Transcript V(r)
V(r) Repulsive region r=σ rm V(rm)=-ε, Fattr=Frep V(σ)=0, Vattr=Vrep r Attractive region εm Short range interactions are dominated by interelectronic repulsion, The quantitative treatment of such interactions necessarily requires QM description (Pauli effects). These interactions substantiate the concept of sterical hindrance. V(r) r σ Vrep(r≤σ)=∞ Vrep(r>σ)=0 r r σ=2r Vatt=0 An analytical function reproducing the results of quantum calculations is: V (R ) C R n n 10,12 C = constant characteristic of the atom-atom pairwise interaction Van der Waals radius = it represents the minimum contact distance between two atoms. It can be approximately considered as the interatomic distance beyond the repulsive energy rapidly rises rVdW ≈ σ The Van der Waals radius also defines a Van der Waals area (SVdW) and volume (VVdW). They represent areas and volumes that cannot be penetrated (excluded areas and volumes). The Van der Waals volume of a molecule is approximately given by the sum of of the Van der Waals volumes of the atoms or groups of atoms forming the molecule. S VdW 2 4 VVdW 3 3 Contact interatomic distances Atom pair C--C Average distance (Å) 3.2 Minimum distance (Å) 3.0 C--O 2.8 2.7 C--N 2.9 2.8 C--H 2.4 2.2 O--O 2.8 2.7 O--N 2.7 2.6 O--H 2.4 2.2 N--N 2.7 2.6 N--H 2.4 2.2 H--H 2.0 1.9 At long range intermolecular distances we can approximately neglect quantum effects and describe nuclei and electrons as point charges following the laws of classical electrostatics (Coulomb law). The interaction energy between a charge qi and a charge distribution qj is simply given by the sum of the pair coulomb contributions: Vi = q i j qj R ij Fcoul 1 q1q2 2 40 R 1 2 1 q1q2 V 40 R 12 The energy of coulomb interaction between ions is of the order of 250 kJ·mol-1. q = electric charge = 1.60·10-19 C = 4.8·10-10 ues 1 ues = 1 electrostatic unit. It is the charge that at 1 cm of distance from an other unit charge exerts 1 dyne force (CGS units). ( H 2O ) 78 ( esano ) 2 The dipole moment is a vector measuring the distance between the center of the positive and negative charge distributions: l - + qr μ Unit positive and negative charges separated by a 1 Å distance: = q · l = 4.8·10-10·1·10-8 = 4.8·10-18 ues·cm = 4.8 D (Debye) 1 Debye = 10-18 ues·cm=3.336·10-30C·m Dipole moment of some molecular bond bond (D) bond (D) H-F 1.9 C-F 1.4 H-Cl 1.1 C-Cl 1.5 H-N 1.3 C-N 0.2 H-O 1.5 C-O 0.1 Which is the polarity of CO? δ-C—Oδ+ H2O: exp=1.85 D (OH) = 1.52D = 2 (OH) cos(52.5°) O H H Isomers of dichlorobenzene: =0 =2.25D =1.48D R12 q1 μ2 q1 2 V= 2 40 R 12 1 The order of magnitude of this interaction is 15 kJ·mol-1. The interaction can be attractive or repulsive depending on the nature of the charge q1 and the charge–dipole relative orientation. l q1 -q1 1 q1 q2 q1 q2 V= l 40 r l r 2 2 l x= 2r r q2 q q 1 1 1 2 40 r 1 x 1 x Per l 1 2r 1 1 x x 2 ...... 1 x 1 1 x x 2 ...... 1 x q1 q2 1 1 q1 q2 2 2 V= 1 x x 1 x x 40 r 1 x 1 x 40 r q1 q2 q1 q2l 1 q2 2 x = 2 40 r 40 r 40 r 2 Q = monopole. Ex. Na+ l - + μ dipole Quadrupole CO2 Octupole The interaction energy decreases with the distance as faster as the order of the multipolar interaction increases. For an n-pole interacting with an m-pole: V 1 r n m 1 The interaction energy can be attractive or repulsive depending on the relative orientation between the two molecular dipoles: + + +- -+ - For two non-rotating dipoles (fixed orientation, like in solids): 1 2 V=3 40 R 12 int 2kJ mol -1 For rotating dipoles (solution, gas) the interaction energy should be averaged among all possible orientations following the Boltzmann distribution. Because attractive energy orientations are slightly favoured with respect to orientations giving rise to repulsive interactions, dipole-dipole interactions in solution or in gas phases are attractive, depending on the temperature: V =-C 12 22 kT R 612 At 25°C for two HCl molecules (μ=1D) at 5Å: int 0.07 kJ mol -1 An apolar molecule under the effect of an external electric field could be polarized. Electric field ------ +++ -----+++ -----+++ Polarized molecule 1 2 indotto α E β : E ...... 2 polarizability hyperpolarizability Polarizability is a molecular property that increases with the number of electrons belonging to the molecule and decreases with the increase of the ionization potential : Z (atomic number) I(ionizati on potential) ' 40 Volume polarizability [m3] polarizability [C2m2J-1] Vacuum permittivity [C2m-1J-1] q=0 z p=0 Q = -7.510-40 Cm2 x CO2 xx= // p// = //E// // = 4.0510-24 cm3 = 2.0210-24 cm3 yy=zz= p= E The electric field generated by a permanent dipole moment gives rise to a dipole moment (induced dipole moment) on a nearby apolar molecule. Cl H (HCl)= 1 Debye At 3Å ε ≈ -0.8 kJ·mol-1 Benzene = 110-29 m3 V=- 12 2 4 0 R 6 int 0.8 kJ mol -1 12 As the orientation of the induced dipole depends on the orientation of the inducing dipole, the dipole-induced dipole interaction does not depend on the thermal energy (kT). A pure quantistic effect arising from the correlation between the electron motions of the interacting atoms at large distances. These interactions, named dispersion or London interactions, occur in all systems, even between apolar molecules. They are always attractive. Semiclassically can be described as the interaction between istantaneous dipoles arising from the fluctuations of electronic charge distributions. Molecular polarizability V =-C 12 22 6 R 12 C Ionization energy 1 2 R 612 I1I 2 I1 I 2 For two CH4 molecules (=2.6·10-30 m3, I7 eV) separated by 3 Å: εint -2 kJ·mol-1. Interaction Ion-ion Distance dependence R-1 Energy (kJ mol-1) 250 Ion-dipole R-2 15 Dipole-dipole R-3 R-6 2 0.3 Dipoleinduced dipole Dispersion (London) R-6 0.3 R-6 2 Type ion Fixed dipoles Rotating dipoles All the molecules In basence of ions and for rotating systems in solution, dipolar interactions are attractive nad depend on the sixth inverse power of the distance. These contributions to the potential energy can be described by an analytical finction as : V=- C R6 The electrostatic origin of H-bond interaction is emphasized by the involvement of strong electronegative atoms in competition with the same H-atom: D-(donor) H+ - - - -A-(acceptor) H-bond interaction can de described as a dipole-dipole interaction between fixed dipoles: VHB = - C 3 R D- H - A The order of magnitude of H-bond interaction is 20 kJ·mol-1 (R 2Å). Length and strenght of H-bond depend on the electronegativity of the donor-acceptor pair and on the geometry of the atomic groups: 1.03 Å N O H 1.9-2.0 Å C r0 12 r0 6 V ( r ) 4 r r 13 7 V ( r ) 24 r0 r0 F 2 r r0 r r The attractive force is maximum at : 26 r r0 1.244r0 7 6 At this distance: Tipically around 10 pN. Fmax -2.397 r0