Intermolecular Interactions and Potentials
Origin of intermolecular force • Is it Gravitational? No , is of the order of 10 -52 J and so negligible • Strong force? No , significant in the range of 10 -4 nm, but molecular dimensions are typically 5 � 10 -1 nm • Weak force? No , although electromagnetic in origin but exits only between nuclear particles • Electromagnetic? Yes , since they are charged particles, attractive at long range and repulsive at short range
Types of intermolecular interaction • Electrostatic • Induction • Dispersion
In terms of the coordinates r, θ of the point P and the axial separations of the charge from O this may be written as � � � � 1 Q Q � � 1 2 Φ = + � � � � 2 2 1 / 2 2 2 1 / 2 πε 4 0 + + θ + − θ ( r z 2 z r cos ) ( r z 2 z r cos ) 1 1 2 2 P r 1 r r 2 � z z 2 -z 1 Q 1 Q 2 O expanded in powers of z 1 /r and z 2 /r { } 2 2 2 + − θ − θ − 1 Q Q (Q z Q z ) cos (Q z Q z )(3 cos 1) ... 1 2 2 2 1 1 1 1 2 2 Φ = + + + 3 2 πε 4 r 2 r o r
previous eq. may be written as { } 2 µ θ θ θ − 1 Q cos (3 cos 1) Φ = + + + ... 2 3 πε 4 r 2 r r 0 Q = Q 1 + Q 2 , total charge of the molecule, zeroth moment of the charge distribution � = Q 2 Z 2 -Q 1 Z 1 , is the dipole moment of the charge distribution � = Q 1 Z 2 1 +Q 2 Z 2 2 , is the quadrupole moment of the charge distribution the above eq. is valid only for r >> z 1 , z 2 . Since we are interested in the long- range interactions this approximation is valid.
Interaction of two linear charge distribution Now we consider the energy of interaction of two charge distribution - Z’ 1 - Z 1 Z’ 2 Z 2 (a) P O Q’ 1 Q 1 Q’ 2 Q 2 Z’ 2 - Z 1 -Z’ 1 Z 2 P O (b) Q’ 2 Q’ 1 Q 1 Q 2 Q’ 2 - Z 1 Z 2 P (c) O Q 1 Q 2 Q’ 1 Q 2 � 1 � 2 (d) � Q 1
For the configuration (a) the electrostatic energy of the two distribution is U a el (r) = Q’ 2 Ф(Q’ 2 ) + Q’ 1 Ф(Q’ 1 ) Ф(Q’ 2 ) is the electrostatic potential at Q’ 2 due to the first (unprimed) molecule, Ф(Q’ 1 ) is electrostatic potential at Q’ 1 arising from the same source Since these electrostatic potentials can be expressed in terms of z’ 2 and z’ 1 { } 1 ' ( µ ' − µ ' 2 µµ ' ( Θ + Θ ' ' (3 Θ ' µ − 3 µ ' Θ ) 6 ΘΘ ' QQ Q Q Q Q → ( ) A a U ( ) r = + − + + + + ... el 2 3 3 4 5 4 πε r r r r r r 0 various term in the electronic energy is the dipole-dipole interaction which varies as r -3 and is given by Q’ = Q’ 1 + Q’ 2 , total charge of the second molecule � ’ = Q’ 2 z’ 2 -Q’ 1 z’ 1 , is the dipole moment � ’ = Q 2 ’z 2 ’ 2 +Q 1 ’z 1 ’ 2 , is the quadrupole moment
various term in the electronic energy is the dipole-dipole interaction which varies as r -3 and is given by ' − µµ 2 a = U ( ) r ' µµ 3 πε 4 r 0 Thus in the configuration the dipole-dipole contribution to the interaction energy is negative and there is an attractive force between the neutral molecules Similar development can be carried out for configurations (b) and (c) that shown earlier for the dipole-dipole interaction of two neutral molecules and can obtain ' + µµ 2 b = U ( ) r ' 3 µµ πε 4 r 0 Which corresponds to repulsive force and U c μμ’ (r) =0. i.e) dipole-dipole interaction energy is zero
These illustrations show that the dipole-dipole interaction energy is proportional to r -3 for a fixed configuration of the molecules, and that it is strongly dependent on orientation varying from attractive to repulsive as one molecule is rotated An analysis of the interaction of two linear charge distributions in the general configuration in fig (d) wherein Ф denotes rotation of the second dipole about the line joining them leads to the dipole-dipole interaction energy ' − µµ θ θ φ = ζ θ θ φ U ( , r , , ) ( , , ) ' 1 2 1 2 3 µµ πε 4 r 0 ζ θ θ φ = θ θ − θ θ φ ( , , ) (2 cos cos sin sin cos ) Where 1 2 1 2 1 2 describes the dependence of the energy on orientation
If it is gas phase the two molecules are free to rotate it is often the average of the dipole-dipole energy over all possible orientations <U el > μμ‘ is required. The probability of observing a configuration of energy U is proportional to the Boltzmann factor exp(-U/KT) 2 '2 µ µ 2 = − + ... U ' el µµ 6 2 πε 3 (4 ) r kT 0 � The leading term of the orientation-averaged dipole-dipole contribution to the electrostatic energy of interaction to two neutral molecules is therefore attractive and inversely proportional to the sixth power of their separation � The temperature dependence of <U el > μμ‘ is a result solely of the orientational averaging with the Boltzmann weighting
