3007
H i l d e b r a n d ' s Equations f o r Viscosity and Diffusivity
Chemical Society, 1155 16th St., N.W., Washington, D. C. 20036. Remit check or money order for $4.00 for photocopy or $2.00 for microfiche, referring to code number JPC-73-3002. References and Notes (1) C. M. Criss and M. Salomon, "Physical Chemistry of Organic Solvents," A. K. Covington and T. Dickinson; Ed., Plenum Press, New York, N. Y., in press. (2) A. J. Parker, Chem. R e v . , 6 9 , l (1969). (3) 0. Popovych. CRC Crit. Rev. Anal. Chem., 1, 73 (1970). (4) J. F. Coetzee and C. D . Ritchie, Ed., "Solute-Solvent interactions," Marcel Dekker, New York, N. Y., 1969. (5) (a) R. G. Pearson, J. Amer. Chem. Soc., 85, 3533 (1963); (b) S . Ahriand, J. Chatt, and N. R. Davies, Quart. Rev., Chem. Soc., 12, 265 (1958). ( 6 ) R. Payne, private communication. (7) W. Dannhauser and A. F. Flueckinger, J. Phys. Chem., 68, 1814 (1964). (8) J. F. Coetzee, D . K. McGuire, and J, L. Hedrick, J. Phys. Chem., 67,1814 (1963). (9) See paragraph of end of paper regarding supplementary material. (10) J, N. Butler, "Ionic Equilibrium," Addison-Wesley, Reading, Mass., 1964.
(11) K. P. Anderson and R. L. Snow, J. Chem. Educ., 44,756 (1967). (12) J. Bjerrum, "Metal-Ammine Formation in Aqueous Solution," P. Haase and Son, Copenhagen, 1957. (13) L. G. Sillen, Acta Chem. Scand., 16, 159 (1962). (14) E. Guntelberg, Z. Phys. Chem., 123, 199 (1926). (15) L. G . Sillen and A. E. Martell, "Stability Constants of Metal-ion Complexes,"The Chemical Society, London, 1964. (16) D. C. Luehrs, R . T. Iwamoto, and J. Kleinberg, lnorg. Chem., 5, 201 (1966). (17) J. N. Butler, Anal. Chem., 39, 1799 (1967); see also J. N . Butler, D. R. Cogley, J. C. Synnott, and G. Holleck, "Study of The Composition of Nonaqueous Solutions," Final Report, Air Force Contract No. AF-19(628)6131,Sept 1969; AD No. 699-589. (18) R. Alexander, E. C. KO, Y. C. Mac, and A. J. Parker, J. Amer. Chem. Soc., 89, 3703 (1967). (19) J. N . Butler, J. Phys. Chem., 72, 3285 (1968). (20) M. K . Chantooni and I . M. Kolthoff, J. Phys. Chem., 77, 1 (1973). (21) R. L. Benoit, A. L. Beauchamp and M. Deneux, J. Phys. Chem., 73, 3268 (1969). (22) (a) R. Alexander, A. J, Parker, J. E. Sharp, and W. E. Waghorne, J. Amer. Chem. Soc., 94, 1148 (1972); (b) M. Salomon, in press. (23) M. Salomon, J. Phys. Chem., 74, 2519 (1970); J. Eiectrochem. Soc.. 118, 1610 (1971). (24) S. Ahrland, Struct. Bonding (Berlin), 1, 207 (1966);5, 118 (1968). (25) J. N. Butler, D. R. Cogley, and W. Zurosky, J. Electrochem. SOC.. 115, 445 (1968). (26) J. C. Synnott and J. N . Bulter, J. Phys. Chem., 73, 1470 (1969).
