Robert H. Cole and John G. Berberian'
Brown University Providence, Rhode Island 02912
II
An Experiment 0" D,ipole Moments and Polarirabilities of Gas Molecules
The determination of molecular dipole moments from dielectric constant measurements is one of the classic methods of physical chemistry, and most laboratory manuals for undergraduate physical chemistry include one or more experiments. The conventional ones are measurements of a series of dilute solutions of nolar molecules in a non~olarsolvent. usually a t a single temperature, or of a polar gas, preferably a t several temperatures. In our experience, neither type of experiment as usually arranged has been very satisfactory for several reasons. The solution experiment suffers from the necessity of analyzing the measurements by a solution theory involving local field effects which are not adequately explained in terms an undergraduate can be expected to understand. The gas experiment can be made free of these difficulties by working a t sufficiently low densities, but more elaborate apparatus is required to measure the very small changes in capacitance and residual effects of variable cell or lead capacitances often produce serious errors. The purpose of this paper is to present an experiment on polar gases in which measurements of the necessary accuracy can be made very easily using commercially available equipment with the exception of the capacitance cell (or cells). The necessary theory can be developed presupposing only an elementary knowledge of statistical mechanics and of electrostatic directly derivable from Coulomb's law.
where n/V is the molar density, No is Avogadro's number, and kT has the usual meaning. The linear term in n/V is the ideal gas result, a simple derivation of which is given in the Appendix, while the term in (n/V)% represents interaction effects of pairs of molecules with @ a parameter to be evaluated from molecular theory, and further terms in the virial type expansion result from higher order interactions. The experiment can be analyzed by assuming that the density n/V is small enough that all terms but the first are negligible and that n/V is given in terms of measured pressure by the ideal gas law n/V = P/RT. Equation (2) can then be written
Values of or and p can then be obtained from values of (e - l)RT/P a t two or more temperatures. A better method is to derive limiting values of (e - 1) RT/P at zero pressure or density by extrapolation. At densities high enough that the term @ ( N I V )in~ eqn. (2) cannot be neglected, the corresponding virial approximation to density in terms of pressure can he introduced by writing
and to the same accuracy
Principles of the Experiment
The basic measurement is simply of differences of the capacitance CXof a test, cell containing the gas of interest from that of a fixed reference capacitor of known value CRas the gas pressure is changed, the experiment being done for several temperatures of the gas. The difference ACx = Cx - CR is obtained directly by a suitably arranged transformer bridge, a convenient one being described below, as is also the difference ACT = Cv - CR when t,he test cell is evacuated and has capacitance C". The desired dielectric constant is given by the ratio Cx/CT, obtained from the measured diff erences by the relation
The analysis of the experiment to obtain dipole moment p and polarizability a is then based on the equation
Present address: Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104.
With this expression, eqn. (2) can be written
(a
+ &T)~,](&,)
P + higher order terms in RT
(5)
Thus a plot of (e - l)(RT/P) against (P/RT) gives the desired polarization function as the intercept a t zero pressure and deviations from ideal behavior expressed by @ and B, are repre~ent~ed by the initial slope. In order to apply the preceding analysis it is necessary to measure very small changes in capacitance, as the dielectric constants of gases a t moderate pressures d i e r only slightly from unity. At 300°K and 10 atmospheres, e - 1for a nonpolar gas such as ethane is of order 0.01, and for a moderately polar gas with p = 1.7 Debye, e - 1is about 0.07. For a cell with convenient vacuum capacitance of 100 picofarads (1 picofarad, pf, = 1 w f ) , the changes ACx are then 1 and 7 pf which should be measured to 0.001 pf, i.e., about 10 parts per million of the cell capacitance. The problems in doing this with ordinary methods can be appreciated from the fact that a 1 cm length of standard coaxial cable has a capacitance of 1pf, with the result that minute changes Volume 48, Number 2, February 1971
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in lead or stray capacitance can produce serious errors. The success of the experiment as we have developed it is largely the result of using a transformer bridge method in which only the direct capacitance bet~veen two sets of electrodes insulated from ground is measured and capacitances of connecting leads to ground have no effect. Of several commercially available transformer bridges adequate for the purpose, one u-e have found convenient, and useful for a variety of other measurements as \yell, is the Wayne Kerr Type B221A Universal Bridge. On its lowest range, a capacitance of 1 pf can be measured with a resolution of 0.001 pf. The bridge in its standard form cannot measure the difference of two 100 pf capacitors needed for the experiment with this resolution, but is easily modified to do so by merely adding two connecting leads to points in the internal circuit.
