dipole moment hs13

Physikalisch-chemisches Praktikum I Dipole Moment - 2013

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Dipole Moment

Summary

The dipole moments of some polar molecules in a non-polar solvent are determined. To do so, relative permittivity and refractive indices have to be measured. The relation between macroscopic quantities and molecular (microscopic) properties is discussed.

Theory

Definition of the dipole moment

A system that contains two separated charges of opposite sign, q1 = q and q2 = -q is called an electric dipole.

It is described quantitatively by the dipole moment µ :

rqrrq =−= )( 12µ , (1)

where r1 and r2 are the vectors that define the position of the two charges in space.

Figure 1: Definition of the dipole moment as a vectorial property

The dipole moment is thus a vectorial property. For a distribution of negative and positive point charges qi the dipole moment is:

µ = qi ri

i

∑ , (2)

where ir are the positions of the charges qi.

The electric dipole moment of a molecule is the sum of the contributions of the positively charged nuclei and the negatively charged electron distribution:

r µ =

r µ + +

r µ −. The nuclei can in good approximation be treated

as point charges: r µ + is thus given by

r µ + = Z ie

r R i

I =1

N

∑ where Zi is the nuclear charge of nucleus Ni, e is the

elementary charge, and r R i is the position vector of this nucleus. For the determination of the electronic part

r µ − the electron distribution has to be known. This can be obtained from quantum chemical calculations.

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 a center of 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. In the SI system, the


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unit of the electrical dipole moment is C⋅m. Since these units result in very small numbers, the unit Debye (1D = 3.33564⋅10-30 Cm) was introduced in honor of the discoverer of the dipole moment, Peter Debye (1911-1912 Professor of theoretical physics, University of Zurich).

Two elementary charges (e = 1.602⋅10-19 C) of opposite sign placed at a distance of 1.28 * 10 -10 m, i.e. the bond length of the HCl molecule, give rise to 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:

X =µ(measured)µ(calculated)

⋅100%, (3)

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 measured and discussed. This is practically done by measuring relative permittivity and refractive indices of solutions and pure solvents. The relation between these two macroscopic properties and the molecular dipole moment is discussed in the following.

Relative permittivity, Polarization, and Polarizability

Between two charged plates of a capacitor there is an electrical field. If the distance between the two plates is much smaller than the surface of the plates, the field can be considered homogeneous except in the border regions (Figure 2). The electrical field strength E0 is given by

000

0 / εσε

==S

qE , (4)

where σ 0 = q /S denotes the surface charge density of one condenser plate (S = surface area, q = charge on

one plate) and ε0 is vacuum permittivity (ε0 = 8.85419 10-12 J-1C2m-1). The voltage U0 between the two plates is proportional to the charge q:

Figure 2: Electrical field in a capacitor

q = C0 ⋅ U0. (5) The proportionality factor C0 is called capacity. The following equation relates C0 and the field strength E0:

C0 = q /U0 = q /(E0 ⋅ d) . (6)


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C0 is thus proportional to 1/E0 provided charge q and plate separation d remain constant.

Experimental observation

As a capacitor is filled with matter, the voltage U between the two plates is reduced from the voltage in vacuum, U0:

U = U0/ε , E = E0/ε, and thus C = ε C0 (7)

with ε > 1. ε is the so-called static relative permittivity of the filled in material (dielectric) and is a material constant.

In order to explain this behavior we introduce the polarization r P of a dielectric in a field

r E 0. It is defined as

follows

r P = ε0(

r E 0 −

r E ) = ε0(ε −1)

r E (8)

and is a vectorial quantity. r E 0,

r E , and

r P are parallel. Inserting (4) into (8) one gets the scalar relation

E = (σ 0 − P)

ε0

=def σ

ε0

. (9)

The electric field E with a dielectric can thus be described as the field in vacuum. Its strength does not

correspond to the surface charge density σ 0 any more but to a reduced density σ (σ = (σ0 – P)). Polarization

can thus be regarded as a surface charge density that is induced by field r E 0 on the interface between

dielectric and capacitor plates. This induced charge density σ partially compensatesσ 0 because these charges

have opposite sign. The so-called free surface charges are caused by three processes, which arise at the molecular level under the influence of an electric field. The observed polarization P can thus be regarded as being the sum of three contributions:

P = PD + PE + Pµ . (10)

Electronic polarization PE :

PE is found in all atoms and molecules. It is caused by the fact that the centers of positive charges (nuclei)

and negative charges (electrons) are pulled in different directions in an external field.

Distortion polarization PD :

This contribution is only found in polar molecules and is caused by changes in bond lengths and angles.

PE and PD are combined to the so-called displacement polarization which is caused by distortion of a

molecule in an external electric field E. Due to such a distortion a molecular dipole moment µi is induced.

