H. Bradford Thompson
Institute for Atomic Research Iowa State University A
I
1
The Determination of Dipole Moments in Solution
The measurement of electric dipole moments in solution is a part of many physical chemistry laboratory programs, and is finding increasing use in undergraduate research projects. Dipole moment studies are well adapted to both purposes, since important lessons concerning the taking and interpretation of data can he illustrated. Unfortunately, these points are missing from the dipole moment exercises in most laboratory manuals. While this deficiency may possibly he barely justifiable in a laboratory exercise, it is particularly serious when the apparatus and techniques described are applied for research purposes. The discussion here is by no means comprehensive; numerous methods and techniques not covered can give good results. Rather, I have tried to prepare a guide to procedures that have general acceptance, and that seem to me most promising for the relatively inexperienced investigator interested in dipole moments. Attention will he concentrated on the experiment; those interested in theory are directed to the annotated bibliography in the final section. Procedures for Calculation of Dipole Moments
One of the several equations hearing the name of Peter Dehye is that relating polarization, P, to the molecular dipole moment, p :
where: M = molecular weight, d = density, N = Avogadro's number, k = the Boltzmann constant, T = absolute temperature, c = dielectric constant, p = molecular electric dipole moment. The term Po = 4?rNp2/9kT is commonly called the orientation polarization. The quantity 4irNao/3 represents polarization not involving orientation (rotation) of the molecule. I t is useful to discuss this in terms of P., polarization by displacement of the electron cloud, and Pa, polarization by distortion of the atomic framework; that is, bending or stretching of bonds. The estimationof Pa and P, is a principal problem in dipole moment determination. Equation 1 should strictly apply only to pure suhstances in the vapor state, since in its derivation no provision for specific interactions of neighboring polar Work m a perfurmed nr rhe Amrs I.nborator? of the C.6. Ato~nic Eneryy Qnnmksion. Contribution So. 1731. A u I ~ ' >p r e w ~ ~ ~ddress:D c p a r t m ! ~ d~Vtwrni~rry ~ Vniversi~v,of \ I , < hican, A m Arbor. ' A number of treatments not starting with equation (I), or purporting to improve upon it, are available. Smyth (6) and Partington (8) discuss these at length.
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Journol of Chemical Education
molecules was included. However, useful treatments of dilute solutions of polar molecules in nonpolar solvents, starting from equation (I), have been made.' For such a solution we can hopefully state: P,,= X,P,
e m - 1 X,Ms + XlMl + X,P* = .a + 2 dm
where X = mole fraction, subscript 1 refers to solvent, subscript 2 refers to solute, and subscript 12 refers to the solution. All P's are total polarizations. Thus P12can he determined from density and dielectric constant measurements on dilute solutions. Since PI can likewise be determined from the corresponding solvent properties, the solute polarization P2 can in principle be calculated. Some variation of this procedure is found in most manuals. Practical considerations have led most authorities to prefer alternative treatments of the data. Among these considerations are the following: 1. Data. an the pure solvent are used for each solution examined, and are thus weighted much more strongly than data on the solutions. 2. Treatment and handling of the solvent is likely to differ from that of the solutions, thus introducing a source of sizeable error in the determination of the fsmallll) . XZI - - term uDon suhtraction of XLPIfrom PC%. 3. If P, is extrapolated to i d n i t e dilution, the most dilute solutions, where the determination of XIPz by difference is least certain, are weighted most heavily.
Procedures that do not rely strongly on measurements on the pure solvent are therefore to he preferred. The Procedure of Holverstodt and Kumler ( 1 31
I n dilute solution, the dielectric constant el2 and specific volume VIZ = l/d12 are very nearly linear functions of solute concentrations: s* = s
v,z =
0,
+ a'X2
+ B'X*
(3) (4)
By rearrangement and substitution in equation (2), we can derive:
Assuming PI and Pz constant, we now differentiate this equation with respect to X2, substituting a' and 8' for the partial derivatives of el2 and vlz respectively:
At the limit of very dilute solution this reduces to
AU the quantities necessary to calculate P2may thus be obtained from graphs of specific volume and of dielectric constant versus mole fraction for a, series of solutions. These plots will generally have the linear form implied by equations (3) and (4) if solute mole fractions are kept below 0.005, unless association occurs. It is best to run five or more solutions. A simpler variation of this procedure can be made by graphing nz and Vlz versus weight fraction wa rather than mole fraction: 68s
= 6,
v ~ n= UI
+ aror +
(8)
=
B/8'
=
Guggenheim's Procedure ( 1 4 )
By far t,he most tedious and troublesome chore in applicat,iou of the Halverstadt-Kumler procedure is the precise measurement of solution densities. Guggenheim has shown that this can be avoided a t the price of equally precise measurement of refractive indices. I n a manner analogous to the derivation of equation (5), we can obtain
(9)
Boz
At the limit of infinite dilution, we can show that ar/ar'
As yet., extensive evidence that this approximation is satisfactory is not available.
