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Physikalisch-chemisches Praktikum I – 2017

Dipole Moment

Summary

In this experiment you will determine the permanent dipole moments of some po- lar 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 at radio 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 , Polarization, and ...... 3 1.2.1 Different Contributions to the Polarization...... 4 1.3 The Debye equation...... 5 1.4 ...... 6 1.5 Relative Permitivity and ...... 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 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

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−

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 element dV and e is the . The of a is the sum of the contributions of the positively charged nuclei and the negatively charged 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 ψ(r) and thus ρe(~r) = |ψ(~r)| . In the SI system, the unit of the electric dipole moment is ·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 , 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 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 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.

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1.2 Relative Permittivity, Polarization, and Polarizability Consider the electric field between two charged plates of a (in ). 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 density of one capacitor plate (S = surface area, −12 −1 2 −1 q = charge on one plate) and 0 is the (0 = 8.85419·10 J C m ). The 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 . Polarized and oriented molecules are shown schematically.

q = C0 · U0 (5)

The proportionality factor C0 is called . 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)

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~ ~ ~ Assuming that the vectors P , E and E0 are parallel, we can insert4 into8 to obtain the 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 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 ~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.

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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 () 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 and angles between charge-carrying units of polar molecules. This is responsible for the so-called atomic Atomic polarization PA. Both PE and PA are due to the distortion of charge and are not very strongly dependent on temperature. The reason why it is not quite so straightforward to turn equation 11 into a relation between measurable quantities, such as  and the molecular polarizability α and perma- ~ 0 nent dipole moment µ0 is the fact that in dense media the local field E , which enters in equation 13 or determines the alignment of polar molecules, is not equal (not even on average) to the macroscopic field E~ of equation8.

1.3 The Debye equation

 ’

Figure 3: The Debye model: A solute molecule experiences the electric field inside a cavity (white) of a dielectric but does not interact with neighboring molecules

The theory developed in 1912 by Debye is based on a model, where the local field E~ 0 seen by the individual molecules is that inside a spherical cavity in a continuous medium of relative permittivity  and electric field E~ : 1 1  + 2 E~ 0 = ( + 2)E~ = P~ (15) 3 30  − 1 The last identity uses the definition of the polarization8. Combining this with equa- tions 11, 14, and 13 and averaging over the thermal distribution of dipole orientations in an external field, one can derive the following formula, known as the Debye equation:

 2   − 1 NA µ0 V = αE + αA + (16)  + 2 30 3kBT

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where V = M/ρ is the molar volume of the substance, NA Avogadro’s number, and µ0 the permanent dipole moment of the molecule in the cavity. The polarizability terms on the right hand side correspond to the electronic and atomic polarization and the last term to the orientation polarization. Note that the latter contribution becomes smaller with rising temperature, a consequence of thermal motion. The Debye model neglects polar interactions of dipoles with their surroundings. It is therefore valid only for polar gases at low pressure or for dilute solutions of polar molecules in non-polar solvents. In the latter case the Debye equation can be re-written:

 2   − 1 1 − 1 NA µ0 [V1 + x(V2 − V1)] − V1(1 − x) = x αE + αA + (17)  + 2 1 + 2 30 3kBT where x is the mole fraction of the solute, and V1 and V2 are the partial molar of solvent and solute. The relative permittivity  is that of solvent with solute, while the solvent alone has a relative permittivity 1. Quantities on the right hand side only refer to solute properties. With the help of this equation, we could in principle determine µ0 of the solute molecules by measuring relative permittivities at different temperatures. In this practical course, however, we will employ a different method which makes use of the fact that  is actually not a constant but is strongly dependent.