As long as the external field is not too strong, we can assume

µi = αE . (11) The proportionality factor α is called the polarizability of a molecule. It indicates how easy a molecule can be distorted. In liquid solutions the induced dipole moment µi is always oriented in direction of the external

field.

Orientation polarization Pµ :


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Pµ is only found in polar molecules with a permanent dipole moment r µ 0 . It is caused by a partial alignment

of the molecular dipole moment into the direction of the external field. In contrast to PD and PE , Pµ varies

strongly with temperature since the thermal motion is opposite to the alignment.

In an external electric field, polar molecules have thus a total dipole moment r µ according to

r µ =

r µ 0 +

r µ i . (12)

Our model of a dielectric between two charged capacitor plates can now be sketched as shown in Figure 3. We hereby denote a molecular dipole moment by . It explains the free surface charges q’ at the dielectric boundary surface S. P can then be written as:

V

dq

dS

dqSqP

⋅=

⋅

⋅==

''/' . (13)

Figure 3: Simple model of a dielectric

P can be regarded as the electrical dipole moment (q’·d) per unit volume V of the dielectric, which due to the

molecular dipole moments jµr

corresponds to the vector sum:

∑=

=lN

j

jV

P1

1µrr

, (14)

where Nl denotes the number of molecular dipole moments in volume V and each individual dipole moment

jµr

comes from equation (2). Equation (14) provides an interpretation of the macroscopic property r P in

terms of molecular (microscopic) properties.

The Debye equation

In 1912 Debye developed a theory for the relation between the polarizability α, the permanent dipole moment µ0 and the relative permittivity ε of a substance. He found the following formula:

ε −1ε + 2

M

ρ=

NL

3ε0

α +µ0

2

3kT

, (15)

where M denotes the molar mass, ρ the density, NL Avogadro’s number, and µ0 the permanent dipole moment. The first term in parentheses corresponds to the distortion polarization and the second term to the orientation polarization. One notes that the latter contribution becomes smaller with rising temperature, a consequence of thermal motion. This so-called Debye equation (15) neglects polar interactions of dipoles with their surroundings. It is therefore valid only for dilute solutions of polar substances in non-polar solvents and for polar gases at pressures smaller than 1 atm.

-+-


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The Debye equation is the basic equation for the determination of molecular dipole moments in non-polar solvents. It order to apply it, however, polarizabilites α of solvent and solute molecules have to be known. To determine these quantities, the refractive indices of solvent and solute molecules have to be measured.

Dispersion of the Polarizability and the Relation between εεεε and n

Dispersion of the polarization

Dispersion generally means the frequency dependence of a physical quantity. The displacement of charge in a molecule due to an external field causes an induced dipole moment µi = αE i. This effect, as well as the

orientation of the permanent dipole moments, is achieved in a finite time. The same is valid for the reverse case, namely if the polarizing field is switched off and the molecules return into their non-polarized disordered initial state. The so-called 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. Typical relaxation times of different polarizations are listed in Table 1. Pµ obviously depends on the viscosity of the medium.

Table 1: Relaxation times at different frequencies and in different media

polarization size of molecule

medium τ [sec] frequency

PE 10-15 UV

DP

10-12-10-14 IR Pµ

gas 10-12 far IR Pµ small solution, low

viscosity 10-10-10-11 microwaves

Pµ large solution, high viscosity

10-6

Pµ glasses up to hours

radio frequencies

The Debye theory is only valid if the polarizing electrical field changes slowly enough such that polarization and orientation of the molecules are always in equilibrium with the field. This condition is fulfilled with certainty in a static field or in solutions of low viscosity. Also, an alternating radiation field with a frequency of 106 Hz fulfils this condition since the relaxation time τ is much smaller than 1 µs as seen in Table 1. From frequencies larger than 1010 Hz also the atomic polarization disappears (see Figure 3) and at frequencies higher than ca. 1014 Hz, only electronic polarization is left. P and thus also ε depend on the frequency and the contribution of electronic polarization can be determined by measuring ε at light frequencies of 4 to 8⋅1014 Hz. Figure 4 depicts schematically the discussed dispersion of the polarization.

Figure 4: The dispersion of polarization


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The relation between n and ε

From Maxwell’s theory of electromagnetic waves follows an expression for the phase velocity ν of a light wave in a medium with relative permittivity ε and a magnetic susceptibility µm:

ν =c

ε ⋅ µm

(16)

where c is the speed of light in vacuum. The refractive index n of this medium is defined by:

n =sinα

sinβ=

c

ν (17)

n is thus always ≥ 1.

§ α = angle of incidence

β = angle of refractive ν = speed of light in medium

Figure 5: Refraction of light

For many organic substances µm deviates only very slightly from 1, we can therefore write in good approximation:

ν =c

ε (18)

or

n2 = ε (19)

The relative permittivity corresponds thus to the square of the refractive index n. Its magnitude and frequency dependence are a measure for the achievable electron polarization and its relaxation time.