MJMa
Xoting that concentrat.ion in moles/ml is given by (10)
and therefore
Cl =
x,
+
(15)
V ~ X I M LX&d
we may combine equations (5) and (14) to obtain: Both theory and experience indicate that plots versus weight fraction are at least as linear as those versus mole fraction. Having thus obtained P2,it remains to determine or estimate the electronic and atomic polarizations, Paand P,, so that we may calculate the dipole moment by using equation (I), which we will slightly rearrange:
The electronic polarization equals the molar refraction (extrapolated to very low frequency) and can thus he estimated in several ways: 1. Experimental determination of the refractive index, n, of the pure solute from which P . = M R = - - -111'- - -- 1 M* (13) nn2 2 dz
+
Whiie the second procedure might appear preferable to the first or third, it requires highly precise refractive indices, involving excellent temperature control and provision against evaporation. I n my opinion, if this much trouble is to be taken with refractive index measurements, Guggenheim's procedure (see below) is to be preferred. The estimation of Pais a much more difficult matter. There appears to be no experimental way of obtaining this quantity that is simple enough for routine use. Fortunately, the atomic polarization is small-approximately 5 1 0 % of the magnitude of P,. For some time this relationship was widely used to estimate P,. Today most workers prefer to neglect P,,since P, as determined from the refractive index for the sodium D line is several percent larger than the extrapolated "zero-frequency" value. The D-line molar refraction is commonly used as an approximation for P. P,. The procedure of Halverstadt and Kumler requires precise measurements of solution densities and dielectric constants. However, the procedure lessens to some extent the need for high absolute accuracy. A detailed discussion of the precision required will be given in a later section. Estok (15) has proposed that for most substances accurate solution densities are unnecessary, since p can be approximated by vz - 81.
+
We can apply equation (12) to each component, noting that for the solvent there is no molecular dipole moment. Then
As before, ureare now faced with the problem of estimating atomic polarizations. We can (as before) neglect them or assume that they have been "included" in the molar refra,ct,iou t,erm. Alternatively, Guggeuheim suggests that the term in brackets will be small and nearly const.ant, since atomic polarizations will vary roughly wit,h the size of the molecule. I n either case, a plot of the term on the left versus cz will have a slope from which t,he dipole moment can be ohtained. Guggenheim's procedure is newer than that of Halverstadt and Kumler, and has not been used as widely. It is more convenient for the worker equipped to measure refractive indices accurately. Where both methods have been used, results appear ta be equivalent. For example, using the same dielectric constant data, moments ohtained by each procedure for five dihasic adds are compared in Table 1. Differences are within the expected error in the moments (0.05 D), with the Guggenheim moments slightly lower. An additional advantage to Guggenheim's method is that only the solute need be weighed. Solutions then can he made up to the mark in volumetric flasks if desired. It has been pointed out (16) that the subtraction of nZfrom r before plotting may tend to obscure irregularities due t,o experimental error. A way of avoiding this Table 1 .
Electric Moments of Dibasic Acids
Moments. according to H. 8 K. (18)
G. (IL) Oxalic acid Malonic acid Suecinic acid Glutaric acid Adi~ieacid
2.66 2.56 2.08 2.47 2.30
2.71 2.62 2.11 2.50 2.36
a THOMPSON, H. B., EBERSON, L., AND DAALEN, J. V., J . Phys. Chem., 66, 1634 (1962).
