J. B. Moffat University of Waterloo Waterloo, Ontario Canada
Thermodynamics from Dipole Moments An experiment in physical chemistry
Experiments which illustrate the interrelationship of molecular and macroscopic properties are often difficult to obtain a t the level of third year physical chemistry. Too often such experiments involve a level of statistical mechanics beyond the capabilitites of most third year students, and the instructor may be forced to supply the necessary equations into which the student "plugs" what he feels are the appropriate numbers. An experiment which appears to circumvent some of these problems has been in use for the past three years in our physical chemistry laboratory. While there may be some difficult points in the theoretical derivation of equations related to the present experiment, most advanced undergraduates probably would have little trouble in satisfying themselves about the degree of validity of the relations used. The experiment involves the determination of the dipole moments of solutions of n-propyl nitrite in benzene. These results are then used to calculate the standard free energy of isomerization of the nitrite, which can exist in the cis or trans form. The experiment should demonstrate to the student (1) the theory and method of obtaining dipole moments in liquid solution, and (2) a method of obtaining thermodynamic functions which is somewhat different from the usual technique of measuring macroscopic properties. In our laboratory a Wissenschaftlich Technische Werkstatten DM 01 "dipolmeter" operating on a heterodyne beat principle is used to measure the dielectric constants. The apparatus is calibrated each time with air, cyclohexane, carbon tetrachloride, and benzene. Four solutions of accurately known concentration of n-propyl nitrite in benzene are prepared by the student. Concentrations of 1-5% by weight are used. The refractive index of the pure components of the solution and of each of the four solutions is also measured. There are a number of different methods whereby the dipole moment can be calculated from the experimental data. Many of the standard texts ( I , 3) contain at least partial developments of the appropriate equations, and hence will largely be omitted here. Usually a t a later point in the derivations, it is found convenient to introduce the so-called molar polarization
where c is the dielectric constant; M, the molecular weight; d, the density; N, Avogadro's number; and a, the polarizability. The molar polarization is then taken to he the sum of a perturbation term Pd and an orientation term P,. 74
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Journal of Chemical Education
Equation (2) ultimately results. P
or
S-1
($)
( ; ~ i )
Pd
=
+ PI.
n' - 1 = n ()-
M ~ w N 7 +3
'I
where lz, the refractive index, is equal to &. It is well to emphasize that equation (2) is valid only for gaseous systems, as a result of assumptions involved in its derivation. If we attempted to measure the dipole moment of a polar substance by measuring the dielectric constant, e, of the pure substance, we would be including interactions between molecules. If, however, we make use of dilute solutions of the polar material in a nonpolar substance, then presumably molecular interactions should be effectively eliminated. Thus, as has been mentioned previously, dilute solutions of n-propyl nitrite in benzene are employed in the present experiment. The presence of two components in the system to be examined requires the introduction of an additional relation P = X,P, X Z 2 (3) where P is the molar polarization of the two-component solut,ion, XI and Xp are the mole fractions and PI and Pp are the molar polarization of components 1 and 2 of the solution. Of course the assumption mentioned in connection with equation (2) is still present in (3). Hence one could plot P values versus X2, where component 2 is the solute, extrapolate to X* = 0, and obtain a value of P for the solute. The appropriate relation for this purpose is
+
Unfortunately the quantity P - PI represents the difference between two numbers of similar size so that P - PI is small. Hence a small error in P is magnified when division by the small number X2 is carried out for the dilute solutions involved. An alternative method, which is employed in the present experiment, involves the extrapolation of the dielectric constant and density. Hedestrand (3) has demonstrated that for a dilute solution the dielectric constant is a linear function of the number of polar molecules per ml. In addition, the density of a dilute solution is linear in the weight or mole fraction of the more dilute component. I. F. Halverstadt and W. D. Kumler (4) have derived an equation for the polarization of the solute a t infinite dilution