Hildebrand's Equations for Viscosity and Diffusivity H. Ertl and F. A. L. Dullien" Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada (Received duly 37, 1973) Publication costs assisted by the National Research Council of Canada
The liquid viscosities and self-diffusion coefficients of benzene, the monohalogenated benzene derivatives, and some normal paraffins have been determined as a function of temperature down to the proximity of the melting points. Based on these data it appears that Hildebrand's equation for viscosity is valid only down to T / T c = 0.46, whereas in its original form the equation suggested by him for self-diffusion coefficient is not valid anywhere in the temperature range studied. A small modification of this equation, however, has been found to describe the experimental diffusivities well in every case.
Introduction Hildebrandl has recently modified a viscosity equation originally suggested by Batschinski.2 Batschinski's equation is 1 7J--IC - - ___ n C where L; is the specific volume and u: and c are constants. w is a nearly constant fraction of the critical volume but Batschinski failed to find a correlation for C. The investigation of eq 1 for organic liquids by Al-Mahdi and Ubbelohde3 showed a wide variation of C with no trend recognizable. It is assumed by eq 1 and also by Hildebrand's modified form, eq 2, that fluidity, 4, (4 = l/viscosity) is governed by the volume increase. The form of the modified equation is
where VOis a constant and is the molar volume a t which viscous flow ceases. For a large number of liquids values of the constant parameters B and VOhave been given in ref l and 4. It has been shown there that VO is approximately a constant fraction of the critical volume. namely, V0/Vc 0.3. B, on the other hand, has been found to be linearly related to the chain length in the case of n-alkanes. A clear statement about the applicable range of eq 2 has not been given in either ref 1 or 4. It has been demonstrated for 6 0 2 and C3Hg that eq 2 represents the experimental results up to the neighborhood of the critical point. However, for even larger volume an increased dependence on temperature develops which has been explained by Hildebrand and Lamoreaux4 as due to the increase of translational momentum. The vast majority of the B and VOvalues have been obtained from viscosity measurements a t variable temperature and atmospheric pressure. It is evident, however, that the effect of pressure The Journal of Physical Chemistry, Vol. 77, No. 25, 7973
3Q08
H. Ertl
is not taken into account by eq 2, since a t constant volume (and variable temperature and pressure) this equation would require the viscosity to remain constant. In actual fact, however, the constant volume viscosity decreases with increasing t e m p e r a t ~ r e . 5 . For ~ C6H6 and 1/T.7 CC14 a t constant volume Eicher and Zwolenski8 have recently applied eq 2 over large temperature intervals including the region close to the freezing point. Based on least squares fits they concluded that eq 2 is not satisfactory in describing the temperature dependence of liquid viscosities over wide temperature ranges. In the reply by Hildebrand and Lamoreauxs it was pointed out that a possible explanation for the failure of eq 2 at temperatures close to the freezing point is given by an effect which Ubbelohde and coworke r P have termed “pre-freezing.” Molecules with a tendency of interlocking, resulting from branched and reentrant portions, give rise to the “pre-freezing” effect. However, as it is shown in this contribution, molecules with shapes which do not favor entanglement can also deviate from the predicted behavior of eq 2. On the other hand, a number of liquids do not exhibit a viscosity anomaly on approaching the freezing point. Part of the present paper will deal with the basic question: what is the inapplicable temperature range for the different liquids? It will be shown that the occurrence of deviations from eq 2 depends on the value of the reduced freezing temperature, TIT,. Hildebrandl has suggested that an equation similar t o eq 2 should also apply for the diffusion coefficient, D, if # is replaced by D. This results in
and F. A. L. Dullien
-
0
I
I
30
40
v$
VOLUME EXPANSION
ARBITRARY
Figure 1. Hildebrand plots for fluidity of
5
-
2
i
.~
O A O
LANDOLT-BORNSTEIN ( 1 4 )
0 A
4
ERTL and DULLIEN ( I / ) MELTING POINT NORMAL BOILING POINT
‘t
Tr=T/Tc=046
V
3
where B’ is a constant. Only for benzene and carbon tetrachloride has Hildebrand provided plots of D us. T which appear to be straight lines. These plots, however, cover only narrow temperature ranges. Over a wider temperature range they may not be straight lines as the more pertinent plots of D LIS. V are not straight lines either (see Figure 3). In the present work we show that the form of eq 3 is not suited to represent diffusivity data even over the limited temperature range for which eq 2 holds. A modification of eq 3 which gives D a stronger volume dependence has been found to represent the experimental data well.