Volume 43, Number 2, February 1966
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is given by Smith (4), who discusses Guggenheim's procedure a t length. Precision and Sources o f Error
Although claims of precision to a few parts per thousand have on occasion been made, most workers report dipole moments determined in solution to *0.05 to 0.10 D. In view of the many approximations involved, these limits seem appropriate. Thus it might seem reasonable to strive to keep error from any individual source below 0.02 D. The accuracy required in various quantities will differ from one compound to another. Table 2 shows the requirements for succinic acid, and can be taken as fairly typical for moments of about 2 D. The requirements in Table 2 clearly fall into two groups: those in the percent range, and those calling for precision to one part in 10,000 or better. The latter are required because small changes in properties with concentration must be determined in order to calculate a, 8, and S. In our example the change in dielectric constant over a 0.004 range in weight fraction would be 0.022. One-percent precision clearly requires measurement to 0.0002. This is, however, only a requirement with respect to the change in el%. The student who understands this adequately will also realize that an error common to all dielectric constant values could perhaps be as large as 1%. The same comment applies to specific volumes or refractive indices. There is, for example, little point in applying air buoyancy corrections to weighings unless relative precisians are nearly an order of magnitude better than those discussed above. Electronic and Atomic Polarizations
+
The high relative error allowable in P , P. tends to justify the approximations necessary in determining this quantity. Where the moment is relatively large (2 D or more) the use of the molar refraction from the pure liquid is probably justified; it would be indeed surprising if the additivity implied by equation (14) failed by more than a few percent. The estimation of
molar refraction from atomic or group refractions is probably somewhat riskier, particularly in molecules containing extensive multiple bonding. Errors from this source are probably not important, provided (a) the moment measured is not small, (b) arguments based on the result do not depend on small differences in moment, (c) other moments with which the moment in question are to be compared are calculated using the same procedure, and (d) care and consideration are given the question of electron-delocalization in cases where there is extensive multiple bonding. It might appear that since the relative uncertainty allowable in the molar refraction is large, great precision in n12would not be required to determine molar refraction from the solutions (option 2 above). Since, however, the molar refraction is considerably smaller than the molar polarization, the same level of precision will be required here as in Guggenheim's method. Conversely, while the allowable uncertainty in P. Pa does not appear in the second section of Table 2, it should not be implied that this source of error has been eliminated. The approximation implicit in equation (17) and the discussion following involve a corresponding uncertainty.
+
Experimenfolly Measured Quantities
Clearly, concentration can easily be determined to much greater accuracy than is required. Even for 20-ml samples with a maximum of 0.5 weight percent solute, the level of uncertainty using a common analytical balance will be 0.1 percent. Error from experimental sources will thus depend on the precision in dielectric constant and in density or refractive index. Each is discussed in greater detail below. A collection of approximate temperature coefficients appears in Table 3. Clearly, temperature regulation to within 0.02 degrees will be sufficient to reduce variations from this source to one fifth or less of the uncertainties specified in Table 2. The absolute temperature clearly need be known much less precisely. Error versus Size of Moment
To find the dipole moment using equation (12), we must subtract P, Pafrom P,. If the moment is small, large relative errors will result a t this stage. Table 4 shows the errors inp that might result from 1.7 ml error
+
Table 2. Uncertainties Corresponding to 0.02 Debye Uncertainty in Dipole Moment (Succinic Acid)
Table 3.
Solvent Benzene Carbon tetrachloride Cyclohexane Dioxsne
4rrN~a/27kT
Method of Guggenheim (14) 0.57 ml
b
de/dt0 (degrees-')
dv/dtc .* (ml/degree)
-0.0020J
0.00136 2z 0.00004
- 0.0020 -0.00161 -0.00171
0.00077 0.00001 0.00158 + 0.00005 0.0011
-0.00058 -0.00054* -0.000431 (p)
data
Compounds " Advances in chemistry ~eries.15and 22, American Chemical s;ciety, Washington, D.C., 1955 and 1959; EGLOFP, G., "Physical Properties of Hydrocarbons," ACS Monograph 78. . ., Reinhold -Pnhlishine Corn.. , New York. 1939: TMMERMANS. J.. "Phvsico-Chemical Constants of Pure oreankc ~omuounds.'' ~isevie;Publishing Co. Inc., Amsterdam, 19
