Z. LOSENICKY
4308
Thermal Conductivity of Binary Liquid Solutions
by Z. Losenickyl Department of Physics, Universitu College, Cape Coast, Ghana
(Received April 2.9, 1.968)
The calculation of the thermal conductivity of binary liquid solutions is discussed and the general equation, which does not include any empirical constants, is derived. The comparison with experimental values for nine various binary solutions shows that the deviation of calculated values from experimental data does not exceed 4%.
Introduction Several different empirical relations have been proposed for the calculation of thermal conductivity of binary liquid solutions at atmospheric pressure. Probably the most common is the additive rule
x = WlXl + wzxz
(1)
where w1and w2are the weight per cent of each component and X1 and x2 are the thermal conductivities of components. An attempt to use the Predvoditelev equation for thermal conductivity of liquids2 to calculate the thermal conductivity of binary liquid solutions was made by Tsederberg,a who also compared his results and the results of additive rule with available experimental data and showed that neither method was reliable (see Table I), since in many cases the calculated values differ from the experimental ones by more than
Table I
Solution components
CeHs, cc14 CzHbOH, HzO
CHaOH, CsHB C6H6, CzHjOH CH,OH, cc14 CC14, CHC13 C&, CHC1, CsHjC1, cc14 CHzOH, C&C1
Temp of oomponents,
--Max
OC
Eq 1
30 0 30 60 15 6 75 15 15 15 30 15
4.4 20.0 24.0 27.6 7.3 -0.1 -0.2 15.6 2.0 3.1 3.6 10.7
- exptl), 56-
dev (calod Eq 3
3.1 -3.9 -2.6 -3.1 2.0 -1.2 -1.3 3.3 2.1 3.4 3.5 1.5
a
14.1 4.5
...
-2.2 5.8 -0.6 -0.4 26.6 1.4 10.6 7.0 8.6
Tsederberg’s attempt to employ the Predvoditelev relation.a
10%. Tsederberg3 suggested certain rules in order to decide which of those two methods is better for a given solution. Other empirical relations for binary solutions were proposed by Barratt and i?;ett1etonj4 Bates,5 and Filippov and Novoselova.6-8 All these relations The Journal of Physical Chemistry
include at least one empirical constant. The least complicated is that proposed by Filippov
x = WlXl + WZXZ - PWlWZIXl - xzl
(2)
where the last term is the empirical correction of the additive rule, constant P having various values depending on particular compounds of a solution (see ref 8). In this paper mutual interactions of different pairs of molecules in a solution are considered, and it is shown that a simple general equation which does not contain any empirical constants can be derived. Theoretical Considerations The terms wJl and w2Xz in the additive rule can be considered as the approximate contributions of the first and the second components, respectively, to the thermal conductivity of the solution. The heat conduction is caused by the interaction among molecules, and in binary solutions there are different interactions between molecules causing the energy transport, namely, 1-1, 1-2, 2-1, and 2-2, where the numbers 1 and 2 indicate the molecules of the first and the second components and the order of these numbers indicates the direction in which the energy is transformed. The intensity of energy transfer depends on the intensity of molecular interactions which is given by the van der Waals interaction energy, the accurate determination of which by methods of quantum mechanics is very difficult, especially in the case of more complicated molecules. However, in the first approxi(1) This work was performed while the author was on leave from the Physics Department, Faculty of Mechanical Engineering, Technical University, Prague, Czechoslovakia. (2) A. S . Predvoditelev, Zh. E k s p . Teor. Fiz., 3, 217 (1933); 3, 230 (1934); 4, 43 (1934); 4 , 68 (1934). (3) N.V. Tsederberg, “Thermal Conductivity of Gases and Liquids,” Edward Arnold Ltd., London, 1965. (4) T. Barratt, “International Critical Tables,” Vol. 5, McGrawHill Book Co., Inc., New York, N. Y., 1929, p 227. (5) 0. K. Bates, G. Hazzard, and G. Palmer, Ind. Eng. Chem., Anal. Ed., 10, 314 (1938). (6) L. P. Filippov and I. 8. Novoselova, Vestn. Mosk. Univ., 3, 37 (1955). (7) L. P. Filippov, ibid., 8 , 67 (1955). (8) L. P. Filippov, Int. J. Heat Mass Transfer, 11, 331 (1968).