20
IO
1
50
1 60 70 v,* [cm3/mo1]
I vn-pentanes __-
-1
,
vo[cm3/rno~]
:I
g
0.4
rJ J LL
Discussion Viscosity. The authors have presented in a previous paper experiment21 results for the viscosities of the monohalogenated benzenes between 20” and their freezing points.ll They also showed a relation between the occurrence of the “pre-freezing” effect and restricted rotation of the molecules.12 It was also pointed out that the viscosity of liquids with a reduced freezing temperature T f , < 0.46 deviates increasingly from the values extrapolated from the “high”-temperature behavior, as the freezing point is approached. This criterion was also found to hold in cases where other classifications according to the entropy of fusion or the molecular sphericity fail.l3 Since the transition from normal to anomalous behavior is a gradual one, the value of 0.46 has to be interpreted as a narrow temperature range around this value. Viscosity is said to exhibit anomalous behavior if it deviates from the behavior a t “high” temperature, i e . , well above the freezing point. For this range it is well known The Journal of Physrcai Chemisfry. Voi. 77, No. 25, 7973
Olt 1
, 2
I
3
,
,
,
4
5
6 7 8 9 1 0
R E L A T I V E V O L U M E EXPANSION
I
20
v-v, [%] VO
Figure 2. Log-log plots of fluidity vs. relative volume expansion of benzene and derivatives.
that an Arrhenius type equation is a suitable form to represent the viscosity-temperature dependence. Other forms of equations, e . g . , eq 2, are equally suitable. Since volume expansion is certainly the primary cause for the ease of viscous flow, eq 2 is superior to the Arrhenius type equation, due to its physical content. Figure 1 depicts the fluidity (4) behavior of three rz-paraffins. The viscosity data were taken from Landolt-Bornstein.14 In order t o save space, the x axis has been drawn as the molal volume minus an arbitrary constant which is indicated for each liquid. A straight line in this plot indi-
3009
Hildebrand’sEquations for Viscosity and Diffusivity
90
94
9s
102
Figure 3.
110
106
M O L A R VOLUME
114
I18
12 2
126
[ crn3/mo1e]
V
Hildebrand plots for diffusivity of benzene and derivatives.
TABLE I: Values of the Parameters in 1 / V = B ( W for r, > 0.46 B. Liquid
Benzene Fluorobenzene Chlorobenzene Bromobenzene lodobenzene n-Heptane n-Decane
cp-:
18.2 (18.5)a 15.5 15.0 12.7 11.2 (18.8) 20 [17.7Ib (15.2) 16.9 [16.6]
- Vo/Vo)
VO? crn3/rnoi
82.0 (82) 84.6 94.0 98.2 105.8 (130) 130.6 [129.1] (183) 183.0 [I831
6,
%
0.76 0.80 0.38 0.39 0.21
1 n
data
-
8 1
4
MELTING POINT NORMAL BOILING POINT
#
Tr = 0.46
4’ /
9
14 21 11
9
0.43
15
1.20
12
l o 6
0
n t-
U
W
R o u n d brackets: from Hildebrand.’ *Square brackets: from Hildebrand and L a r n o r e a ~ x . ~
8
cates agreement with eq 2 . For each liquid an anomaly is present. Similarly to the case of a n Arrhenius plot,12 the anomalies appear for reduced temperature values T , < 0.46. The magnitude of the deviations as seen in Figures 1 and 2 is misleading with respect to 7. Due to the form of eq 2 a given deviation of 1/11 will result in increasing deviations for 7 as l / ~gets smaller. The failure of eq 2 is also clearly shown in Figure 2 which is a log-log plot of the fluidity, 4, of benzene and the halogenated benzenes US. relative volume expansion, ( V - Vo)/Vo.The VO’S used were obtained from a least-squares fit to the 7 data for T r > 0.46. For these liquids also anomalous behavior is observed when Tr < 0.46. It is interesting that no anomalous behavior is observed for benzene as this substance freezes a t a high enough temperature (Tfr = 0.496 > 0.46) to