS FOR THE HARMONIC OSCILLATOR , DIATOMIC MOLECULES, THE QUANTUM RIGID ROTOR, FIGURE 18 3 ENERGY LEVELS WITHIN A SIMPLE ROTOR , MICROWAVE SPECTROSCOPY: BOND STRENGTH AND BOND LENGTH, ELECTRONIC SPECTRA, FIGURE 18 4 ELECTRONIC PROMOTION IN ALKENES , FIGURE 18 5 ABSORPTION WAVELENGTHS OF CONJUGATED POLYALKENES , DIPOLE MOMENTS, FIGURE 18 6 A CHARGED PARALLEL PLATE CAPACITOR , FIGURE 18 7 A CHARGED CAPACITOR WITH A DIELECTRIC , DIELECTRIC CONSTANT, POLARIZABILITY, BOND MOMENTS, FIGURE 18 8 THE TOTAL DIPOLES OF TWO DICHLOROETHENE ISOMERS , NUCLEAR MAGNETIC RESONANCE (NMR), SPIN–SPIN COUPLING, FIGURE 18 9 SCHEMATIC NMR SPECTRUM OF ETHANOL, CH3CH2OH , MAGNETIC RESONANCE IMAGING (MRI), ELECTRON SPIN RESONANCE, EXAMPLE 18 1, FIGURE 18 10 SCHEMATIC DIAGRAM OF A VIBRATION–ROTATION BAND , EXAMPLE 18 2, FIGURE 18 11 THE VIBRATION–ROTATION SPECTRUM OF CO , CLASSICAL MOLECULAR MODELING ENTHALPY: ADDITIVE METHODS, FIGURE 19 1 ENTHALPIES OF FORMATION OF “ADJACENT” N-ALKANES , BOND ENTHALPIES, BOND ADDITIVITY, FIGURE 19 2 BOND ENTHALPIES CALCULATED IN CH2, FROM THE REFERENCE STATE OF GASEOUS ATOMS (TOP), AND RELATIVE TO ELEMENTS IN THEIR STANDARD STATE (H2(G) AND C(GRAPHITE)) , STRUCTURE, FIGURE 19 3 STRUCTURALLY DISTINCT ALKANE CONFORMERS RESULTING FROM THE TETRAHEDRAL SYMMETRY OF CARBON , FORCE CONSTANTS AND PARAMETERS, ENERGY EQUATIONS, FORCE FIELDS, THE ALLINGER MM METHOD, GEOMETRY AND ENTHALPY: MOLECULAR MECHANICS, MOLECULAR MODELING, THE GUI, FIGURE 19 4 VISUALIZATION OF THE OUTPUT FOR THE ETHANE MOLECULE (PCMODEL 8 0 C?) , FINDING THERMODYNAMIC PROPERTIES, FILE 19 1 PARTIAL MM4 ENTHALPY OUTPUT FOR ETHANE , THE OUTSIDE WORLD, TRANSITION STATES, EXAMPLE 19 1, EXAMPLE 19 2, FILE 19 2 AN INPUT FILE FOR WATER , FILE 19 3 THE MM4 GEOMETRY OUTPUT TAPE9 MM4 FOR WATER , EXAMPLE 19 3, FILE 19 4 MM4 INPUT GEOMETRY FOR METHANE , FILE 19 5 MM4 OUTPUT GEOMETRY FOR METHANE , QUANTUM MOLECULAR MODELING THE MOLECULAR VARIATIONAL METHOD, THE HYDROGEN MOLECULE ION, FIGURE 20 1 THE HYDROGEN MOLECULE ION, H+ 2 , FIGURE 20 2 BONDING AND ANTIBONDING ORBITALS FOR H+ 2 , FIGURE 20 3 BONDING AND ANTIBONDING SOLUTIONS FOR THE H+ 2 , HIGHER MOLECULAR ORBITAL CALCULATIONS, SEMIEMPIRICAL METHODS, AB INITIO METHODS, THE GAUSSIAN BASIS SET, FIGURE 20 4 THE 1S STO (SOLID LINE) AND A GAUSSIAN APPROXIMATION (DOTTED LINE) , FILE 20 1 (INPUT) A FOUR-PARAMETER GAUSSIAN FILE FOR THE HYDROGEN ATOM , FIGURE 20 5 COMPARISON OF THE 1S STO OF HYDROGEN WITH AN ARBITRARILY PARAMETERIZED TWO-GAUSSIAN FUNCTION ?