1.4 Dispersion Dispersion generally denotes the frequency dependence of a . The dis- placement of charge in a molecule due to an external field as well as the re-orientation of the permanent dipole moments require a finite amount of time. The same is true for the reverse case, when the polarizing field is switched off and the molecules return to their non-polarized disordered initial state. The relaxation time τ indicates how much time is needed until the polarization effects are reduced to 1/e after the external field has been switched off. This time is not purely a characteristic of the molecules but can strongly depend on solvent properties such as . Orientation of dipoles or the distortion of polar molecules will only contribute to the polarization of a medium if the polarizing electric field changes slowly enough so the molecules can adapt to changes of field strength or direction. This condition is fulfilled in static fields but usually also in alternating fields up to MHz frequencies, since the relaxation times τ are much smaller than 1 µs. At frequencies larger than 1010 Hz (10 GHz) re-orientation of the permanent dipoles can no longer follow the fast sign changes of the external field, and orientation polarization is no longer observed. At frequencies higher than 1014 Hz (100 THz), i.e. beyond the vibrational frequencies of molecules, atomic motion also stops to respond to the rapid field changes and only electronic polarization is possible. The strong frequency dependence of the polarization is schematically depicted in Figure4.

1.5 Relative Permitivity and Refractive Index The plane wave solution of Maxwell’s equations in a medium with relative permittivity  (and relative magnetic susceptibility µm = 1) yields a wave with phase (the

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Radio

Orientation THz IR Visible UV Atomic

Electronic

10 3 10 6 10 9 10 12 10 15 Frequency (Hz)

Figure 4: Schematic representation of the frequency dependence of polarization. product of and frequency): c v = λf = √ (18)  where c is the in vacuum. Defining the refractive index n as the ratio of phase velocity in the dielectric and the phase velocity in vacuum (n = c/v) and we thus have n2 =  (19) i.e. the relative permittivity corresponds to the square of the refractive index n. Its magnitude and frequency dependence are a measure for the electronic polarization and its relaxation time. You are probably much more familiar with the refractive index than

λ1 = 1 n1 n 2 λ = 2 2

Figure 5: Refraction of light at the interface between two media with refractive indices n2 > n1. Blue lines indicate wave maxima, which can only match at the interface if the propagation direction changes. with the permittivity. For example, a light beam is diffracted at the interface between two materials with different n according to Snell’s law (see Fig.5): sin α n = 2 (20) sin β n1

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This is a direct consequence of the requirement that the electric field must not have dis- continuities at the boundary between the two media in which the wave travels at different speeds, i.e. where the spacing between successive maxima of the wave are different. A nice demonstration of the refraction of waves at an interface can also be found in this Mathe- matica demonstration project. Here we will use this effect to determine  at high (optical) frequencies with a re- fractometer. In the region of visible light (4 − 8 · 1014 Hz) we can neglect the atomic 2 polarizability αA and the static dipole term and replace  by n in equation 17 to obtain:

2 2 n − 1 n1 − 1 NA 2 [V1 + x(V2 − V1)] − 2 V1(1 − x) = x αE (21) n + 2 n1 + 2 30

1.6 The Guggenheim Method In a paper which appeared in 1949,[1] E. A. Guggenheim combined equations 17 and 21, which essentially eliminates the electronic polarizability αE. He proposed to measure and 2 plot  and n as a function of the solute concentration, yielding the slopes b and bn2 . Under the assumption that the atomic polarizability αA of a molecule is proportional to its molar volume, he could show that

2 2 (1 + 2)(n1 + 2) NA µ0 b − bn2 = (22) 3 30 3kBT The corresponding (approximate) formula for the dipole moment of a polar molecules in non-polar solvents is thus:

2 27kBT 1 µ0 = 0 2 (b − bn2 ) (23) NA (1 + 2)(n1 + 2) You will use the Guggenheim method and analyze your experimental data with this 2 equation. The slopes b and bn2 are obtained by plotting  and n against concentration. Calculate concentrations by multiplying the weight fractions w2 = m2/(m1 + m2) with M2/ρ1, the ratio of the molar mass of the solute and the density of the pure solvent. The density (ρ1), the relative permittivity (1) and the refractive index (n1) of the pure solvent must be determined independently, or taken from literature.

2 Experiment

2.1 The Dipole Meter To determine the relative permittivity  at low frequencies, where all contributions to the polarizability are effective (see Fig.4) we use a dipole meter, which operates in the region of radio waves (≈ 106 Hz). A dipole meter consists of an electric circuit with an L and capacitor into which the solution can be filled (Figure6). It oscillates at its resonant or eigenfrequency:

1 r 1 f = (24) 2π LC

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(a) +V (b) +V B A

R C R C L L

L’ switch at A switch at B 180° Phase shift (charging) (discharging)

1 ∆ =

Figure 6: (a) When discharging through an inductance L, he voltage across a capacitor with 1 q 1 capacitance C oscillates with a characteristic eigenfrequency 2π LC . (b) Schematic represen- tation of an oscillator circuit in a dipole meter, which leads to continuous oscillations at the eigenfrequency. For more information see for example this link.