preclude a noticeable reduction of rotational energy of the molecules in the liquid. As the example of the monohalogenated benzene derivatives clearly indicates, a relatively small loss in symmetry will lower the reduced freezing temperature sufficiently for the molecules to be closely crowded and the linear relation between fluidity and molal volume is violated. The values of the parameters B and VO are given in Table I. Also included in the table is the mean standard error, e , and the number of data points, n, employed in the least-squares fit. e = (100/n) X [ ( Y , - L ~ ~ / Y ~ ] ’ ”where , J and j are the experimental and best fit values, respec-
v, 3
a
P i LL
I
)
Figure 4.
Hildebrand plots for diffusivity of n-paraffins
tively. As seen from the numbers in brackets, quite close agreements have been obtained with the values of B and VO quoted by Hildebrandl and Hildebrand and Lamor e a ~ x Inspection .~ of the magnitude of t indicates excellent fit of eq 2 to the temperature range Tr > 0.46. The occurrence of viscosity anomalies of lower temperatures has been previously interpreted as due to a scarcity of free space which does not permit free rotation of molecules in the 1iquid.l’ This may lead to the formation of “clusters” which have to be considered as short-lived dynamic arrangements of adjacent molecules having coordinated movements. It is obvious from Figures 1 and 2 that performing a least-squares fit over the whole normal liquid range, as done by Eicher and Zwolenski,a is inappropriate for liquids extending into Tr < 0.46. The Journal of Physical Chemistry, Vol. 77. No. 2 5 , 1973
301 0
H. Ertl
TABLE I!:
Relevant Temperature
2-Methylpenlane 3-Methylpentane 2,2-Dimethylbutane n-Hexane p-Xylene rn-Xylene
ata for
the Isomeric Hexanes and p- and m-Xylene
-153.7
0.24 0.219
60.3
0.669
63.3
0.667
-99.9
0.354
-95.3
0.35 0.463
49.7 68.7 138 139
0.66 0.673
- 162.9
13.2 -47.4
0.365
0.665 0.665
224.9 231.2
-15.6
0.515
- 17.6
0.506
216.2 234.7 345 346
-13.7
0.53
TABLE Ill: Valuesof
v,
4"
POINT
Benzene Fluorobenzene Chlorobenzene Bromobenzene lodobenzene n-Heptane n-Decane
[crr13/mol]
C,HSF C6H5CP C, H 5 Br
810 91 4 96 7
C6h,I n-C7H,, n - CIOHP2
1038 1246 179 7
V
4
I , 1 , I 2 3 4 5 6 7 8 9 1 0 I L A T I V E VOLUME EXPANSION (V-VoV VO
1
'9 7 %
25
Figure 5. Log-log plots of diffusivity vs. relative volume expansion for n-paraffins,benzene, and derivatives. The same authors have recently reported accurate experimental viscosity data for three isomers of n-hexane.15 Hildebrand and Lamoreaux4 have employed their viscosity data and found straight lines in the 4 us. V plots. This, however, is no indication for the validity of eq 2 for temperatures down t o the freezing point since Table 11 clearly shows that the lowest experimental temperature, Tiow,is still far above the temperature, TO.46, corresponding to Tr = 0.46. As is known for the case of n-hexane, from the inspection of the Tf, values we also expect an anomalous viscosity behavior for the isomers, as Tfr is appreciably smaller than 0.46 for each liquid. Hildebrand and coworkers4 have found earlier that the molal volumes and viscosities of p - and rn-xylene have almost identical values over their common liquid range with no evidence of "pre-freezing" of the para isomer above its melting point of 13.2". The characteristic temperature data of the two liquids have also been included in Table 11. Again, the nonanomalous viscosity behavior observed for p-xylene and the anomaly found for the meta isomer are plausible in the light of their Tf, values. Tf,= 0.365 The Journal of Physical Chemistry, Voi. 77, No. 25. 7973