manifold utth st,orage vessels 4 and 5 containing two unknown gases to be measured, a vacuum pump, and a 0-10 atm Rourdon gauge readable to 0.1% of full scale. The cells are first evacuated and t,he differences ACv determined for each cell. One of the gases is then admitted to the cells to the maximum pressure desired and the differences ACg determined after equilibrium is reached. A series of five or six lower pressures is t~hen produced by successive coolings of the storage vessel with liquid nitrogen, and the process is repeated for the second gas from the other storage vessel. The two gases we have used are SFs as an example of a nonpolar gas for which one verifies that the limiting value of (c - 1) (RTIP) is independent of T, and CHF, as a polar gas with dipole moment which can be determined from the measurements a t the three temperatures. Typical plots of (e - l)(RT/P) obtained with the apparatus are shown in Figure 3; the derived value of dipole moment from the data is @ = 1.62 Debye in good agreement with the value p = 1.649 Debye from more precise electrical and pressure measurements by H. Sutt.er2 using t,he same basic method. The entire experiment is easily done in an ordinary laboratory period. Other Experiments
-
REFERENCE
Figure 1. The rchemotic diagram of the Wayne Kerr B221A bridge showing modiflcotionr ond cell connections.
A schematic diagram of the essential parts of the bridge is shown in Figure 1, the additions being a lead from the end of t,he oscillator transformer (1000th turn) to a shielded BKC connector labeled BNC-A and a lead from the last turn of the detector transformer on t,he "knourn" side of the bridge (-100th turn) to a second BNC connector labeled BNC-B. The output voltage of the oscillator transformer is t,hen applied to the cell of capacitance Cx apd the st,andard CR and the currents from them are fed to t,he detector t,ransformer windings. The detector signal proportional to t,he difference of Cx and Cn is t,hen balanced to a null by adjusting the bridge standards, and their readings give ACX or ACv with a full scale range of 1pf. The cells used in the experiment were originally constructed for high pressure measurements and consist of s parallel plate assembly in a thick-walled stainless steel bomb for use to 200 atm. Considerably simpler designs should be adequate for the pressures below 10 atm of the experiment and could make use of plate stacks of commercially available fixed air capacitors, such as the General Radio Type 1403-D 100 pf standard used as reference capacitor Cn.
The choice of SFs and CHF3 as representative gases was dictated by availability and convenience, but many others could be studied with the same arrangement, for example COz and various halogenated hydrocarbons. The basic bridge method can also be applied to solution dipole moment measurements by comparing a suitable cell containing solution to a similar one filled with pure solvent. (Such measurements of alkyl amides in benzene are described by Meighan and Coles.)
Figure 2.
The gar handling system.
~ x ~ e r i m e n t Procedure al
The gas handling system is shown in Figure 2. Three 100 pf test cells in thermostats a t 50, 100, and 150°C or other temperatures as desired are connected to a S U T T ~H. . ~G. , Thesis, Brown University, 1969. a MI:IGHAN, R. hl., AND COLE,R. H. J. P h w C h n . , 68, 503 (1964).
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10%lm~l 2.6 Figure 3.
2.7
A plot of (e
2.8
-
2.9
3.0
l)RT/P verrvr 108/TlDKl far CHFI.
31
The transformer bridge employed in the experiment is a useful laboratory instrument for a wide range of capacitance measurements a t audio frequencies and can also measure a wide range of conductance, with the advantage over circuits often used that it is direct reading in conductance. As a result, it can conveniently be used for such experiments as ionization constants of weak electrolytes, conductimetric titration, and kinetic studies of slow ionic reactions if these result in conductance changes. This versatility, together with the fact that generator and detector for a frequency 1592 Hz are contained in the instrument, makes i t easier to justify the initial cost.
