T H E RELATIONSHIP BETWEEN THE VISCOSITY OF A LIQUID AND T H E VAPOR CONCENTRATION GRAHAM W. MARKS Hears1 Mining .Building, University of California, Berkeley, California Received July 69, 1938
Further knowledge of the relationships between and among various physical and chemical properties of pure liquids and solutions is needed for a better understanding of the liquid state. Certain of these properties, for example, fluidity and vapor pressure, are only indirectly related to each other, because they are in turn dependent upon some other property. In 1868 Rellstab (6) compared the viscosities of various liquids a t temperatures a t which the vapor pressures were the same and also at a given temperature, but found no simple quantitative expression for his data. Later Porter ( 5 ) pointed out that in general, when comparison is made a t ordinary temperatures, liquids which are quite viscous have relatively low vapor pressures, whereas those which are quite fluent have relatively high ones. Bingham (2) has shown that if the fluidities of certain aliphatic ethers are plotted against their corresponding vapor pressures, the points fall on or near a single curve. For a number of substances he plotted the reduced fluidities, which were calculated by using the fluidity of heptane at the boiling point as a standard, against the corresponding vapor pressures and obtained curves which were quite similar. An expression connecting the fluidity and the corresponding vapor pressure of a pure liquid can readily be deduced from a theoretical foundation. The equation tl =
a~Se(7/T+dT)
(1) in which is the absolute viscosity, T is the absolute temperature, e is the base of the natural system of logarithms, and a, 8, y, and 6 are constants, has been derived from a thermodynamic and kinetic basis (1). I n the integrated form the Clausius-Clapeyron equation is 1nP =
AH -+K RT
Combining this with equation 1 p = R ~ K IT ~ ( K s / T- KaT + K4) where K I
. . . . K , are constants. 549
550
GRAHAM W. MARKS
Since P = CRT, the Clausius-Clapeyron equation can be written as lnC= -A/T-InT+B (4) where A = A H / R , B = K - In R, and C is the concentration of the vapor in moles per liter. Combining equations 1 snd 4 where - @ A - y = K", -1nK liquid. Obviously
+ Bfl = K',
and 9 is the fluidity of the
c = 7 ~ l e ( ~ -n K/ Or + R ~ )
(6)
Equations 3, 5, and 6 define both the liquid and its saturated vapor. EMPIRIC.4L FORMULAF
Since the above expressions are rather cumbersome for ordinary purposes of calculation, the writer has sought and found simpler relationships which are quite satisfactory for general usage. When the logarithm of the fluidity in rhes of a large number of liquids is plotted against the logarithm of the corresponding vapor concentration in mules per liter, calculated from the gas law C = P/RT, in which P is the vapor pressure, curves result which, in general, are straight lines over wide temperature ranges, The equation for such curves is 9 = 90
c"
(7)
m are constants characteristic of a given liquid. The former will be termed the "reference fluidity" and is, theoretically, the fluidity when the concentration in the vapor phase is 1 mole per liter. It is noted that
90and
The linearity of the curves for certain substances is shown in figure 1. In table 1 are given values of the constants m and 9 0 for fifty-two pure liquids. Vapor pressure data in millimeters of mercury and absolute viscosities in poises were obtained from the International Critical Tables and the Landolt-Bornptein Tabellen. In general, the procedure used was to read the viscosities from a viscosity versus temperature curve of a given liquid for those temperatures a t which vapor pressures were actually given in the afore-mentioned tables. After making calculations for fluidities and vapor concentrations, the best possible straight line was drawn through the log-log plot and the values of 9 0 and m calculated from data read from the curve. The method of least squares was not used in
VISCOSITY AND VAPOR CONCENTRATION
551
deducing the constants, as the above procedure was considered sufficiently accurate. By means of the above empirical equation fluidities can be calculated which agree with experimental values with an accuracy, in general, of within about 1 2 per cent for the liquids listed and over the temperature ranges given. Many of the available vapor pressure data are inaccurate.
. LOG, C
FIG. 1. Curves €or a number of liquids showing the linear relationship between the logarithm of the fluidity, 'p, in rhea and the logarithm of the vapor concentration, C, in moles per liter. Curves: 1, o-toluidine; 2, n-butyl alcohol; 3, ethyl alcohol; 4, isobutyl alcohol; 5 , ethyl propyl ether; 6, aniline; 7, methyl ethyl ketone; 8, formic acid; 9, isoamyl alcohol; 10, iodine.