(R) = 0 40E?1 0 R2 + 0 60E?0 25 R2 , FILE 20 2 (OUTPUT) THE STO-2G BASIS SET WRITTEN AS A 1S ORBITAL CONSISTING OF FUNCTIONS WITH ARBITRARILY SELECTED EXPONENTS 1 00 AND 0 25 , STORED PARAMETERS, FILE 20 3 (INPUT) AN STO-2G INPUT FILE USING A STORED BASIS SET , FILE 20 4 (OUTPUT) STORED PARAMETERS FOR THE STO-2G BASIS SET , FIGURE 20 5 APPROXIMATION TO THE 1S ORBITAL OF HYDROGEN BY 2 GAUSSIANS , MOLECULAR ORBITALS, FILE 20 5 (INPUT) A MOLECULAR ORBITAL INPUT FILE FOR H2 , Z-MATRIX FORMAT, FILE 20 6 (INPUT) A GAUSSIAN INPUT FILE FOR H2 , GAMESS, FILE 20 7 (INPUT) GAMESS FILE FOR HYDROGEN MOLECULE , METHANE, FILE 20 8 ONE OF MANY POSSIBLE STO-2G OPTIMIZED COORDINATES SETS FOR METHANE , SPLIT VALENCE BASIS SETS, POLARIZED BASIS FUNCTIONS, HETEROATOMS: OXYGEN, FILE 20 9 INPUT ?LE FOR A GAUSSIAN C?STO-3G CALCULATION ON METHANOL , FILE 20 10 OPTIMIZED GEOMETRY FROM A GAUSSIAN C? STO-3G CALCULATION ON METHANOL (INTERNAL AND CARTESIAN COORDINATES) , FINDING F H 298 OF METHANOL, FIGURE 20 6 THE G3MP2 THERMOCHEMICAL CYCLE FOR DETERMINATION OF F H 298 OF METHANOL , FURTHER BASIS SET IMPROVEMENTS, POST-HARTREE–FOCK CALCULATIONS, PERTURBATION, COMBINED OR SCRIPTED METHODS, SCHEME 20 1 A COMPUTATIONAL CHEMICAL SCRIPT , FIGURE 20 7 ADDITIVE EXTRAPOLATIONS IN THE G3(MP2) SCRIPTED METHOD , FILE 20 11 PARTIAL GAUSSIAN G3(MP2) OUTPUT , DENSITY FUNCTIONAL THEORY (DFT), PROBLEMS AND EXAMPLES, EXAMPLE 20 1, FILE 20 12 A Z-MATRIX INPUT FILE FOR METHANE , EXAMPLE 20 2, FILE 20 13 CONTROL LINES FOR A GAMESS CALCULATION , EXAMPLE 20 3, PHOTOCHEMISTRY AND THE THEORY OF CHEMICAL REACTIONS EINSTEIN’S LAW, QUANTUM YIELDS, TABLE 21 1 SOME EXPERIMENTAL QUANTUM YIELDS , FIGURE 21 1 MECHANISM FOR FLUORESCENT AND PHOSPHORESCENT LIGHT EMISSION , LIPID PEROXIDATION, OZONE DEPLETION, BOND DISSOCIATION ENERGIES (BDE), LASERS, ISODESMIC REACTIONS, THE EYRING THEORY OF REACTION RATES, THE POTENTIAL ENERGY SURFACE, FIGURE 21 2 EYRING POTENTIAL ENERGY PLOT FOR THE REACTION H + H–H ?H–H + H , FIGURE 21 3 ACTIVATION OF THE SYMMETRICAL REACTION H + H–H ?H–H + H , FIGURE 21 4 THE ENTHALPY OF ACTIVATION OF AN EXOTHERMIC REACTION , OPTICAL INVERSION, FIGURE 21 5 AN OPTICALLY ACTIVE SPECIES , FIGURE 21 6 INVERSION OF OPTICAL ACTIVITY , THE STEADY-STATE PSEUDO-EQUILIBRIUM, ENTROPIES