Correlating Viscosity and Vapor Pressure of Liquids DONALD F. OTHMER AND JOHN W. CONWELL Polytechnic Institute, Brooklyn, N . Y .
M
ANY physical properties of liauids. of solids, and of liquid solutions of other liquids, solids, or gases have been correlated by a simple method of logarithmic plotting and the correspondingly simple algebraic equations (14). For vapor pressures and latent heats, l o g P = (L/L')log P'
+- c
Viscosity data may be plotted directly to give straight lines on log paper against a temperature scale readily calibrated from the vapor pressures of a reference substance. The method follows from that previously suggested for vapor pressures (14), vapor compositions, and other properties (17). A thermodynamic derivation indicates the soundness of the plot and of the governing equation: log fi = -(E/L)log P'
+c
The use of reduced temperatures correlates even better, since it tends to make the lines for all liquids converge in a narrow range at the extrapolated points corresponding to the critical. Fluidities, the reciprocal of viscosities, may be correlated by similar plots, either against the temperature scale or against the reduced temperature scale obtained by this method.
where at the same temperatures, P and P' are vapor pressures and L and L' molal latent heats, respectively, of two compounds (the latter in each case being that of a standard or reference substance), and C is a constant. Log P' really serves as the temperature variable and is obtained directly from vapor pressure data of the standard substance. LIL' is nearly independent of temperature. A log plot gives a substantially straight line whose slope (L/L')provides latent heat data a t any temperature for the compound in question from that of the reference substance. This vapor pressure plot was extended (16)for use with reduced pressures at reduced temperatures in order to increase the precision. Further applications were made to gas solubilities and partial pressures (19) and to the pressures of adsorbed materials from adsorbents (18). Other properties, such as vapor composition, equilibrium constant, activity coefficient, relative volatility, and electromotive force, were also shown to correlate as straight lines by this method (16,17); and the slopes of the lines were identified with the heats of vaporization and of solution. I n most cases the equation is not needed, and vapor pressure (or other function) is plotted on log paper (or logarithms are plotted on ordinary graph paper) by three steps: (a) Corresponding ternperatures and vapor pressures of the reference substance are read from a table; temperatures are indicated on the X-axis a t appropriate values of pressures, with ordinates erected accordingly, ( b ) Pressure (or other function) is plotted from the logarithmic scale of the Y-axis on the respective temperature ordinates. The same units do not have to be used on both the X and Y axis, since there is a constant ratio between any two units; this ratio would merely move the line up or down on the plot without changing its form of slope. (c) Points so obtaincd are connected by a line, usually straight.
(6.13). .
I
The equation of the straight lines in the upper graph of Figure 1 is :
+
log p - A log P C (1) where p is viscosity and P is vapor presaure of any liquid, both expressed in any desired units; A and C are constants. I n the usual case this equation will be less useful than the simple graph which, for a new compound (in most cases), gives a single straight line right up to the boiling point. Only two points are thus needed to establish the entire range of values, but a third intermediate point, also on the line, will check the continuity of the straight line. If breaks are shown, two points will be required for each straight line section. The plotted viscosity data are a t atmospheric pressure; in general, lines of such a plot must be isobaric. If there is a pressure change, there is a change in viscosity. For example, data at the boiling point may tend to fall slightly away from the indicated straight lines, since the experimental method may have included some application of superatmospheric pressure to prevent vaporization from interfering with the experimental determinations. Another useful function is fluidity, +, the reciprocal of viscosity; when a comparable plot of fluidity of the same liquids is made in the lower graph of Figure 1, a similar series of straight lines is obtained; because of the properties of logarithmic plotting, this plot is the mirror image of the upper graph of Figure 1, with the lines having identical algebraic slope but negative sign:
+ = l / f i ; - log 4 = log
P
From Equation 1: -log+= - A l o g P + C log + = A log P C
APPLICATION TO VISCOSITY DATA
Plotted by the s a n e method previously standardized for many functions, Figure 1in the upper graph gives viscosity of different substances w. temperature, indicated in the regular way from the corresponding vapor pressure of water as the reference substance. The correlation is good, and viscosities may be so plotted as straight lines for any of these materials; many others have also been checked. I n some cases the data for a substance are best expressed as a series of two or three connected straight lines.
While each section is straight, to indicate the validity of the relation over the particular temperature range, the breaks (and corresponding changes in slopes) are due to the change of the physical and often chemical nature of the material at that particular temperature. With water, for example, there is an abrupt breakin the line for viscosity between3OOand40"C.; this corresponds t o the postulated change in the molecular structure of water at 40" C.