Eq. A shows the variation of energy as r -4 for dipole-quadrupole interactions r -5 for quadrupole-quadrupole interactions for fixed orientation but for rotational case the corresponding interaction energy is given by 1 { } 2 ' 2 ' 2 2 Θ = − µ Θ + µ Θ + U ... Dipole-quadrupole el µ 8 2 πε r kT (4 ) 0 2 '2 Θ Θ 14 = − + U ... Quadrupole-Quadrupole ' el ΘΘ 10 2 πε 5 r kT (4 ) 0 Above expressions show that these two contributions to the orientation-averaged electrostatic interaction energy are also attractive
Induction energy Any molecule is placed in a uniform, static electric field E there is a polarization of its charge distribution ++ Electric field E - - Process of induction ++ - - - - ++ Induced dipole moment in the molecule is given by μ ind =α E (a) α is static polarizability of the molecule Energy of a neutral dipolar molecule in an electric field E is π = − � µ ⋅ (b) U dE Using eq (a) in (b) 0 Since μ and E are parallel 1 π 2 = − � α = − α EdE E U ind (c) 2 0
Interaction of dipolar molecule with non-polar molecule � Neutral molecule possessing permanent dipole moment can induce a dipole moment in a second molecule which is nearby whether the second molecule is itself polar or not Non-polar molecule P Interaction of a dipolar molecule with r � non-polar molecule O � µ θ cos 1 Φ = � 2 The electrostatic potential due to the dipole at P is r πε 4 0 1 1 � � ( ) ( ) 2 2 2 ∂ φ ∂ φ µ 1 2 [ ] 2 2 � � = = + = θ + θ E | E | 4 cos sin � � 3 ∂ ∂ θ r r 4 πε r 0
� Magnitude of the dipole induced in the molecule at P is thus α'E and the energy is -(1/2) α'E so that the interaction energy of induction is ( ) 2 2 θ + µ 3 cos 1 1 ' U = − α (i) ind 6 2 2 (4 πε ) r 0 which is attractive for all configurations and inversely proportional to the sixth power of the intermolecular separation for a fixed orientation � If the above potential energy is averaged over all possible orientations, giving each orientation the Boltzmann weight exp(-U/KT), the average induction energy between a dipolar molecule and a non-polar molecule is, at sufficiently high temperatures 2 ' − µ α = U ind 2 6 πε (4 ) r 0 � This term is not temperature dependent unlike the corresponding term for the electrostatic energy
Interaction of two polar molecule • If interaction is between two polar molecules each molecule induces a dipole moment in the other so that there are two contributions to the total interaction energy of the eq. (i). • The orientationally-averaged induction energy for two dipolar molecules is { } 1 2 2 ' ' = − µ α + µ α + U ... ' ind µµ 2 6 πε (4 ) r • For two identical polar molecules this reduces to 0 2 − αµ 2 . = + U .. ' ind µµ 2 6 πε (4 ) r 0 • The general treatment of induction forces is very much more complicated than that given here. This because for many molecules the polarizability is not isotropic but is tensorial character . In addition higher-order multipoles in polar molecules can induce higher-order multipoles in other molecules and there can be other contributions to the induced dipole moment.
Dispersion energy • So far the we analysed contributions to the long range energy by means of classical electrostatics and classical mechanics • For the interaction of two molecules possessing no permanent electric dipole or higher-order moments, electrostatic and induction contributions are absent and the interaction arises solely from the dispersion energy • The dispersion energy cannot be analysed by classical mechanics since as we shall see its origins are purely quantum mechanical • Consequently dispersion energy is well explained by the Drude Model
Drude model � Considering each molecule is composed of two charges +Q and –Q. � We imagine charge +Q to be stationary and that the negative charge oscillates about the positive charge with an angular frequency ω 0 in the Z-direction which is along the line joining the positive charges of the two molecules as show in fig. r 1-d Drude model of the dispersion interaction z a z b +Q -Q -Q +Q Animation � If we denote the displacement of the negative charge of molecule a from its positive charge by z a , we note that at any time t the moelcules possess instantaneous dipole moments μ a =Qz a (t) and μ b =Qz a (t)
Recommend
More recommend