OF ACTIVATION, THE STRUCTURE OF THE ACTIVATED COMPLEX, EXAMPLE 21 1, EXAMPLE 21 2, REFERENCES ANSWERS TO SELECTED ODD-NUMBERED PROBLEMS (BOYLE, 1662) AND V = K2T (CHARLES, 1787) THE CONSTANTS K1 AND K2 CAN BE APPROXIMATED SIMPLY BY AVERAGING A SERIES OF EXPERI- MENTAL MEASUREMENTS, ?RST OF PV AT CONSTANT TEMPERATURE T FOR THE BOYLE EQUATION, THEN OF V/T AT CONSTANT PRESSURE P FOR CHARLES’S LAW ALL THIS CAN BE DONE USING SIMPLE MANOMETERS AND THERMOMETERS THE SUMMED FORCE F IS GIVEN IN UNITS OF NEWTONS (N), AND THE AREA IS IN SQUARE METERS (M2) THE N M?2 IS ALSO CALLED THE PASCAL (PA) THE PASCAL IS ABOUT ?VE OR SIX ORDERS OF MAGNITUDE SMALLER THAN PRESSURES ENCOUNTERED IN NORMAL LABORATORY PRACTICE, SO THE CONVENIENT UNIT 1 BAR ?105 PA WAS DE?NED THE LOGICAL UNIT OF VOLUME IN THE MKS (METER, KILOGRAM, SECOND) SYSTEM IS THE M3, BUT THIS ALSO IS NOT COMMENSURATE WITH ROUTINE LABORATORY PRACTICE WHERE THE LITER IS USED ONE THOUSAND LITERS EQUALS 1 M3, SO THE MKS NAME FOR THIS CUBIC MEASURE IS THE CUBIC DECIMETER—THAT IS, ONE-TENTH OF A METER CUBED (1 DM3) BECAUSE THERE ARE 1000 CUBIC DECIMETERS IN A CUBIC METER AND 1000 LITERS IN A CUBIC METER, IT IS EVIDENT THAT 1 L = 1 DM3 THE UNIT OF TEMPERATURE IS THE KELVIN (K), AND THE UNIT OF WEIGHT IS THE KILOGRAM (KG) FORMALLY, THERE IS A DIFFERENCE BETWEEN WEIGHT AND MASS, WHICH WE SHALL IGNORE FOR THE MOST PART CHEMISTS ARE FOND OF EXPRESSING THE AMOUNT OF A PURE SUBSTANCE IN THE MOLE THE MOLE THE CONCEPT OF THE MOLE (GRAM MOLECULAR WEIGHT IN EARLY LITERATURE) ARISES FROM THE DEDUCTION BY AVOGADRO IN 1811 THAT EQUAL VOLUMES OF GAS AT THE SAME PRESSURE AND TEMPERATURE CONTAIN THE SAME NUMBER OF PARTICLES THIS SOMEWHAT INTUITIVE CONCLUSION WAS DRAWN FROM A PICTURE OF THE GASEOUS STATE AS BEING CHARACTERIZED BY REPULSIVE FORCES BETWEEN GASEOUS PARTICLES WHEREBY DOUBLING, TRIPLING, AND SO ON, THE WEIGHT OF THE SAMPLE TAKEN WILL DOUBLE, TRIPLE, AND SO ON, ITS NUMBER OF PARTICLES, HENCE ITS VOLUME IT WAS ALSO KNOWN AT THE TIME THAT ELECTROLYSIS OF WATER PRODUCED TWO VOLUMES OF HYDROGEN FOR EVERY VOLUME OF OXYGEN, SO AVOGADRO DEDUCED THE FORMULA H2O FOR WATER ON THE BASIS OF HIS HYPOTHESIS OF EQUAL VOLUME FOR EQUAL NUMBERS OF PARTICLES IN THE GASEOUS STATE BY AVOGADRO’S TIME, IT WAS ALSO KNOWN THAT THE NUMBER OF GRAMS OF OXYGEN OBTAINED BY ELECTROLYSIS OF WATER IS 8 TIMES THE NUMBER OF GRAMS