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Dipole Moment – 2017 Physikalisch-chemisches Praktikum I Dipole Moment { 2017 Dipole Moment Summary In this experiment you will determine the permanent dipole moments of some po- lar molecules in a non-polar solvent based on Debye's theory and the Guggenheim approximation. You will combine concentration-dependent measurements of refrac- tive index for visible light and of relative permittivities at radio frequencies and develop an understanding of the connection between these macroscopic quantities and molecular properties. Contents 1 Introduction1 1.1 Definition of the dipole moment........................1 1.2 Relative Permittivity, Polarization, and Polarizability............3 1.2.1 Different Contributions to the Polarization..............4 1.3 The Debye equation..............................5 1.4 Dispersion....................................6 1.5 Relative Permitivity and Refractive Index..................6 1.6 The Guggenheim Method...........................8 2 Experiment8 2.1 The Dipole Meter................................8 2.2 The Refractometer...............................9 2.3 Measurements.................................. 10 2.4 Practical Advice................................. 11 3 Data Analysis 12 1 Introduction 1.1 Definition of the dipole moment Two separated charges of opposite sign, q1 = −q and q2 = +q form an electric dipole. The dipole moment is defined by: ~µ = q(~r2 − ~r1) = q~r (1) where ~r1 and ~r2 are the vectors that define the position of the two charges in space. The dipole moment is thus a vector quantity. For a distribution of negative and positive point charges qi the dipole moment is: X ~µ = qi~ri (2) i Page 1 of 13 Physikalisch-chemisches Praktikum I Dipole Moment { 2017 − 1 + () 2 Figure 1: Definition of the dipole moment for two point charges and a continuous charge distri- bution where ~ri are the positions of the charges qi. For continuous charge distributions ~µ = e R ~rρ(~r)dV , where ρ(~r)dV is the probability of finding charge at position~r in a volume element dV and e is the elementary charge. The electric dipole moment of a molecule is the sum of the contributions of the positively charged nuclei and the negatively charged electron distribution (~µ = ~µ+ + ~µ−). The nuclei can in good approximation be treated as PN ~ point charges: ~µ+ is thus given by ~µ+ = i=1 ZieRi, where Zi is the nuclear charge of ~ nucleus i at position Ri. The electronic part ~µ− is determined by the electron distribu- tion. It may be obtained from quantum chemical calculations, which yield the electronic 2 wavefunction (r) and thus ρe(~r) = j (~r)j . In the SI system, the unit of the electric dipole moment is Coulomb·meter. Since these units result in very small numbers, however, the unit Debye (1D = 3:33564 · 10−30 Cm) is often used (in honor of Peter Debye, who was, from 1911-1912, professor of theoretical physics at the University of Zurich). According to general convention, the dipole moment points from the center of the negative charge distribution to the center of the positive one. If the two centers do not coincide the molecule has a permanent dipole moment. Its existence is strongly related to the symmetry properties of a molecule. Molecules with inversion symmetry like benzene, acetylene, or nitrogen, for instance, have no permanent dipole moment. In the case of HCl, however, the centers of the two charge distributions do not coincide. If we place two elementary charges (e = 1:602 · 10−19 C) of opposite sign at a distance of 1:28 · 10−10 m, i.e. the bond length of the HCl molecule, we obtain a dipole moment of 6.14 D. This represents a purely electrostatic model for the ionic HCl structure. In practice, however, only a dipole moment of 1.08 D is found. The molecule is thus only partially ionic. The ionic character X describes partially ionic chemical bonds: µ X = exp · 100%; (3) µcalc For example X of HCl is 17.6%. Experimental dipole moments provide information about the electron distribution in a molecule. In this experiment dipole moments of some polar molecules in non-polar solvents are determined. This is done by measuring relative permittivity and refractive indices of solutions and pure solvents. The connection between these two macroscopic properties and the molecular dipole moment is explained in the following sections. Page 2 of 13 Physikalisch-chemisches Praktikum I Dipole Moment { 2017 1.2 Relative Permittivity, Polarization, and Polarizability Consider the electric field between two charged plates of a capacitor (in vacuum). If the distance between the two plates is much smaller than the surface of the plates, the field is approximately homogeneous except in the border regions (Figure2). The electric field strength E is given by 0 q E0 = = σ0/0 (4) S0 where σ0 = q=S denotes the surface charge density of one capacitor