Whether the relationship applies over aider temperature ranges than those given can only be determined when further overlapping values for viscosities and vapor pressures are available. Deviations of certain of the curves from straight lines occur a t lower temperatures and are, perhaps, due to association. Water, which shows an anomalous behavior in so many properties, likewise does so in this case. It would be of interest to compare the log fluidity-log vapor concentration
.
552
QRAHAM W MARKS
TABLE 1 Values for certain pure liquids of the constants PO and m i n the exvression 'p = (OO@ LIQUID
I
TEMPERATURE RANQm
m
PO
0.192 0.255 0.241 0.310 0.046 0 . 325 0.545 0.208 0.232
366 328 167 536 136 1070 645 398 996 1003 1079 1130 1046 930 1100 1146 1112
.
*C
Chlorine . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bromine . . . . . . . . . . . . Iodine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sulfur. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mercury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carbon dioxide . . . . . . . . . . . . . . . . . . . . . . . Carbon disulfide . . . . . . . . . . . . . . . . . . . . . Hexane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heptane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Octane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isopentane . . . . . . . . . . . . . . . . . . . . . . . . . . . Diisopropyl . . . . . . . . . . . . . . . . . . . . . . . . . . Benzene . . . . . . . . . . . . Toluene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethylbenzene . . . . . . . . . . . . . . . . . . . . . . . . o-Xylene7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . m-Xylene . . . . . . . ........ p-Xylene$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclohexane . . . . . . . . . . . . . . . . . . . . . . . . . Methylene chloride. . . . . . . . . . . . . . . . . . . Chloroform . . . . . ............... 1,l-Dichloroethane . . . . . . . . . . . . . . . . . . . 1,2-Dichloroethane . . . . . . . . . . . . . . . . . . .
1,1,2,2-Tetrachloroethane ............ Tetrachloroethylene . . . . . . . . . . . . . . . . . . Trichloroethylene . . . . . . . . . . . . . . . . . . . . Methyl iodide . . . . . . . . . . . . . . . . . . . . . . . . Ethyl iodide . . . . . . . . . . . . . . . . . . . . . . . . . Iodobensene . . . . . . . . . . . . . . . . . . . . . . . . . Methyl sulfide . . . . . . . . . . . . . . . . . . . . . . . . Methyl alcohol ....................... Ethyl alcohol. . . . . . . . . . . . . . . . . . . . . . . . Propyl alcohol . . . . . . . . . . . . . . . . . . . . . . . n-Butyl alcohol. ..................... Isobutyl alcohol . . . . . . . . . . . . . . . . . . . . . . Isoamyl alcohol . . . . . ........... Ethyl ether . . . . . . . . . . . . Methyl propyl ether . . . . . . . . . . . . . . . . . Ethyl propyl ether . . . . . . . . . . . . . . . . . . .
* Does not fit below 40°C .
t Does not fit below 20°C . $ Does not fit below 3OoC.
(-80) .(-40) (-7.3) .35 114 .170 120 .150 0-340 40 .100 5 .29
0-50 0 .70 0 - 100 0 .130 (-30) .50 0 .30 90 7.8 . 0 .110 0 .130 20 .140 10 .140 30 .140
20-40 0. 30 (-10) .61 0 - 100 55 7. 0 .81 25 .80 40 .110 80 25 . 0-ao 0-60 5 - 150 0-40
0-60 0 .70 0-100 20 . 75 60-100 10 .130 (-100) .50 0. 35 0-80
0.228 0.237 0.272 0.304 0.319 0.291 0.289 0.287 0.228 0.292 0.434 0.271 0.255 0.352 0.280
0.304 0.326 0.234 0.215 0.255 0.241 0.237 0.235 0.296 0.038 0.421 0.330 0.530 0.444 0.230 0.255 0.259
1000 1264 1108 675 598
664 745 680 779 576 599 572 593 661 832 803 73.6 898
616 1388 1261
999 1078 1111
553
VISCOSITY AND VAPOR CONCENTRATION
TABLE 1-Concluded LIQUID
II -
TEMPERATURE RANQE
m
‘PO
0.223 0.268 0.401 0.295 0.235 0.248 0.186 0.244 0.329 0.361 0.253
873 1017 736 1425 696 766 502 759 947 1427 712
‘0.