OF HYDROGEN BY HIS 2-FOR-1 HYPOTHESIS, AVOGADRO REASONED THAT THE LESS NUMEROUS OXYGEN ATOMS MUST BE 2(8) = 16 TIMES AS HEAVY AS THE MORE NUMEROUS HYDROGEN ATOMS THIS THEORETICAL VISION LED DIRECTLY TO THE CONCEPT OF ATOMIC AND MOLECULAR WEIGHT AND TO THE MASS OF PURE MATERIAL EQUAL TO ITS ATOMIC WEIGHT OR MOLECULAR WEIGHT, WHICH WE NOW CALL THE MOLE 2 VARIOUS EXPERIMENTAL METHODS HAVE BEEN USED TO DETERMINE THE NUMBER OF PARTICLES COMPRISING ONE MOLE OF A PURE SUBSTANCE WITH THE RESULT 1GENERAL PRACTICE IS TO WRITE EXPERIMENTALLY DETERMINED QUANTITIES IN ITALICS AND UNITS IN ROMAN LETTERS, BUT THERE IS SOME OVERLAP AND WE SHALL NOT BE STRICT IN THIS OBSERVANCE 2THE WORD IS MOLE, BUT THE UNIT IS MOL DALTON’S LAW RT V CONST + = I NI RT V CONST = RT V CONST I NI EACH GAS ACTS AS THOUGH IT WERE ALONE IN THE CONTAINER, WHICH LEADS TO THE CONCEPT OF A PARTIAL PRESSURE PI EXERTED BY ONE COMPONENT OF A MIXTURE RELATIVE TO THE TOTAL PRESSURE THIS IDEA IS EMBODIED IN DALTON’S LAW FOR THE TOTAL PRESSURE OF A MIXTURE AS THE SUM OF ITS PARTIAL PRESSURES: PTOTAL = I PI APART FROM EMPHASIZING AVOGADRO’S IDEA THAT THE IDEAL GASEOUS STATE IS CHARACTERIZED BY THE NUMBER OF PARTICLES, NOT BY THEIR INDIVIDUAL NATURE, DALTON’S LAW ALSO LEADS TO THE IDEA OF A PRESSURE FRACTION OF ONE COMPONENT OF A MIXTURE RELATIVE TO THE TOTAL PRESSURE EXERTED BY ALL THE COMPONENTS OF THE MIXTURE: X PI = PI I PI 3MANY REAL GASES ARE NEARLY IDEAL UNDER NORMAL ROOM CONDITIONS MOL?1, WHICH FOLLOWS DIRECTLY FROM MEASUREMENTS OF P AND V ON KNOWN AMOUNTS OF A GAS AT SPECI?C VALUES OF T, ONE CAN DETERMINE THE ATOMIC OR MOLECULAR WEIGHT OF AN INDEPENDENT SAMPLE WITHIN THE LIMITS OF THE IDEAL GAS APPROXIMATION ANOTHER WAY OF ?NDING THE MOLECULAR WEIGHT OF A GAS IS THROUGH GRAHAM’S LAW OF EFFUSION, WHICH STATES THAT THE RATE OF ESCAPE OF A CON?NED GAS THROUGH A VERY SMALL HOLE IS INVERSELY PROPORTIONAL TO ITS PARTICLE WEIGHT—THAT IS, ITS ATOMIC OR MOLECULAR WEIGHT THIS BEING THE CASE, MEASURING THE RATE OF EFFUSION OF TWO GASES—ONE OF KNOWN MOLECULAR WEIGHT AND THE OTHER OF UNKNOWN MOLECULAR WEIGHT—GIVES THE RATIO MWKNOWN/MWUNKNOWN AND HENCE EASY CALCULATION OF MWUNKNOWN ASIDE FROM IMPORTANT MEDICAL APPLICATIONS (DIALYSIS), GRAHAM’S WORK ALSO FOCUSED ATTENTION ON THE RANDOM MOTIONS OF GASEOUS PARTICLES AND THE SPEEDS WITH WHICH THEY MOVE WE CAN RATIONALIZE GRAHAM’S LAW AS THE RESULT