The Guggenheim approximation

For a completely independent determination of the polarizability α in (15) the contribution of distortion polarization should be determined. This is however difficult to do. The Guggenheim approximation 2,3 allows approximate calculation of dipole moments of polar molecules in non-polar solvents. It is based on the

Debye equation and makes the determination of r µ 0 possible solely by measuring ε and n

2. Hereby, it is

assumed that the distortion polarization of a molecule is proportional to its molar volume. The Guggenheim approximation reads

µ02 =

27kT

NL

ε0

1ρ1(ε1 + 2)(nD1 + 2)

⋅ M2 ⋅ (bε − bn ) (20)

with

bε =dε

dw2

and 2

2 )(dw

ndb D

n = (21)


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The physical properties of the pure solvent are labeled with 1 and those of the dissolved substance with 2. nD is the refractive index at the frequency of the sodium D-line and w2 = m2 / (m1 + m2) is the weight fraction of the dissolved substance. The obtained experimental data is analyzed according to this Guggenheim equation. To do so, one needs the density (ρ1), the relative permittivity (ε1) and the refractive index (nD1) of the pure

solvent. The slopes εb and nb are obtained by plotting ε and nD

2 against the weight fraction w2.

Experimental Part

Dipole- and refractometer needed used during the experiment have to be kept very accurately at the same constant temperature.

The Dipole Meter

In this part of the experiment, we want to determine the relative permittivity ε at low frequencies, i.e. in the region of radio waves (~106 Hz). In this spectral region, all contributions to the polarizability are effective (see Fig 4). To do so, we use a dipole meter, which consists of a resonant circuit that oscillates at its eigenfrequency:

LC

121π

ν = (22)

The eigenfrequency is measured with a frequency counter. Filling the capacitor with a liquid changes C according to (7):

C = εC0

We can thus rewrite Eqn. 22 to

ν =1

2π

1LC0ε

= 1

4π 2LC0ε

= k

ε (23)

by merging all the constant terms into a constant factor k. This finally reveals the relation between the frequency readout and the relative permittivity:

ε =k

ν 2 (24)

After calibration with a liquid of known relative permittivity ε, the proportionality constant k can be determined.

Experiment

The eigenfrequency of the pure solvent with known relative permittivity ε (heptane, εheptane = 1.924 – 0.00140[T – 200C] is measured first (at least three measurements). With the obtained frequency ν, the constant k of the experimental setup is calculated. Then the eigenfrequency of each solution is measured at three different concentrations (only once) and the corresponding permittivities are calculated as described above. Then, ε is plotted against the weight fraction w2 (not the percentage) and the slope bε is obtained:

bε =dε(ν radio)

dw2

=

dk

ν 2

dw2

(25)


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The Refractometer

According to Maxwell’s theory, the relative permittivity ε(ν) in the visible region of the electro-magnetic spectrum can be easily determined using optical methods by measuring the refractive index at a given frequency. As shown in Fig. 4, in this part of the spectrum, the electronic polarization dominates and no contributions from orientation polarization or distortion polarization exist.

The refractive index is measured with a refractometer as proposed by Abbé. The technique is based on the phenomenon of total reflection. A light beam with an incidence angle α in medium 1 hits a phase interface to an optically less dense medium 2. The angle of refraction β in medium 2 is thus larger than α.

Using a relative angle of refraction

n =n1

n2

=sinβ

sinα>1

For normal refraction the relation

0 ≤ n ⋅ sinα = sinβ ≤1

always holds. An incident angle α0 < 900 must thus exist with

n/1sin 0 =α

If light hits a phase interface with an angle α0, the refracted light streaks the interface (β = 900). In case of angles α > α0 the light beam is reflected at the interface. α0 is called the critical angle for total reflection.

The Abbé refractometer (Figure 5) contains two prisms made of vulnerable flint glass of high optical density (large n1), a prism for illumination and one for detection, a traversable telescope adjusted to infinity, and a lamp. The sample is in a layer with a thickness of 0.1 mm between the two prisms.

Figure 6: Sketch of an Abbé refractometer

The incident light through the lighting prism is scattered on the matte finished bottom side such that the probe is irradiated largely isotropically. Some light hits the detection prism under the critical angle of total reflection. For larger observation angles, the detection prism appears dark. The angle distribution of the leaving light is observed using a traversable telescope that is adjusted to infinity. In contrast to a microscope, not all the light that originates from one point of the examined object is mapped on one point of the eye’s retina but all the light that enters the telescope with the same angle. Since white light is applied and since the


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refractive index varies with frequency (dispersion) no clear bright-dark edge appears in the visual field of the telescope, but a blurred coloured fringe. Using a compensator this fringe can be turned into a distinct bright-dark edge.