Acetone, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (-90) - 56.3 Methyl ethyl ketone.. . . . . . . . . . . . . . . . 20 - 50 Formic a c i d . . . . . . . . . . . . . . . . . . . . . . . . . 10 - 100 20 - 100 Acetic a c i d . . . . . . . . . . . . . . . . . . . . . . . Propionic a c i d . . . . . . . . . . . . . . . . . . . . . . . 20 - 140 Butyric a c i d . . . . . . . . . . . . . . . . . . . . . . . . 20 - 155 Valeric a c i d . .. . . . . . . . . . . . . I 6 0 - 100 Isobutyric acid. . . . . . . . . . . . . . . 30 - 155 . . . . . . . . . . . . . . . . . . . . . . 50 - 144 40 - 100 80-180
5 Does not fit below 80°C. curve for deuterium oxide with that of ordinary water, since it seems that the former is less associated (3). However, available data for both vapor pressures and viscosities of heavy water do not overlap sufficiently in temperature range to make a fair test. Molten sulfur exists in a t least two modifications and the temperatureviscosity curve has a maximum, whereas the vapor pressure rises continuously with temperature increase. The log-log curve is shown in figure 2, but the linear relationship holds only up to about 15OOC.
LOG-C
FIG.2. Liquid sulfur. Plot of the logarithm of the fluidity, ‘p, in rhes versus the logarithm of the vapor concentration, C, in moles per liter.
554
SRAHAM W. MARKS
The equation rp =
CPOC"
can be written in the form
P = RT(~/~,)'/*
(9)
since C = P/RT. Unless we consider the possible existence of supersaturated or subsaturated vapors, there is only one independent variable in this expression, for fixing one determines the magnitude of the other two.
7
2
VAPOR CONCENTRATION, MOLS/ LITER
FIG.3. Plot of the fluidity, c, in rhes a t the boiling point versus the vapor concentration, C , in moles per liter for a number of compounds. Points from left to right: Curve A, nonane, octane, heptane, and hexane: curve B, di-n-butyl, dipropyl, ethyl isobutyl, ethyl propyl, methyl propyl, and diethyl ethers. Curve F: open circles, p-xylene, m-xylene, toluene, and benzene; closed circles, o-xylene and ethylbenzene. Curves C, D, and E: chlorides, bromides, and iodides, respectively; dotted curves, 1 t o 6 inclusive, phenyl, isobutyl, propyl, allyl, isopropyl, and ethyl, respectively. Point a t 7, methyl iodide.
Assume, for a given liquid, that conditions are such that a supersaturated vapor exists, then
where p is equal to mqo/RT. Next consider, theoretically, such conditions that the vapor pressure does not change with change in temperature, then
where
x
= mqo(P/R)*.
Thus the total differential is
drp = pP"-'dP
- xT-"-'dT
VISCOSITY AND VAPOR CONCENTRATION
555
Bingham (2) found that the curves connecting the plotted fluidities of certain aliphatic hydrocarbons, ethers, and halides a t their boiling points are linear. In figure 3 are plotted data showing the linear relationship between vapor concentration and fluidity a t the boiling point for certain organic halides, members of the homologous series CnH2n+2, aromatic hydrocarbons, and ethers. Although points for quite diverse chlorides, bromides, and iodides lie on or near their respective straight lines, the chloride, bromide, and iodide of any given radical do not. Fluidities a t the boiling points were obtained by extrapolation of the fluiditytemperature curves. I5 14 13
2
0 12
14
;13
sll
w &
> 10
Y
12
2 It
L S
0
38
aP IO
& W
$ 9
7 S 6
g* 8 2 7
; 5 t;4
36
e 3
5 5
2
24
I
w
2 3 c
0 I 2 3 4 NUMBER OF CARBON ATOMS
~ 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 REFERENCE FLUIDITY, b .IO'
FIG.4 FIG 5 FIG.-1. Curves showing the alternation in the reference fluidity, (00, with odd and Curve 2, alcohols even number of carbon atoms. Curve 1, acids C.Hi.+iCOOH. C,H~,+IOH. The carbon atom of the carboxyl group of the acids has been excluded. FIG. 5 . Variation of the internal pressure in atmospheres with the reference fluidity,
(PO,
for a number of liquids.