THERMAL CONDUCTIVITY OF BINARY LIQUIDSOLUTIONS mation, starting from London’s ideag about the origin of interaction forces, we can suppose that the interaction energy is approximately proportional to the product bld2, where d, are average cross sections of particular atoms or molecules. The intensity of interaction of a molecule of the first component with other molecules of the same component will be indicated as Pll, and PI2 will be the intensity of interaction of a molecule of the first component with molecules of the second component. Similarly, for a molecule of the second component intensities of interaction will be P22 and Pzl. If nl and n2 are numbers of molecules of particular components of a, solution, then the following approximate relations can be written for the mentioned intensities: Pll = kDlznl, Plz = kDlD2n2,Pz2= kd22nz, and P21 = kbldznl, where IC is a proportionality factor. With respect to the above discussion, the contribution of the first component to the thermal conductivity of a solution will be proportional rather to (P11 P1z)wl than simply to w1 and, similarly, the contribution of the second component will be proportional to (Pz2 Pz1)w2. Thus for the corrected additive rule we have
4309 m
n
where Pi,= kdidjn,and
(P11
+ P12)WlXl + (Pzz + P2l)WZX2
Table I1 : Cross Sections of Molecules
Formula
+ PlJWl + (P2z + P2l)WZ
=
1
Av cross section
-Cross sections in directions (IOZO), m2A B c
Ht0 CnHbOH CHIOH C& CGHsCl CCI4 CHCL
9.13 19.3 14.9 36.0 41.4 27.0 27.0
8.04 17.1 14.5 18.5 21.1 26.0 20.5
6.56 14.4 11.4 20.8 23.3 27.9 22.5
7.91 16.9 13.6 25.1 28.6 27.0 23.3
“\ H\
/OH
H -C-OH
H -C-C-H
H’
\H
H
CIH,CI
0
CaH,OH
/
‘“OH
(3)
where the proportionality factor IC in the intensity terms is given simply by the common condition used in the case of additive rules
(P11
n
CPipi = 1.
i - 1 j-1
+ +
=
n
(4)
The average cross sections, d, can be determined from the actual Eihape and dimensions of molecules, and in order to simplify these calculations we assume that all bonds are localized, and no correlations due to the delocalization in conjugated molecules are taken into account. The effective sizes for nonbonded contacts between atoms are given by packing radii, and the interatomic distance for two atoms connected by a bond is equal to the sum of their covalent radii.1° The determination of bond angles is based on the principle of maximum overlapping, and no interaction among bonds was considered. Thus in the case of carbon only “pure” states of hybridization were considered, namely, the sp3 state of hybridization with four tetrahedral hybrid orbitals and bond angles of 103.5” and the sp2 state of hybridi,zation with three hybrid orbitals situated in the same plane at an angle of 120” to each other. I n the case of the water molecule, the angle between the two OH bonds was taken as 104.5’ instead of 90”. The average cross section, d,was then determined as an arithmetic mean of molecular cross sections in three mutually perpendicular directions (Table 11). Considerations leading to eq 3 are the same in the case of solutions of n components, and so the generalization of eq 3 for n components seems to be straightforward
Figure 1. Cross sections of molecules were obtained by the projection in directions A, B, and C. The direction A is perpendicular t o the plain of the sketch, the directions B and C are indicated.