This compensator consists of two parts, each one consisting of three Amici prisms that can be twisted against each other. This construction reveals a system of variable dispersion that can be used to calibrate the dispersion of lighting and detection prisms. To do so, the cross wire is adjusted on the bright-dark edge and the refractive index can directly be read on the shown scale.

Experiment

First the refractive index of the pure solvent is measured as a reference. Then, the refractive index of each solution is measured for three different concentrations (the same solutions as for the dipole meter measurement are used). With the obtained refractive indices n, the relative permittivity ε = n2 is the visible is calculated. Then ε is plotted against the weight fraction w2 (not the percentage) and the slope bn is determined:

2

2

2

)()(dw

nd

dw

db vis

n ==νε

(26)

With bn and bε all variables are known in order to calculate µ0 according to the Guggenheim approximation.

Carrying out the experiment

At the beginning, turn on the dipole meter and thermostat. The apparatus has to reach a thermal stationary state before measurements can be made. This takes about 15 min. The temperature of the thermostat may not change by more than 0.1° C during the experiment. Otherwise, the measurements of refractive index and capacity are not sufficiently precise.

Meanwhile, prepare three solutions of the substance received from the assistant in heptane according to the following procedure: In a 50 mL graduated flask weigh about 49 mL (accurately!) 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 35 mL are stored well closed and used later on for determining ε and n. Dilute the 15 mL with solvent to 50 mL and weigh again in order to exactly determine the weight fraction. It should be approximately 1%. Apply the same procedure to prepare a solution with a weight fraction of about 0.3%.

It is very important that the solutions are prepared exactly according to this instruction!

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

As a first measurement, determine the relative permittivity of the pure solvent. For this purpose fill the measuring cell with heptane. Read off the frequency at least three times and take an average result. From the measurement, determine the constant k in Eqn 24. In all subsequent measurements of solutions, it has to be ensured that the cell is clean and dry before a solution is filled in so that the concentrations are not changed. The cell has to be blown out with the dryer first hot, then cold, until no solvent can be smelled. After one measurement, the cell is cleaned with solvent and the solvent is measured again. Finally, a plot is generated according to Eqn. 25.

In parallel, determine εvis by measuring the refractive indices of solvent and solutions. Using a pipette solvent or solution is put on the detection prism. Then, put on the lighting prism using the bayonet nut connector. Clean the prisms using Linsoft tissues and acetone or ethanol. Take care not to scratch the prisms!


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Make a plot according to Eqn. 26. The density ρ1 of the solvent is given below and the molar weight of the dissolved substance M2 is known. All variables in order to calculate µ0 according to the Guggenheim equation are now known.

Practical Advice

Preparing the solutions

• 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.

• 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.

General remarks

• All constants are in the SI system:

e = 1.602177 10-19 C

ε0 = 8.85419 10-12 J-1C2m-1

• Begin with low concentrations and move consequently to higher ones.

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

• Check carefully the temperature during the whole experiment.

Wait ten min. after a solution has been filled in such that thermal equilibrium is reached.

• Repeat every measurement three times and average the results.

• ε1 = εheptane = 1.924 – 0.00140[T – 200C]

ρ1 = ρheptane = 0.681 g/cm3

M1=Mheptane= 100g/mol

d = bond length C-Cl = 1.77Å

ϑ2 = 55°53’

ϑ3 = 72°0’

Tasks

1. Determine the dipole moments of chloro-cyclohexane, dichloromethane, trichloro-methane (chloroform) and monochloro-benzene. Compare your values to reference values from the literature.

2. The calculated dipole moment of chloro-cyclohexane (µ1) is used to determine the ionic character of the C-Cl bond:


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J =µ1

d ⋅ e

(d = bond length C-Cl, e = elementary charge)

3. The deviation from the vector sum due to alternating induction should be determined for the dipole moments of dichloro-methane (µ2) and trichloro-methane (µ3).

∆µ=µ2 - 2µ1 cos(ϑ2)

∆µ=µ3 - 2µ1 cos(ϑ3)

ϑ is the angle between the individual bond dipoles and the vector sum

4. Determine the reduction of the dipole moment (µM) due to resonance in monochloro-benzene:

∆µ = µ1 − µM

Literature:

1. G. Wedler, Lehrbuch der Physikalischen Chemie, 2. Auflage, VCH Verlagsgesellschaft, Weinheim, 1985

2. P.W. Atkins, Physical Chemistry, 5. Edition, Oxford University Press, 1994 3. C.F.J. Böttcher, Theory of Electric Polarization, 2. Edition, Elsevier, New York, 1973 and

1978. 4. H.B. Thompson, The Determination of Dipole Moments in Solution, J. Chem. Educ. 1966,

Vol.43, pp. 66 - 73

dipole moment hs13

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