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Atkins & de Paula: Atkins’ Physical Chemistry 9e Chapter 17: Molecular Interactions Chapter 17: Molecular Interactions ELECTRIC PROPERTIES OF MOLECULES 17.1 Electric dipole moments electric dipole, two electric charges +q and –q separated by a distance R. electric dipole moment, μ, the vector that points from –q to + q with magnitude μ = qR. 1 D = 3.33564 × 10-30 Cm Chapter 17: Molecular Interactions polar molecule, a molecule with a permanent electric dipole moment. nonpolar molecule, a molecule without a permanent electric dipole moment. Chapter 17: Molecular Interactions Resultant electric dipole moments, μres2 = μ12 + μ22 + 2μ1μ2 cos θ. Chapter 17: Molecular Interactions Calculation of electric dipole moments, μ2 = μx2 + μy2 + μz2 (μi =Σqjij) μ = 2.7 D Chapter 17: Molecular Interactions Self-test; calculation of the μ of formaldehyde Chapter 17: Molecular Interactions 17.2 Polarizabilities induced dipole moment, μ*, the dipole moment induced by an applied electric field. polarizability, the constant of proportionality α in μ* = αE. (unit = C2 m2 J-1) polarizability volume, α = α/4πε0. (unit of ε0: C2 m-1 J-1) polarizability, perturbation expression. 2 z ,0 n 2 n 0 En E0 , where z ,0 n : transition in thez - direction En E0 mean valueE (HOMO- LUMOgap) μz,0 n eR α increases as the molecular size increases α increases as the HOMO-LUMO gap decreases e2 R 2 2 E E α ≈ R3 e2 4 0 R Chapter 17: Molecular Interactions 17.3 Polarization polarization, P, the electric dipole moment density, P = μN. dielectric, a polarizable, nonconducting medium. μ = 0, where no electric field is applied μz = μ, where a strong electric field is applied μz = μ2E/3kT, where a weak electric field is applied dp e E ( ) / kT sin d e E ( ) / kT x E / kT , E ( ) E cos ; energyof dipole ; probability of dipole orientat ion sin d 0 z cosdp cosdp e 0 x cos e cos sin d y cos , dy sin d z x cos sin d x e 1 xy dy e e x x z L( x) x 1 ye 1 xy dy e e x e x ex 1 L( x) x x e e x e x 1 x 1 x 2 1 x 3 x e e x2 ye xy dy 1 1 e 1 0 1 1 x x ; Langevinfunct ion 1 2 6 x L( x) 13 x z 2E 3kT xy dy Chapter 17: Molecular Interactions orientation polarization, the polarization arising from the permanent dipole moments; is lost at microwave frequency distortion polarization, the polarization arising from the distortion of the positions of the nuclei by the applied field; is lost at IR frequency electronic polarizability, the polarizability due to the distortion of the electron 2 distribution; is still alive at Vis frequency z ,0 n 2 frequency dependence of polarizabilities n 0 En E0 2 ħωn0 = En – E0 2 n 0 z , 0 n ( ) 2 n n 0 2 n0 ( ) ( ) 2 2 z ,0 n 2 n0 2 n z ,0 n 0 as 2 n0 2 n Chapter 17: Molecular Interactions 17.4 Relative permittivities permittivity, the quantity ε in the Coulomb potential energy, V = q1q2/4πεr. relative permittivity (dielectric constant), εr = ε/ε0. (ε0 = vacuum permittivity) Debye equation, (εr – 1)/(εr + 2) = ρPm/M. molar polarization, Pm = (NA/3ε0)(α + μ2/3kT). Clausius–Mossotti equation, (εr – 1)/(εr + 2) = ρNAα/3Mε0.; no contribution from permanent dipole, μ Nonpolar molecules or high frequency of applied field N A 2 slope 9 0 k Example 17.2 εr = C/C0 Pm Debye eqn. refractive index and relative permittivity, nr = εr1/2. refractive index, nr = c/c. c: speed of light in vacuum, c: speed of light in medium intercept μ N A 3 0 α Chapter 17: Molecular Interactions INTERACTIONS BETWEEN MOLECULES van der Waals interaction, an interaction between closed-shell molecules that varies with separation as 1/r6. 17.5 Interactions between dipoles multipole, an array of point charges. n-pole, an array of point charges with an n-pole moment but no lower moment. monopole, a point charge. quadrupole, an array of point charges that has neither net charge nor dipole moment. octupole, an array of point charges that sum to zero and which has neither a dipole moment nor a quadrupole moment. Chapter 17: Molecular Interactions multipole–multipole potential energy, V 1/rn+m-1. Chapter 17: Molecular Interactions , 17.5 (a) The potential energy of interaction point dipole, a dipole in which the separation between the charges is much smaller than the distance at which the dipole is being observed; l << r point dipole-point charge interaction V 1q2 4 0 r 2 1 q1q2 q1q2 x l / 2 r qq 1 1 1 2 4 0 r 12 l r 12 l 4 0 r 