The eigenfrequency is measured with a frequency counter. Filling the capacitor with a liquid changes the capacitance C according to7:

C = C0 (25)

We can thus rewrite Equation 24 and obtain

1 r 1 r 1 rκ f = = 2 = (26) 2π LC0 4π LC0  where all constant terms have been merged into a constant factor κ. This yields the relation between the frequency readout and the relative permittivity: κ  = (27) f 2 For calibration, the proportionality constant κ can be determined using a liquid of known relative permittivity .

2.2 The Refractometer √ The refractive index n =  should be measured at frequencies, where the electronic polarization dominates and contributions from orientation or atomic polarization can be neglected. This is conveniently done in the visible region of the electro-magnetic spectrum with a refractometer as proposed by Abbe. It uses the fact that there is a maximum angle of refraction βmax under which light traveling in a medium with refractive index n1 can enter a medium with a higher refractive index n2 (Figure7a). This angle is reached when the incidence angle α approaches 90◦ (sin α = 1) and is thus given by

n1 sin βmax = (28) n2

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An Abbe refractometer (Figure7b) contains two prisms made of flint glass of high optical density (large n2), one prism on top with a rough finished side for illumination and one polished prism for detection (bottom). The sample forms a layer with a thickness of 0.1 mm between the two prisms. Light entering the refractometer through the top prism

(a) (b) 

n1 Illuminating Telescope n2   prism Sample (n1)

Amici prism Detection compensator prism (n2)

(c)

Figure 7: Abbe Refractometer. (a) Enlarged view of the interface between sample and detection prism. Light propagation in this prism is limited by a critical refraction angle βmax. (b) Complete view of the refractometer. (c) Light distribution in the focal plane of the telescope. Light entering the detection prism under the angle βmax is collected at the line separating bright and dark areas. is scattered at the rough interface such that it is traveling in arbitrary directions inside the sample. Because the refractive index of the bottom prism is higher than that of the sample, however, the angles under which light can enter the prism and propagate to the observer are limited by the critical angle βmax. When viewed through a telescope, adjusted to infinite distance, light propagating at different angles appears at different positions in the focal plane (Figure7c) . We thus observe a bright region for β ≤ βmax and a dark region for β > βmax. Usually, a reflecting mirror is rotated to place the birght-dark edge at the center of a cross wire in the telescope objective. The refractive index can then be read off on a calibrated scale. When white light is applied, the frequency-dependence of the refractive index (disper- sion) and, as a consequence, of βmax blurs the sharp separation of dark and bright regions, unless it is compensated. A typical compensator consists of a pair of double Amici prisms that can be rotated against each other. This construction constitutes a system of variable dispersion that can be adjusted to balance the dispersion of the detection prism.

2.3 Measurements • Turn on the dipole meter and the thermostat. The apparatus has to reach a stable temperature before measurements can be made. This takes about 15 min. Dipole- and refractometer have to be kept very accurately at the same constant temperature. Otherwise, the different measurements cannot be properly combined in the final analysis.

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• Prepare three solutions of the substance indicated by the assistant in heptane. It is very important that you keep to the following instructions and make proper notes in your lab journal: In a 100 mL graduated flask weigh about 100 mL of heptane. Then, using a syringe, add as much sample as you need to obtain a weight fraction of about 3%. 15 mL of this solution is put into a second flask (weighed exactly!). The remaining 85 mL are stored well-closed and can be used by other students or to repeat measurements. Di- lute the 15 mL with solvent to 50 mL and weigh again in order to exactly determine the weight fraction. It should be below 1%. Apply the same procedure to prepare a solution with a weight fraction below 0.3%. Carefully note all weights (para and netto) in your notebook!. You will calculate your exact weight fractions from the actual, measured weights.