OF A VERY LARGE ENSEMBLE OF THE MAXWELL–BOLTZMANN DISTRIBUTION PARTICLES COLLIDING WITH THE WALL OF A CONSTRAINING CONTAINER, SUPPOSING THAT THE WALL HAS A HOLE IN IT ONLY A FEW PARTICLES ESCAPE THE CONTAINER BECAUSE THE HOLE IS SMALL ESCAPE PROBABILITY IS DETERMINED BY HOW FAST THE PARTICLE IS MOVING FAST PARTICLES COLLIDE WITH THE WALLS OF THE CONTAINER MORE OFTEN THAN DO SLOW ONES BY A STANDARD DERIVATION (EXERCISE 1 2), ONE ?NDS PV = X WHERE ¯UX IS THE AVERAGE SPEED OF AN ENSEMBLE CONSISTING OF ONE MOLE OF AN IDEAL GAS NOTICE THAT BECAUSE PV = RT HAS THE UNITS J K?1 MOL?1 K = J MOL?1, PV IS A MOLAR ENERGY INCREASING THE TEMPERATURE OF A GAS REQUIRES AN INPUT OF ENERGY WE USUALLY WRITE THE KINETIC ENERGY EKIN OF A SINGLE MOVING MASS SUCH AS A BASEBALL AS EKIN = 2MV2, WHERE V IS ITS SPEED AND M IS ITS MASS CONSIDER A HYPOTHETICAL ONE-DIMENSIONAL X-SPACE ALONG WHICH POINT PARTICLES CAN MOVE WITHOUT INTERFERENCE IF THE KINETIC ENERGY OF MOLECULAR PARTICLES FOLLOWS THE SAME KIND OF LAW AS MORE MASSIVE PARTICLES, WE OBTAIN X IN PV = 3 NA ¯EKIN BUT PV ALSO EQUALS RT FOR ONE MOLE OF A GAS, SO PV = RT = THE MAXWELL–BOLTZMANN DISTRIBUTION ALL PARTICLES OF A CON?NED GAS DO NOT MOVE WITH THE SAME VELOCITY EVEN IF T IS CONSTANT RATHER, THEY MOVE WITH A VELOCITY PROBABILITY DENSITY ?V WHICH IS RANDOMLY DISTRIBUTED ABOUT V = 0 AND WHICH FOLLOWS THE FAMILIAR GAUSSIAN DISTRIBUTION E?V2 THE PROBABILITY DENSITY FUNCTION DROPS OFF AT LARGE VALUES OF ±V BECAUSE THE PROB- ABILITY OF ?NDING PARTICLES WITH VELOCITIES VERY MUCH DIFFERENT FROM THE MEAN IS SMALL THE CURVE IS SYMMETRICAL BECAUSE, PICKING AN ARBITRARY AXIS, THE PARTICLE MAY BE GOING EITHER TO THE LEFT OR TO THE RIGHT, HAVING A VELOCITY V OR –V THE PEAK AT V = 0 IS SOMEWHAT IDEAL GAS LAWS A MAXWELL–BOLTZMANN DISTRIBUTION OVER DISCONTINUOUS ENERGY LEVELS PARTICLES ARE NOT STATIC; THEY EXCHANGE ENERGY LEVELS RAPIDLY THE LEVELS NEED NOT BE EQUALLY SPACED A DIGRESSION ON “SPACE” = GIE?EI/KBT THE DEGENERACY IS ALWAYS AN INTEGER AND IT IS USUALLY SMALL ALSO, FROM EKIN = 2 RT FOR ONE MOLE, WE CAN ?ND THE EXPECTATION VALUE ??KIN?OF THE KINETIC ENERGY PER REPRESENTATIVE OR AVERAGE PARTICLE ??KIN?= AND ??KIN?