1 x 1 x 1 1 l r x 1 1 x x2 1 x x2 1 x 1 x qq 2 xq1q2 qq l q 1 q1l V 1 2 (1 x ) (1 x ) 1 2 2 1 2 2 4 0 r 4 0 r 4 0 r 4 0 r V Chapter 17: Molecular Interactions , Calculating the interaction energy of two dipoles 1 2 V 2 0 r 3 1 q1q2 q1q2 q1q2 q1q2 x l / r q1q2 1 1 2 4 0 r l r r r l 4 0 r 1 x 1 x 1 1 l r x 1 1 x x2 1 x x2 1 x 1 x 2 x 2 q1q2 V 1 23 4 0 r 2 0 r V Self-test 17.4 Chapter 17: Molecular Interactions , 17.5 (b) Dipole-dipole interactions electric field of point charge, E = q/4πε0r2. electric field of point dipole, E = μ/2πε0r3. potential energy of two parallel point dipoles 1 2 f ( ) V 3 4 0 r [ f(θ) = 1 – 3 cos2 θ ] See Further information 17.1 Chapter 17: Molecular Interactions , Keesom interaction, the interaction of two freely rotating point dipoles: first contribution to the vdW interaction C 212 22 V 6 C 3 r 3(4 0 ) 2 kT <V> = μ1μ2<f>/ 4πε0r <f>; weighting factor in the averaging; probability that a particular orientation will be adopted by a dipole p e-V/kT, V=μ1μ2 f/4πε0r3 p 1- V/kT + ∙∙∙, when V ‹‹ kT f fpd 0 d 1 fe 0 V / kT d 1 f (1 V / kT )d 0 0 f 1 0 1 fd 0 1 f (V / kT )d 0 1 fd 0 1 2 1 2 fd f d f 4 0 kTr 3 0 0 1 2 f 2 d 3 0 4 0 kTr 1 1 2 f2 3 4 0 kTr 0 , where 0 denotesan unweightedsphericalaverage f 0 12 22 f 2 212 22 (1 3 cos ) sin d 0 V V 0 (4 0 ) 2 kTr 3(4 0 ) 2 kTr 6 1 2 0 6 f2 0 23 Chapter 17: Molecular Interactions , C 212 22 Keesom interaction, the interaction of two rotating point dipoles: V 6 C 2 r 3 ( 4 ) kT 0 first contribution to the vdW interaction Negative sign: the average interaction is attractive. V 1/r6 : a van der Waals interaction. V 1/T : the greater thermal motion overcomes the dipole interactions at higher temperatures. V 1/r6 : arises from V 1/r3 weighted by the energy in the Boltzmann term ( 1/r3) Chapter 17: Molecular Interactions , 17.5 (c) Dipole-induced-dipole interactions C V 6 r 12 2 C 4 0 Independent on the temperature; thermal motion has no effect on the averaging process Chapter 17: Molecular Interactions , 17.5 (d) Induced-dipole-induced-dipole interactions dispersion interaction (London interaction) London formula C V 6 r I1 I 2 C 1 2 I1 I 2 3 2 Chapter 17: Molecular Interactions , 17.5 (e) Hydrogen bonding hydrogen bond, an attractive interaction between two species that arises from a link of the form A–H∙∙∙B, where A and B are highly electronegative elements (N, O, or F) and B possesses a lone pair of electrons. = c 1 A + c2 H + c3 B anti-bonding Net effect: lowering of energy nonbonding bonding Chapter 17: Molecular Interactions , 17.5 (f) The hydrophobic interaction hydrophobic, water-repelling; possessing a positive Gibbs energy of transfer from a nonpolar to a polar solvent. ΔtransferG > 0, ΔtransferH < 0, ΔtransferS < 0 hydrophobicity constant, π = log(S/S0) S: ratio of the molar solubility of R-A in octanol to that in water S0: ratio of the molar solubility of H-A in octanol to that in water hydrophobic interaction, an effective interaction that is due to the increase in entropy of the surrounding solvent. A hydrocarbon molecule in a water cage Chapter 17: Molecular Interactions , 17.5 (g) The total attractive interaction total attractive interaction between rotating molecules; dipole-dipole, dipole-induceddipole, and dispersion interactions. V = –C6/r6 limitation of V = –C6/r6; consider only dipolar interactions, assume freely rotating molecules, and consider only the interactions of pairs of molecules Axilrod–Teller formula, total dispersion energy of three closed-shell molecules C6 C6 C6 C V 6 6 6 rAB rBC rCA rAB rBC rCA 3 C = a(3 cos θA cos θB cos θC + 1), where a ≈ 3/4αC6 Chapter 17: Molecular Interactions Interactions between dipoles; impact on medicine (molecular recognition & drug design). See I17.1 Chapter 17: Molecular Interactions , 17.6 Repulsive and total interactions hard-sphere potential, V = for r d; V = 0 for r > d. Mie potential, V = Cn/rn – Cm/rm. Lennard-Jones potential, V = 4ε{(r0/r)12 – (r0/r)6}. ε:depth of the well, r0:seperation exp–6 potential, V = 4ε{e–r/r0 – (r0/r)6}.; better than L-J(12,6) potential where V=0 Chapter 17: Molecular Interactions GASES AND LIQUIDS 17.7 Molecular interactions in gases molecular beam, a collimated, narrow stream of molecules travelling though an evacuated vessel. hydrodynamic flow, net flow arising from intermolecular collisions. molecular flow, collision-free flow. Chapter 17: Molecular Interactions Chapter 17: Molecular Interactions supersonic, a stream of molecules in which the average speed of the molecules is much greater than the speed of sound for the molecules that are not part of the stream. supersonic beam, a beam obtained when the region of hydrodynamic flow is skimmed from a supersonic jet and the excess gas pumped away. crossed beam technique, a technique in which two molecular beams are incident at right angles. Low translational T Chapter 17: Molecular Interactions differential scattering cross-section, σ, the constant of proportionality between the change in intensity (dI) and the intensity of the incident beam (I), the number density of target molecules (N), and the infinitesimal path length dx through the sample: dI = σIN dx. impact parameter, b, the initial perpendicular separation of the paths of the colliding molecules. Chapter 17: Molecular Interactions Scattering patterns depend on the impact parameter (b) for the impact of two hard spheres Chapter 17: Molecular Interactions for real molecules; scattering patterns depend on the intermolecular potential, molecular shape, and relative speed of approach as well as the impact parameter. repulsive core long range attractive potential Chapter 17: Molecular Interactions quantum oscillation, the modification of the scattering in the forward direction by interference between the wavefunctions of a particle along two different paths. rainbow scattering, strongly enhanced scattering in a nonforward direction. rainbow angle, θr, the angle for which dθ/db = 0 and the scattering is strong. van der Waals molecules, complexes of the form AB in which A and B are held together by van der Waals forces or hydrogen bonds. Chapter 17: Molecular Interactions 17.8 The liquid–vapour interface 17.8 (a) surface tension surface tension, γ, the constant of proportionality between the increase in surface area of a liquid and the work needed to create the increase: dw = γdσ (=dA, where constant T). dA < 0 (dσ < 0); spontaneous process: surfaces have a natural tendency to contract. Example17.4 dw =2γlh Chapter 17: Molecular Interactions 17.8 (b) curved surfaces bubble, a region in which a vapour is trapped by a thin film. cavity, a vapour-filled hole in a liquid. droplet, a small volume of liquid Laplace equation, pin = pout + 2γ/r. outward force; pressure × area = 4πr2pin inward force; force from pout & surface tension force from pout ; pressure × area = 4πr2pout force from surface tension; 8πγr dσ = 4π(r+dr)2 - 4πr2 = 8πrdr dw = 8πγrdr 4πr2pin = 4πr2pout+ 8πγr pin = pout + 2γ/r Chapter 17: Molecular Interactions 17.8 (c) capillary action capillary action, the tendency of liquids to rise up capillary tubes. capillary rise and surface tension, γ = (ρβ –ρα)ghr/2 Same pressure at same height in a same phase P1=P6, P2=P5, P2=P3 P5=P3 Curved surface: P4 < P5=P3 P4 < P3 ;capillary rise At equilibrium P1=P6, P2=P3 P8=P5, P3=P4 P8-P3= P5-P4 P8-P2= P5-P4 P8-P2= (P5-P7)+(P7-P4) P8-P2=-ραgh, P7-P4 =-ρβgh, P5-P7=2γ/r (θc=0) -ραgh = 2γ/r - ρβgh γ = (ρβ –ρα)ghr/2 Chapter 17: Molecular Interactions nonzero angle between the edge of meniscus and the wall; γsg = γsl+ γlg cos θc contact angle and interfacial tension, cos θc = (γsg – γsl)/γlg. superficial work of adhesion, wad = γsg + γlg – γsl (work of adhesion/area of contact) cos θc = wad/γlg – 1 criterion for surface wetting, 1< wad /γlg < 2. criterion for non-surface wetting, 0< wad /γlg < 1. Chapter 17: Molecular Interactions 17.9 Surface films; will be covered in Chap. 18 17.10 Condensation Kelvin equation for the vapour pressure of droplets, p = p* e2γVm/rRT supersaturated phase, a phase that is thermodynamically unstable with respect to the liquid. spontaneous nucleation centre, a location at which a sufficiently large number of molecules congregate into a droplet. nucleate, provide surfaces to which molecules can attach and thereby induce condensation. superheated, a liquid that has not boiled but is above its boiling temperature. supercooled, a liquid that has not frozen but is below its freezing temperature. Chapter 17: Molecular Interactions Impact on nanotechnology Spontaneous Assembly of a Monolayer of Charged Gold Nanocrystals at the Water/Oil Interface Angew. Chem. Int. Ed. 2004, 43, 458. Directing Self-Assembly of Nanoparticles at Water/Oil Interfaces Angew. Chem. Int. Ed. 2004, 43, 5639.