• Use a 20 mL syringe and plastic tube to fill the dipole meter with solution or solvent. Do it carefully and make sure, that no liquid is spilled at the top of the cell. Any air captured between the capacitor plates would perturb the measurement. After filling in solvent or solution, always wait about 5-10 min until the solvent is in thermal equilibrium with the apparatus. Then read off the eigenfrequency from the frequency counter.

• As a first measurement, measure the pure solvent, i.e. fill the measuring cell with heptane. Read off the frequency at least three times and take an average. Make sure that the cell is clean and dry before any new solution is filled in so that the concentrations are not changed. Use a dryer first hot, then cold, until no solvent can be smelled. After each measurement, the cell should be rinsed with solvent and the solvent measured again to check for drifts.

• In parallel, determine vis by measuring the refractive indices of solvent and all solutions with the refractometer. Pipette drops of solvent or solution on the detec- tion prism, then close with the lighting prism. After every measurement, clean the prisms using soft tissue and acetone or ethanol. Take care not to scratch the prisms!

• When you have finished with the first substance, repeat the with the (stock-)solutions prepared by the other groups.

2.4 Practical Advice • The indicated concentrations do not have to be exactly those specified, but they have to be known exactly.

• Check carefully that pipettes, flasks, etc are clean and dry.

• Use different pipettes for different substances.

• Always close the flasks to avoid change of concentration due to evaporation.

• Keep the solutions after the measurements in case you have to repeat some of them.

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• Begin with low concentrations and move to higher ones.

• When the concentration is changed, the cell has to be rinsed twice with pure solvent and dried with the dryer until you cannot smell any solvent.

• Monitor the temperature carefully during the whole experiment. Wait approxi- mately ten minutes after every change of sample such that thermal equilibrium can be reached.

• Repeat every measurement three times and average the results.

3 Data Analysis

• Use the measurement of the pure solvent with known relative permittivity Heptane = 1.924 − 0.00140(T [◦C] − 20) to calculate the calibration constant κ in equation 27. Also determine the experimental uncertainty for κ.

• For each substance, determine the relative permitivitiy of your different solutions from the measured eigenfrequencies (mean values and uncertainties) as well as the corresponding weight fractions. Plot  against weight fraction and calculate the slope by linear regression. Specify units and experimental error for the slopes b.

• Repeat this analysis for the refractive index measurements to obtain the slopes bn2 .

• Use the Guggenheim equation 23 to calculate the static dipole moment µ0 of the solute molecule. Use error propagation to establish the experimental uncertainty. The density ρ1 of the solvent is given in Table1 and the molar weight of the dissolved substance M2 should be known. • Compare your experimental values for the dipole moments of chloro-cyclohexane, dichloromethane, trichloro-methane (chloroform) and monochloro-benzene to refer- ence values from the literature. What does the difference between chloro-cyclohexane and monochloro-benzene indicate about their charge distributions.

• Use the measured dipole moment of chloro-cyclohexane (µ1) to estimate the ionic character of the C-Cl bond: µ X = 1 (29) d · e (d = bond length C-Cl, e = elementary charge).

[1–5]

References

[1] E. A. Guggenheim, A proposed simplification in the procedure for computing electric dipole moments, Trans. Faraday Soc. 1949, 45, 714–720 [link].

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[2] G. Wedler, H.-J. Freund, Lehrbuch der Physikalischen Chemie, 6th ed., Wiley-VCH, Weinheim, 2012 [link].

[3] P. W. Atkins, J. D. Paula, Physikalische Chemie, 5th ed., Wiley - VCH, Weinheim, 2013 [link].

[4] C. J. F. B¨ottcher, P. Bordewijk, Theory of Electric Polarization, vol. II, Elsevier Science, Amsterdam, 1978.

[5] H. B. Thompson, The determination of dipole moments in solution, J. Chem. Educ. 1966, 43, 66–73 [link].

Table 1: Useful values, all constants are in the SI system

e 1.602177 · 10−19 C −12 −1 2 −1 0 8.85419 · 10 J C m ◦ Heptane(1) 1.924 − 0.00140(T [ C] − 20) 3 ρHeptane(ρ1) 0.681 g/cm

MHeptane(M1) 100 g/mol bond length C-Cl 1.77 A˚

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