1384 IndexTurbulent flow, 444, 444f
Turning point, harmonic oscillator and, 627
Turnover rate, with enzyme catalysis, 578
Two-body problem, 1272–1274
Two-component ideal solutions, 243–244
Two-dimensional graph, 6
Two-phase simple system, 200–202, 200f
U.SeeInternal energy
Ultraviolet catastrophe, 642
Ultraviolet radiation
electronic energy level transitions and, 949
photoelectron spectroscopy, 991–993,
992f
Uncertainty broadening, 716
Uncertainty principle of Heisenburg.See
Heisenburg uncertainty principle
Uncertainty product, 712
for different energy states, 713
of free particle, 714–715
for particle in one-dimensional box,
711–712, 712t
Uncertainty relation, 715–716
Undetermined multipliers, 1052
Ungerade, eigenfunction, 832
Uniform harmonic motion, of harmonic
oscillator, 626, 626f
Unimolecular process
gases, 524, 527
Lindemann mechanism for, 545–547
liquids, 532
Unit cells, of crystal lattice, 1154
Unitary matrix, 1251
United atom, 826
Universe
entropy of, 151
heat death of, 121
Unperturbed energy eigenvalue, 799
Unperturbed wave function, 799
Unsöld’s theorem, 817
Upper consolute point, in phase diagram,
283–284
Upper critical solution point.SeeUpper
consolute point
Upper triangular matrix, 1251
Urease, 575–576V.SeeVoltage
Valence shell, chemical properties and, 816
Valence shell electron pair repulsion (VSEPR),
877–880
Valence-bond method, 823, 866
for heteronuclear diatomic molecules,
863–865, 866
for homonuclear diatomic molecules,
849–850, 866
for hydrogen molecule, 840–842, 840t
resonance and, 885–886
Valence-bond wave function, for benzene,
885–886
Valence-bonds, 881–884
Valence-state ionization potential (VSIP), in
extended Hückel method, 905
Van der Waals equation of state, 21, 30f, 31–34
Van der Waals gas, molar heat capacity of, 70
van’t Hoff equation, 297, 336
van’t Hoff, Jacobus Henricus, 297
Vapor phase, 200, 216, 216f
Vapor pressure, 206, 206f
electrolyte solutes, activity and activity
coefficient determination, 371
of hydrochloric acid, 328–329
Laplace equation and, 228
lowering, 296
partial
of benzene and toluene, 240, 240f
of hydrochloric acid, 329–330, 330f
Raoult’s law, 238
total pressure and, 212–213
Trouton’s rule and, 212
Vaporization, 27–28, 27f, 28.See alsoPhase
transitions
entropy changes of, 145–146
latent heat of, 54
Variable-change identity, 1086
Variables, types of, 5, 13–14
Variance, of standard deviation, 705
Variation energy, 790
Variation function.SeeVariation trial function
Variation method, 790–791
extended Hückel method, 905
for ground state energies, 789
helium application of, 791–793
perturbation methodvs., 802
shielding, 793–795
Variation theorem, 790
extended, 803
Variation trial function, 790–791
for helium, 793
Vector addition, 1241, 1242f
of angular momentum, 775, 776f
Vector derivatives, 1243–1244Vector product, 1243
Vectors, 40, 40f, 1241–1244, 1242f
Velocity
in classical mechanics, 621
of flexible string, 633–634
of point-mass particles, 387
probability distribution of, 394–404
derivation of, 395–397, 395f
identification of parameter b, 398–404,
399f, 404f
Velocity space, 391, 391f
spherical polar coordinates in, 405–406,
406f
spherical shell in, 407, 407f
Velocity vector, 391, 391f
Vibration(s)
of diatomic molecules, 922–929, 922f, 926f
population of, 944–945
normal modes of, 937–939, 939f
of polyatomic molecules, 937–939, 937f,
939f, 945–947
Vibrational energy
of dilute gas, statistical thermodynamics,
1091–1092
potential, 926, 926f
Vibrational partition functions, calculation of
diatomic gases, 1069–1070
polyatomic gases, 1074–1075
Vibrational spectra, of polyatomic molecules,
976–977, 977f
Vibrational states, four harmonic oscillators,
1041, 1042t
Vibrational transitions
rotation transitions and, 965–966, 966f
spectroscopy of, 949, 997
Raman, 987–991, 989–990f
Vibrational wave function, 924
Vibration-rotation spectra, of diatomic
molecules, 964–968, 966–967f
Virial coefficients, 22, 1148–1149
Virial equation of state, 22, 1148
Virial theorem, of mechanics, 751
Viscosity.See alsoViscosity coefficient
of hard-sphere gas, 465–466
of liquids, 1192
temperature dependence of, 470–472
types of, 1198–1199
Viscosity coefficient
hard-sphere gas, 465
in Newton’s law of viscous flow, 452
for non-Newtonian fluids, 453