c,
I I
0
I
20
1
40
60
I
80
c
YO0 O/O
Figure 2. Thermal conductivity of ethanol-water solution: 0 , experimental data; - - - -, eq 1; , eq 3. (9) E’. London, 2. Phys., 63, 245 (1930). (10) L. Pauling, “The Nature of Covalent Bond,” Cornel1 University Press, Ithaca, New York, 1963. Volume 73, Number 13 November 1968
NOTES
4310
Results and Discussion The necessary values for covalent and packing radii were taken from Pauling,lO and they are given in Table 111. The values of Dgwith more particulars concerning how they were obtained are given in Table I1 and Figure 1. Table I11 radii, &----
-------Covalent
Single bond Double bond Triple bond
H
C
N
0
c1
0.30
0.77 0.62 0.55
0.74 0.62
0.72
0.99
r _ _ L -
Packing radii, A---
H
C
N
0
CI
1.15
1.65
1.5
1.4
1.8
Results of eq 3 were compared with experimental data of nine different binary solutions published by Filippov and Novoselova,6 Filippov, l 1 and Tsederberg. 12-14 Figure 2 shows Tsederberg's experimental data for the solution of water and ethanol compared with results of eq 1 and 3. Table I gives the maximum deviations of the values calculated by eq 1 and 3 from the experimental data, and it shows that in the case of eq 3 the maximum deviations do not exceed 4%. The reliability of used experimental data is 2-3%, The verification of generalized eq 5 has not been made because of the lack of experimental data. (11) L. P. Filippov, Vestn. Mosk. Univ., 6, 59 (1954). (12) N. V. Tsederberg, Tr. Mosk. Energ. Inst., 25, 13 (1955). (13) N. V. Tsederberg, Teploenergetika, 9, 42 (1956). (14) N. V. Tsederberg, Nauchn. Dokl. Vysshei Shkoly, Energ., 4, 189 (1958).
NOTES
Nitrogen-15 Magnetic Resonance Spectroscopy. Coupling Constants i n Hydrogen Cyanide
by Gerhard Binsch and John D. Roberts Gates and Crellin Laboratories of Chemistry,'t2 California Institute of Technology, Pasadena, California 91109 (Receiaed April 16, 1968)
Development of a promising new approach to the calculation of nuclear spin coupling constants,s capable of extension to atoms other than carbon and hydrogen, prompts us t o report the coupling constants in hydrogen cyanide-'5N which were measured in connection with a general study of spin coupling in nitrogen-15 labeled compound^.^ One and one-half grams of potassium cyanide-16N (96% W; R'Ierck Sharp and Dohme of Canada) was dissolved in 1.5 ml of water and 5 ml of 20y0 sulfuric acid added dropwise at room temperature. The liberated hydrogen cyanide was carried into a trap cooled t o -78" by a slow steam of nitrogen, After addition of the sulfuric acid was complete, the generating flask was first heated to 80" for 30 min and then to 105" for 15 min. The distillate containing the hydrogen cyanide was cooled to liquid nitrogen temperature and degassed. The hydrogen cyanide was then distilled into a bulb containing 5 g of Drierite and, after The Journal of Physical Chemistry
30 min at room temperature, condensed into an nmr tube which was sealed under reduced pressure. The proton nmr spectrum of the neat liquid taken immediately after preparation showed a sharp doublet centered at 3.60 ppm downfield from external TMS with a splitting of 8.7 f 0.1 Ha. The carbon-13 satellites (2 doublets) were separated by 274 i= 1 Hz. After 6 hr, a faint yellow color developed in the sample and the splitting of the main proton resonance and the carbon-13 satellites themselves disappeared-from then on only one single sharp line was observed. Apparently some catalytic agent formed, possibly associated with the oligomerization of hydrogen cyanide, which induced rapid intermolecular proton exchange. The corresponding coupling constants in acetylene are J l k ~= 248.7 Hz and J l k C H = 49.3 Hz.5 Multiplying J l a C C H in acetylene by the ratio y i 6 N / y l a C and assuming analogous bonding in acetylene and hydrogerr cyanide, one would predict 19.8 Hz for JWCHin (1) Contribution No. 3673. This paper is number VI11 in a series of papers on nitrogen-15 magnetic resonance spectroscopy. (2) Supported in part by Public Health Service Research Grant 11072-03 from the Division of General Medical Sciences. (3) J. A. Pople, J. W. McIver, Jr., and N. S. Ostlund, Chem. Phys. Lett., 1 , 465 (1967). (4) G. Binsch, J. B. Lambert, B. W. Roberts, and J. D. Roberts, J. Amer. Chem. Soc., 86, 5564 (1964). (5) R. M. Lynden-Bell and N. Sheppard, Proc. Roy. Soc., A269, 385 (1962).