plate (S = surface area, −12 −1 2 −1 q = charge on one plate) and 0 is the vacuum permittivity (0 = 8:85419·10 J C m ). The voltage U0 between the two plates is proportional to the charge q: + − + − + 0 - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - U U 0 Figure 2: Electric field inside a capacitor with charge q on the capacitor plates and plate spacing d. Left: in vacuum. Right: filled with a dielectric. Polarized and oriented molecules are shown schematically. q = C0 · U0 (5) The proportionality factor C0 is called capacitance. Capacitance and field field strength E0 inside the capacitor are related via: C0 = q=U0 = q=(E0 · d): (6) When the capacitor is filled with (non-conducting) matter and charge q and plate sep- aration d are kept constant, the capacitance increases and the voltage is reduced (right hand side of Fig.2). As a result, the field between the capacitor plates is smaller than in vacuum: C = C0;U = U0/ ) E = E0/ (7) The constant > 1 is called relative permittivity of the material (also called dielectric). ~ ~ It is useful to introduce a quantity called polarization P of a dielectric in a field E0, defined by: ~ ~ ~ ~ P = 0(E0 − E) = 0( − 1)E (8) Page 3 of 13 Physikalisch-chemisches Praktikum I Dipole Moment { 2017 ~ ~ ~ Assuming that the vectors P , E and E0 are parallel, we can insert4 into8 to obtain the scalar relation σ − P σ E = 0 def= (9) 0 0 The electric field E in a dielectric can thus be treated the same way as a field in vacuum. However, its strength no longer corresponds to the surface charge density σ0 as in equa- tion4, but to a reduced density σ = (σ0 − P ). Polarization can thus be regarded as a ~ surface charge density that is induced by the field E0 on the interface between dielectric and capacitor plates. This induced charge density partially compensates σ0 as it is of opposite sign. 1 If we write P explicitly as a surface charge density we obtain a more intuitive picture of polarization: q0 · d q0 · d P = q0=S = = : (10) S · d V The right hand side of this equation is an electric dipole moment q0 · d per unit volume V of the material, which is thus an alternative way defining polarization. Now consider a material which consists of individual dipoles ~µ(j), for example molecules in a liquid. The dipole moment per unit volume P should then correspond to the vector sum: N 1 X P~ = ~µ(j) (11) V j=1 where N is the number of molecular dipole moments in volume V . Equation 11 provides an interpretation of the macroscopic quantity P~ in terms of molecular properties. 1.2.1 Different Contributions to the Polarization With this microscopic picture in mind, we can distinguish three contributions to the polarization P , which arise at the molecular level from different types of response to an external electric field: P = PE + PA + Pµ (12) Orientation polarization Pµ is only observed in solution when the molecules have a permanent dipole moment ~µ0, due to a partial alignment of the molecular dipole moments in an electric field E~ 0. Without an external electric field, the dipoles' orientations would be random and there would be no net dipole moment. Pµ varies strongly with temperature because thermal motion prevents the full alignment of the molecular dipoles. Reorientation of permanent dipole moments is, however, not the only cause of po- larization in the presence of an electric field E~ 0. The field can also distort the charge distribution of a molecule, in other words, the electric field induces a molecular dipole moment ~µI . As long as E0 is not too strong, we can write I ~ 0 ~ 0 ~µ = αE = (αE + αA)E : (13) 1 There can be no real change of charge on the capacitor plates, since the material we consider is insulating. As illustrated on the right hand side of Figure2, there can, however, be a net surface charge density, when molecular dipoles re-orient or positive and negative charges inside the material are pulled apart in the external field. Page 4 of 13 Physikalisch-chemisches Praktikum I Dipole Moment { 2017 The proportionality factor α is called the polarizability of a molecule. It indicates how easily the charge distribution of a molecule can be distorted. Here we assume that the induced dipole moment ~µI is always oriented in the direction of E~ 0, i.e. that α is a scalar. If the molecules have a permanent dipole moment ~µ0, the total dipole moment in and external field is: I ~µ = ~µ0 + ~µ (14) The fact that the centers of positive charges (nuclei) and negative charges (electrons) are pulled in different directions in an electric field gives rise to the electronic polarizability αE and hence the Electronic polarization PE, which is observed for any material. An additional contribution (αA) to the polarizability can arise from changes in bond lengths and angles between charge-carrying units of polar molecules.
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