Liquid
-44 -42 -49 .-39
the Parameters in D = B [ ( V - V o ) / V o I m B X lo4,
MELTING POINT
[l BOILING
and F. A. L. Dullien
Vo,
cm2/sec cm3/moi
4.81 6.18 5.17 4.71 4.04 4.05 4.09
81.3 81.0 91.4
96.7 03.8 24.6 79.7
m
t,
%
1.34
1.2
1.80 1.58 1.49 1.46 1.51
1.9 1.4 4.2
3.3 2.9 4.4
1.38
n data
19 18 15 17 18 27 19
for m-xylene and thus is appreciably smaller than 0.46, forp-xylene is 0.463. whereas Tf, Diffusion. Figure 3 is a D L I S . V plot of experimental diffusivity datal1 of benzene and the monohalogenated benzenes. It is important that, unlike in the case of fluidity, a curvature is present at any temperature for all liquids and benzene is not an exception in this case. Therefore, the deviation from linear behavior cannot, in the case of diffusivity, be explained in terms of the rotational hindrance becoming important as the temperature is lowered. The curvature in the plots is a clear sign of the invalidity of the diffusivity equation, eq 3, suggested by Hi1debrand.l The same conclusions are recognizable from the plot for two n-paraffins, Figure 3. For comparison with the viscosity plots the position at which T , = 0.46 are also shown. It has been found11 that a modification of eq 3 in the form
D
=B
vf - (v,~
T
(4)
represents the experimental data well. Important as this difference is between eq 3 and 4, both equations propose that log D us. log [ ( V - V O ) / V Ois]linear. The exponent m is found to be larger than 1. This implies a greater dependence of diffusivity on volume expansion as compared with viscous flow. In Table 111 are listed the parameters of eq 4 which were obtained by nonlinear regression technique. Comparison of the VOvalues of Tables I and I11 shows that the values are very nearly identical but with VO of the diffusivity equation consistently larger. With respect to the E values the same trend is obtained for the halogenated benzenes. In agreement with the findings of Hildebrand and L a m ~ r e a u x the , ~ B values decrease with increasing molecular size and rotational inertia. No explanation is readily available for the values of B obtained for benzene and the two n-paraffins (Table 111) which do not fit into a pattern. Figure 5 is a plot similar t o Figure 2 except that it shows the dependence of diffusion, rather than that of fluidity, on relative volume expansion. It is of interest to notice a basic difference between the plots in Figures 5 and
301 1
R e a c t i o n s of B e a m s of LiCl and L i B r with KCI S u r f a c e s
2, respectively, While the lines in the log 4 us. log [(V Vo)/ Vo] plot are virtually parallel in the "high" temperature region, the lines in the diffusivity plot appear to converge. In spite of the different molecular shapes, the diffusion coefficients of the halogenated benzenes and normal paraffins are within a narrow band a t the boiling points. No trend of D for the halogenated benzenes a t fixed volume expansion is observable. The same is true with respect to the volume expansion a t the freezing point. On the other hand, the volume expansion a t the boiling points has about the same value for molecules of similar shape. Conclusions We can summarize the foregoing by the following points. (1)Equation 2 is not universally applicable for liquids. (2) Equation 2 is restricted to temperatures corresponding to Tr greater than about 0.46 at which unhindered molecular rotation is possible. (3) Equation 3, suggested by Hildebrand for self-diffusion, does not conform to experimental results. (4) The modified form of eq 3 (eq 4) represents the diffusivity well.