where €1 and 61 are the dielectric constant and specific volume of the solvent, respectively, and 7 and are the slopes of the straight lines obtained by plotting the dielectric constant and specific volume against the weight fraction of the solute, w2. Hence the permanent dipole moment may be obtained from
which is a form of equation ( 2 ) . Some interesting responses have been obtained from the students by posing an apparently simple question at this point. Is the dipole moment which has been measured for n-propyl nitrite that which is applicable to an assemblage of interacting molecules, or to a single molecule? Molecules of normal propyl nitrite are capable of existing in two extreme orientations, the cis and trans isomers (6)
R' trans
cis
as a result of hindered rotation about the 0-X single bond. Hence the cis and trans isomers will have diierent dipole moments. If it be assumed that the majority of the molecules of n-propyl nitrite exist in either the extreme cis or extreme trans form and that the two forms are in equilibrium with respect to the process of interconversion, then the dipole moment of such a mixture may be taken to be =
x.L.r*..P+ Xtrsaa..ra..a
(7)
where X.i, and X., are the mole fractions of cis and trans isomer, respectively, and hi. and p*,, are the permanent dipole moments of the cis and trans isomers. Hence, if the effective dipole moment of the equilibrium mixture of isomers can be measured and if the dipole moment of the individual isomers can be obtained, then the mole fractions of each which are present in the mixture may be found. Ideally the individual moments of the isomers would be available from other sources, such as microwave spectra. Unfortunately no evidence of such values has been found in the literature. Another somewhat less desirable method involves the use of bond moments such as those tabulated by Smyth (6) to calculate the dipole moments of the separate isomers.. Of course this method suffers from the inevitable disadvantages inherent in such group contribution calculations. Unfortunately the value of the bond moment given by Smyth for N=O is one which applies when the N=O bond is not linked to oxygen. Hence the validity of employing such a value for nitrites is in doubt. I n addition there has been some question as t o the accuracy of the bond moment value for N=O as given in Smyth's tables. To avoid such difficulties, Grant et al. (7) used an experimental dipole moment measurement (8) on gaseous tbutyl nitrite. By using the value of 93% as obtained from
infrared studies (9) for the concentration of the trans isomer in gaseous tbutyl nitrite, they arrive a t a value for the bond moment of N=O. I n order to obtain such a value the bond moments for R-0 and N-0 were taken as shown in the following table. Bond
Bond Mmmt
The 0-N-0 and R-0-h' angles were assumed to be tetrahedral. By making use of these bond moments the dipole moments of the individual isomers of n-propyl nitrite, trans and cis, can be calculated. Hence, the mole fractions of the isomers can be obtained from eqn. (7). The standard free energy of isomeriaation may be taken to be where K x = Xt..../XCi,is the so-called relative isomer abundance ratio and X is the mole fraction. The student must recall the source of equation (8) and in so doing be aware of the assumptions involved in its use here. Results obtained by the students for K g and hence AG,., are of the same order of magnitude as those re ported in the literature. No attempt has as yet been made t o have the students find values for AS and AH, although this presumably could be easily done. I t was believed that the additional laboratory time involved in producing results at other temperatures would not be justified by the small benefit to the student. Comparison of Typical Student Results* with Literature Values
Effective dipole moment of n-pro 1 nitrite (25"8(~)
"Equilibrium constant" Kx (25T)
Free energy of isomeriaation (cal. mole-') (25%)
Literature values 2.39 (liq) (7) 2.41 (gas) (6) 2.37 (liq) (6) 2.28 (dil soh, 20°C) (7)
2.8 (7) 1.94(6) 1 . 7 0( 6 )
-607 (7) -390(6) -310 (6)
.The
...
...
resultq obtain4 by W. Cruick$hank, I). J. MeZI~u~lnon, and H. E. Popkic in tlrr aurhor's undrrwaduate physical cheutietry Ialmmtor?. la ~rxtefullywknoalrtlgcd. w e of
Literature Cited
BARROW, G. M., "Physical Chemistry," 1st ed., McGraaHill Book Co., New York, 1961. MOORE,W. J., "Physical Chemistry," 3rd ed., PrenticeHalL Inc., New York, 1962. HEDESTRAND, G., 2.Phgsik. Chem., 2B, 428 (1929). I. F., AND KUMLER, W. D., J.Am. Chem. Soc. HALYERSTADT, 64,2988 (1942). M. J., Trans. Faroday Soe., 59,347 GRAY,P., AND PEARSON, 1196.1) ~-.~-,.
SMYTH, C. P., J. Am. Chem. Sac., 60,183 (1938). D. W., A N D GRAY,P., J . C h m . GRANT,R. F., DAVIDSON, Phys., 33,1713 (1960). Bull. Chem. Sac. Japan, 28,505 (1955). CHIBA,T., TARTB,P., J . Chem. Phys., 20,1570 (1952). SMITH,J. W., "Electric Dipole Moments," Butterworths Scientific Publications, Limited, London, 1955, p. 92. Volume 43, Number 2, Februory 1966
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