When the reference fluidities of alcohols and acids belonging to the homologous series C,H2,+10H and C,Hz,+lCOOH, for which data are available, are plotted against the number of carbon atoms present, alternations occur in the rwulting curves (figure 4). It is to be expected that in the case of two liquids, in one of which the attractive forces acting among the molecules are relatively greater than in the other (when comparison is made under like conditions), the fluidity of the latter liquid will be the greater. That such is the case is shown by the curve of figure 5, in which the internal pressures of a number of liquids
556
GRAHAM W. MARKS
are plotted against the fluidities,l PO. The internal pressures of liquids are not accurately known. Data used were those of Hildebrand (4). VARIATION OF FLUIDITY WITH TEMPERATURE
Since
‘In’ -dT
- AH/RTZ
and
P
= RT(p/qJ”m
then
and
Thus a semiempirical expression showing the variation of fluidity with absolute temperature is obtained. A test of this equation is given in table 2. AH,in international joules per mole, has been calculated for ten TABLE 2 Test of the app&ation of 14 of text - euuation ..
LIQUID
(PI
Pt
PI
Pa
TI
1 1 T1
--Carbon disulfide . . . p-Xylene. . , . . , . . , . m-Xylene... I . . . . . Benzene.. . . . . . . . . Trichloroethylene.. . . . . . , Aniline . . . . , , . . . . . . Ethyl iodide. , . , . . . Chlorine.. . . . . . . . . o-Toluidine. , . . . . . Ethyl propyl ether
rhea
rhea
291.5 289.9 142.3 129.8
333 432.7 427.4 142.0 401.6 3.4! 238.0 40
854.0 624.9 448.9 300
303 353 283 280.9
323 403 393 325.7
190.5 91.4 138.9 131.6 52.1 282.5
260.4 153.1 225.2 198.0 90.2 420.2
453.0 96.6 364.0 594 10.5 472.1
303 353 273 193 323 283
343 393 323 233 353 323
mm.
94 18.0 41.5
58.7 2.1 89.1
1
PERCEm AH (CLAW- DEVIATIOA PBom 01u& (EQUACLAPEY- CLAUBIUBCLAPEY110N14) RON EQUARON TION) REEULT
A€€
joule8 permole
percent
29,230 34,530 41,060 34,760
27,680 35,070 40,950 34,210
f5.59 -1.54 +0.27 +1.64
34,110 48,370 31,870 21,600 50,870 31,650
34,000 48,480 31,860 21,630 50,860 31,8801
f0.32 -0.23 +0.03 -0.14 4-0.02 -0.09
1 In making comparisons i t would probably be better to reduce the vapor concentrations t o standard conditions, and thus values of powould be given when the vapor concentration is 1 mole per liter under such conditions. However, this procedure changes VU only slightly for ordinary liquids and for purposes of plotting is of no significance.
557
VISCOSITY AND VAPOR CONCENTRATION
liquids chosen a t random, using the above equation and the integrated form of the Clausius-Clapeyron equation. Column 10 shows the percentage deviations from the Clausius-Clapeyron results to be small. For other equations expressing the variations of viscosity or fluidity with temperature, the reader is referred to Bingham (2) and to Souders (7). It has been found that the relationship p =
DIT"
(15)
where p is the density of the liquid and D is a constant characteristic of the liquid, approximately expresses the variation of density with absolute temperature. This equation utilizes the constant m defined by equation 8. TABLE 3 Test of the applicafion of the equation (T2/TlIm= LIQUID
(T~TI)"' p ~ / p t
11 ~
Carbon tetrachloride Chloroform Formic acid Methyl iodide Ethyl iodide n-Propyl alcohol Methyl ethyl ketone n-Butyric acid Iodobenzene
l'
0
l I
o O
'
0
1
20
1
25 5
1
50 80
150
PER CENT DIFFERENCE
_ _ ~ 1 0493 1 0498 -0.05 1 0579 1 1 0870 -2.68 1 0563 1 0407 +1.54 10269 1 0363 -0.90 1 0490 1 1 0749 -2.47 1 1404 1 1194 $1.87 1 0265 I 1 0409 -1.39 1 0429 1 0607 -1.68 1 1046 1 1377 -2.91
1
40 55 40 30 60 loo
l - 10o
pl/p2
I
1'
A test of this relationship for a number of liquids is given in table 3. It has been applied in the form PlIP2 =
(2)"'