(8) L. D.Eicher and B. J. Zwoienski, Science, 177,369 (1972). (9) J. H. Hildebrand and R . H. Larnoreaux, J. Phys. Chem., 77, 1471 (1 973). (IO) A. R. Ubbelohde, "Melting and Crystal Structure," in "Phase T;ansitions," Solvay Institute, 14th Chemistry Conference, Interscience, London, 1971. (11) H. Ertl and F. A. L. Dullien, A.l.Ch.E. J., in press. (12) H. Ertl, and F. A. L. Dullien, Proc. Roy. SOC.,Ser. A, in press. (13) D. B. Davies and A. J. Matheson, Trans. Faraday SOC., 63, 596
(1967). (14) H. H. Landolt and R. Bornstein, "Zahlenwerte und Funktionen," 6th ed, Vol. I I, Part 5a, Sprlnger-Verlag, Berlin, 1969. (15) L. D. Eicher and 8.J. Zwolenski. J. Phys. Chem., 76,3295 (1972).
Appendix. Nomenclature
B B' C D m n u
V Vo
T
= constant, mol-I cm sec
= constant, cm2 sec-l = constant, cm2 sec-I = self-diffusion coefficient, cm2 sec-l = exponent (eq 4) = number of data points = specific volume, cm3 mol-I = molar volume, cm3 mol-1 = constant (eq 2 , 3 , and 4), cm3 mol-I = temperature
Greek Letters References a n d Notes J. H. Hildebrand, Science, 174,490(1971). A. J. Batschinski, 2. Phys. Chem., 84,643 (1913). A. A. K. AI-Mahdi and A. R. Ubbelohde, Trans. Faraday SOC., 51,
361 (1955). J. H. Hildebrand and R . H. Larnoreaux, Proc. Nat. Acad. Sci. U.
I$
= absolute viscosity, mol cm-I sec-l = mean standard error = fluidity (=1/7),mo1-I cm sec
w
= constant (eq I), cm3 mol-l
7 e
S.,
69,3428(1972). E. A. Moelwyn-Hughes, "Physical Chemistry," Pergarnon Press, New York, N, Y . , 1961,pp 717-719. A. A. Miller, Macromolecules, 4,757 (1971). A. A. Miller, J. Phys. Chem., 67,2809 (1963).
Subscripts c r
= critical
= reduced
Reactions of Beams of Lithium Chloride and Lithium Bromide with Potassium Chloride Surfaces Robert 8. Bjorklund and Joseph E. Lester*' Department of Chemistry and Materials Research Center, Northwesfern University. Evanston, lllinois 60201 (Received July 5, 1973) Publication costs assmted by the Materials Research Center, Northwestern University
As a test of an equilibrium treatment of gas-solid reactions, we have used a mass spectrometer to measure the reaction products of LiCl and LiBr molecular beams with KCl single crystals as a function of beam flux and crystal temperature. The data exhibit an increase in the dimerization of beam molecules and certain reaction products with decreasing crystal surface temperature. Using the calculated equilibrium partial pressures of the reaction products, one can correctly predict the trends of the experimental data with varying crystal temperature and beam flux, but not the relative measured concentrations of products. The agreement is better at higher temperatures.
Introduction Interpretation of the evaporation rates of the products of the vaporization of solids and of the products of gassolid reactions by the use of equilibrium thermodynamic properties was first attempted by Langmuir. His analysis
of vaporization rates using the kinetic theory of gases was quite successful in correlating the vaporization rates of several metals with their equilibrium thermodynamic properties. Recently, Batty and Stickney have applied a quasiequilibrium approach to analyze the data from the The Journal of Physical Chemistry, Vol. 77, No. 25, 1973