= A DIGRESSION ON “SPACE” THE TERMS DENSITY AND PROBABILITY DENSITY WERE USED IN SECTION 1 8 THESE ARE DIFFERENT BUT ANALOGOUS USES OF THE WORD DENSITY IN THE ?RST CASE, DENSITY WAS USED IN THE USUAL SENSE OF WEIGHT OR MASS PER UNIT VOLUME, M/V IN THE SECOND CASE, THE PROBABILITY DENSITY IS DE?NED AS THE PROBABILITY IN A SPECI?ED SPACE ANY VARIABLE MEASURED 4LIKE ALL ENERGIES, THIS ZERO POINT IS ARBITRARY THE SUM-OVER-STATES OR PARTITION FUNCTION ADDING UP ALL THE PARTICLES IN ALL THE STATES OF A SYSTEM GIVES THE TOTAL NUMBER OF PARTICLES IN THE SYSTEM: NI = N THE SUM-OVER-STATES OR PARTITION FUNCTION GIE??I/KBT = GIE??I/KBT Q WHERE WE HAVE GIVEN THE SYMBOL Q TO THE SUMMATION GIE??I/KBT THIS IMPORTANT SUMMATION APPEARS FREQUENTLY AND IS GIVEN THE NAME SUM-OVER-STATES OR PARTITION FUNCTION REWRITING THE RATIO NI/N, WE HAVE NI Q = NGIE??I/KBT ?XED AT T = CONST WE SEE THAT, FOR A GIVEN NUMBER OF MOLECULES N AT TEMPERATURE T, THE RIGHT HAND SIDE OF THE EQUATION IS ?XED FOR ANY SPECI?C ENERGY STATE, THAT IS, NI Q = CONST THE SUM-OVER-STATES IS THEN A SCALING FACTOR, DETERMINING THE RELATIVE POPULATION OF A STATE, NI = CONST/Q IF Q IS LARGE, THE STATE IS SPARSELY POPULATED IF Q IS SMALL, IT IS DENSELY POPULATED ANOTHER WAY OF LOOKING AT Q IS THAT IT IS AN INDICATOR OF THE NUMBER OF QUANTUM STATES AVAILABLE TO A SYSTEM FOR MANY AVAILABLE STATES (LARGE Q) A GIVEN STATE IS LESS DENSELY POPULATED THAN IT WOULD BE IF ONLY A FEW STATES WERE AVAILABLE (SMALL Q) THE OCCUPATION NUMBER OF A QUANTUM STATE RELATIVE TO THE TOTAL NUMBER OF PARTICLES NI/N IS, STRICTLY SPEAKING, A PROBABILITY; HOWEVER, GIVEN THE IMMENSE NUMBER OF PARTICLES IN A MOLE OF GAS, WE MAY TREAT IT AS A CERTAINTY THE SUMMATION OF ALL POSSIBLE FRACTIONS NI/N MUST BE 1: I NI N = I NI N = N N = THE PROBABILITY DISTRIBUTION OF MOLECULAR VELOCITIES IN 3-SPACE IS A COLLECTION OF RANDOMLY ORIENTED VECTORS AWAY FROM AN ORIGIN OF THE ZERO (LEAST SQUARES) ESTIMATE PLOTTED IN VELOCITY-SPACE, THE PROBABILITY DENSITY APPROACHES ZERO NEAR THE ORIGIN THIS COMES ABOUT WHEN WE USE SPHERICAL POLAR COORDINATES TO EXPRESS OUR PROBABILITY DENSITIES BECAUSE, NEAR THE ORIGIN, THE VOLUME OF THE SPHERICAL SPACE BECOMES VERY SMALL AS THE RADIAL DISTANCE IS INCREASED, THE SPACE BECOMES LARGER BUT THE PROBABILITY PROBLEMS AND EXERCISES WITH UNITS OF BAR DM3 K?1 MOL?1 IF THE VOLUME IS EXPRESSED IN M3, THEN R IS IN BAR M3 K?1 MOL?1 = 100(0 0831) = 8 310 J K?1 MOL?1 REMEMBER THAT THE UNIT J K?1 IS FOR A MOLAR GAS CONSTANT NOTICE THAT THE NUMERATOR IS AN ENERGY THE TABULATED VALUE IS 8 3144725 J K?1 MOL?1 (CR