SOLUTIONS: LIQUID CONSTITUENTS ONLY The fluidity, @, of an ideal solution consisting of two components is best given, according to Bingham (2), by the equation = W]
+
- k(a1 - W)(ZII -
(16) where al = volume fraction of component A, 'p1 = fluidity of pure component A, u1 = specific volume of A, w = weight fraction of 4,and k = a constant. Subscript 2 refers to component B. For two or more components equation 16 can be written in the form 3 = alrpl
+
UZ(P~
+ .... +
~ Z W
ancpn
~ 2 )
- ~(Au)
(17)
558
GRAHAM W. MARKS
It is assumed, as by Bingham (2), "that if the volume is decreased for any reason by an amount Au, the fluidity will be decreased by an amount which is some function of this, ~ ( A u ) " . Since
Consider the case when Raoult's law holds, and when ml = mz = . . . . mn,which may be true under ideal conditions and is approximately true for a number of liquids, as is shown in table 1. p1 = P ~ z Ietc. ,
Then
Hence
When volume changes occur with temperature and are expressed as functions of the absolute temperature,
+ . . . + knfrt(T)PI - f(Av)
= klf,(T)P;" 4- kz.f*(T)Pz"
(21)
s
TABLE 4 Application of the equation 9 = p0Cn1to data concerning solutions of ( a ) benzene and carbon tetrachloride and ( b ) ethyl alcohol and water* (a)
B E N Z E N E A N D CARBON TETRACHLORIDE. TEMPERATURE RANGE 0-40°C.
'
1
Weight per cent benzene I
11 40 22 37 43 79 67 03
1
I1
0 0 0 0
374 376 366 352
1 1
1
Bo
808 867 921 935
I
(b)
ETHYL ALCOHOL A N D WATER. TEMPERATURE R A N O E 20-75'C.
1 Weight percent ethvlalcohol 1 50 i
o
60 70 80 90
0 0 0 0
~
1
1
1 1
' 583 561 524 486 441
1 I
1 1
1272 1178 1047 962 901
* For values of m and qpo for the pure components refer t o table 1. C was calculated by using the sum of the vapor pressures of the two constituents.
VISCOSITY AND VAPOR CONCENTRATION
559
The fluidity of the ideal solution is given in terms of the partial pressures of the constituents and the absolute temperature. I t is of interest to learn whether or not equation 7 is applicable to solutions consisting of liquid components. Mixtures of benzene and carbon tetrachloride are known to approach ideality (2). Equation 7 was found applicable to data covering a temperature range of 0' to 40°C. for these mixtures (table 4). However, in the case of mixtures of ethyl alcohol and water (20' to 75°C.) the equation does not fit when the weight per cent of alcohol is less than about 50 per cent. The curves are then similar to those for water, which is t o be expected, as the mole fraction of water present is about 0.75 or higher. SUMMARY
Empirical and theoretical expressions have been given showing the relationship between viscosity or fluidity and vapor pressure or vapor concentration. A semi-empirical equation has been deduced for the variation of fluidity with absolute temperature. Theoretical relationships have been given concerning the fluidity of solutions. REFERENCES (1) ANDRADE: Nature 126, 309 (1930). DRWCKER: Z. physik. Chem. 92, 287 (1918). DUNN:Trans. Faraday SOC.22, 401 (1926). DE GUZMAN: Anales soc. expafi. fis. qufm. 11, 353 (1913). IYER:Indian J. Phys. 6(14), 371 (1930). BENDALL AND MONROE:J. Am. Chem. SOC.39, 1799 (1917). LEDERER:Kolloid-Beihefte 34, 270 (1931). MAXWELL: Phil. Mag. 36, 129 (1868). RAMAN:Kature 111, 532, 600 (1923). SAEPPARD: Nature 126, 489 (1930). (2) BINGHAM: Fluidity and Plasticity, 1st edition. McCraw-Hill Book Co., Inc., New York (1922). (3) BINGAAM AND STEARNS:J. Chem. Phys. 2, 107 (1934). International. Critical Tables, Vol. IV, p. 19. hlcGraw-Hill (4) HILDEBRAND: Book Co., Inc., New York (1928). (5) PORTER: Phil. Mag. 23, 458 (1912). (6) RELLSTAB : Uber die Transpiration hornologer Fliissigkeiten. Inaugural dissertation, Bonn, 1868. (7) SOUDERS: J. Am. Chem. SOC.69, 1252 (1937).