PLATE 2
PLATE I r
-
z
-7'"' Zf
PLATE 2
PLATE 2
101
1bl
Limitations
Figure 4. The coordinate diagram for the cdculation of the fleld due to: la) the charger on the capacitor ploter, Ibl the molecular dipole moments.
A limitation of this procedure for obtaining dipole moments is based on the extraction of the dipole moment from the (e - 1) (RT/P) versus l/T. The analysis is valid only if the dipole moment is independent of temperature. R/lolecules with temperature dependent hindered rotational degrees of freedom do not, in general, conform to a linear relationship between (c - 1) (RTIP) and 1/T; thus, a simple molecular dipole moment cannot be readily determined from the above analysis for hindered-rotor molecules.
angles between z and radius vectors RI and R2 to the elements of area dA, and dAz, and are necessary to give the non-vanishing z component of El. For the coordinates pl and p2 of Figure 4a, R12 = (1 - aI2 PI', Rz2 = aZ pzz, dA1 = 2npldpl, dAz = 2spzdp2, and eqn. (Al) can be written
+
+
Appendix
Two mathematical developments are needed to justify the working equations of the experiment: a derivation of the electrostatic relations between polarization, dielectric constant, and capacitance, and a statistical derivation of the relation of polarization in an applied field to polarizability and dipole moment. Derivations found in most physical chemistry texts and laboratory manuals sufferfrom use of such concepts as surface polarization and local fields which are hard to follow if the reader is inexperienc~d(and sometimes if he is not). In what follows, we give derivations which proceed directly from Coulomb's law and dipole fields of molecules. These have, we believe, the considerable advantages that dangers of confusing macroscopic and molecular concepts are reduced and that the origins of local field effects can be indicated without becoming embroiled in attempts to justify or make plausible such concepts as Lorentz and Onsager local fields. For those interested in extension to situations where these complications must be considered, we indicate how the Loreutz model can be introduced. A. The first problem is to determine the average electric field E in the interior of a parallel plate capacitor and the resulting potential difference, giGen the charge densities on the plates and the average electric moments m of the molecules at a knoaSndensity filling the space between the plates. By symmetry, the field E is perpendicular to the plates, i.e., along the z axis in Figure 4a, and the charges u and - u per unit area on the plates are uniform. The part El, of the field due to these charges is then by Coulomb's law given by the integral over the two electrode surfaces
where the bracketed factors are the cosines of the
The integrals are available in tables, or can be reduced to simpler type forms by the substitutions XI = p12 (1 - a)%,dxl = 2 pldpl and xz = p2? a2,dx2 = 2pzdp2, with the result on evaluation that
+
+
El,
=
270
+ 2rn = 47s
(A21
To obtain E , there must be added to El, the field Ef. resulting from the molecular dipoles m distributed through the volume between the electrodes. The average moment f i must by symmetry be parallel to z, and the potential at the field point P with coordinates X p , Yp, Zp of the dipole m with coordinates X , Y, Z as shown in Figure 46 is
where R2 = (Z - ZP)' and the field of f i a t P is
+ (Y - YP)' + ( X - XP)"
The summing of the fields of all the dipoles riz is simplified by observing that
and hence after taking the origin of coordinates at P
If there are N dipoles in volume V, the number in a volume element dV is NdV/V and the total average field of all dipoles liz is
Integrating with respect to Z from -a to 1
- a gives
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The integrals over the electrodes (obtained by transformation of the volume integral rather than by arguments involving surface polari~at~ion charge densities) can be evaluated in the same way as for El,, with the result E 1"
-
4rrNm --
and the macroscopic field E, is given by E,
=
E,,
+ EP, = 4T ( - !g!) (r
(A31
-
Since the field isuniform, the potential difference V , V z betu~eeuthe electrodes is
VI- V2
s-. 1-0
= -
Ezdz
= -
47
(
@
where dw = 2a sin 0d0 is the element of solid angle for molecular rotation coordinates. The part of the energy W depending on 0 corresponds to the potential of average torque on the molecule from the fields at the molecule. As Onsager pointed out in his classic paper, the effective field for orienting a permanent dipole differs from both E and the average field F, effective in inducing a dipole if there are significant interactions wit,h the other polar molecules. To take account of this, we write W = -pF, cos 0 where F, can be replaced by E only in the limit of low density of molecules. For small fields such that pF,/kT