-
(2)
THERMODYNAMIC BACKGROUND
The Clausius-Clapyron equation relates P (pressure), T (temperature), and L (latent heat of vaporization per mole) with R (the gas constant): dlnP L dT RT2 -i-
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TEMPERATURE IN DEGREES C.
Gueman (io) indicated a similar deriva-
tion:
where E is related t o the heat of fusion of the liquid or the activation energy of viscosity, If these equations are combined, there results at the same temperature : dlog9 E d log P = Or
-
dlogp
W P
-
-E
L
(3)
This relation indicates that a b g plot of viscosity or of fluidity against pressure at the same temperatures for a given substance would be a straight line with slope equal to the ratio of the energy of activation and the heat of vaporization. Others (1-4) have also contributed t o this idea of the relation of viscosities and vapor pressures of the same substance. It was previously shown (i4), however, that dlog P L
TEMPERATURE
IN
DEGREES C.
Zi5p-n
where P’ is the pressure and L’ the molar heat of vaporization of a second fluid. When this is combined with Equation 3:
VAPOR PRESSURE OF WATER IN mm. Figure 1. Logplot of Viscosity (above) and of Fluidity (belaw) of Eleven !Representative Liquids against Temperature Which is Obtained from the Corrwponcling Vapor Pressures of Water
RELATION TO
These may be integrated to give approximately: logp = - $ l o g P ’ +
E log 6 = E, log P’
+c
c (6)
Hence it follows that Equation 1 is correct, assuming the constancy of E/L; the proposed relation and method of plotting are thermodynamically sound within the assumptions of the derived equations. The relation E / L or its reciprocal L / E was discussed by Glasstone (8) and indicated to be a constant for a given substance by Eyring and Ewe11 (6,7,8). For large groups of substances it may be regarded as substantially constant, depending on molecular structure and related properties. As indicated here, this ratio may oonveniently be found or checked as the relation between the slopes of the lines on a log plot of vapor pressures and of a log plot of viscosities when both are made against temperatures as determined by the vapor pressurea of the same reference substance.
OTHER CORRELATIONS
A method which follows from the equation of Andrade (i,$, 3) is to plot logarithms of viscosity or logarithms of fluidity against 1/T, which is comparable t o the familiar plot of log P against 1/T. The latter does not give lines which are nearly so straight as does the plot of log P against log P’, as has been shown (1.6). The plot of log p or log 4 against 1/T also does not give lines which are nearly so straight as does a plot of log p or log 6 against log P‘. The reason is t h a t the 1/T plot assumes the constancy of E, which is not so correct as the assumption of the constancy of E / L in the present plot. I n other words, E varies as does L; while neither is constant, the ratio is practicrtlly constant. Numerous other more or less empirical relations of viscosity, temperature, and vapor pressure have been presented. One closely related to the present method is the so-called Porter rule @ I ) , a plot of temperature of one liquid us. temperature of another liquid where it has the same viscosity. This is directly analogous t o the Diihring plot for the temperatures at which two liquids have the same vapor pressure. It has been shown (1.4, $0) that reciprocal temperatures should be used for the Diihring plot; i.e., the reciprocal of the temperature of one substance should be plotted against the reciprocal of the temperature of
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Vol. 37, No. 11
INDUSTRIAL AND ENGINEERING CHEMISTRY
6ooF
400
300
z c
40
/-A I
I
Figure 2.
2
3
I 4
I
5
I 1 1 1 1 I I I I I I I I I 8 IO 20 30 40 60 80 IO0 VAPOR PRESSURE OF WATER IN mm.
6
I I 200 300
Log Plot of Fluidity of Eleven Representative Liquids against Reduced Temperature Which is Obtained from the Corresponding Vapor Pressures of Water
The rimcomity ia calibrated directly on a reciprocal male on the right,
80 that either fluidity or vieoosity may be read from the same chart. "him plottins against reduced temperature reduce. conaiderably tha width of the band of lines, as compared with Figure 1.
another where the vapor pressure is the same. This is also true for Porter's method of plotting, and a better approximation to straight lines is found when the reciprocal temperatures are used. I n that case exactly the same plot will be obtained on a different background grid work of coordinates as is obtained in the present plot. This follows directly and was explained in detail for the case of vapor pressure and temperature (14). It is, however, much more simple to plot on log paper than on reciprocal ruling paper; several other advantages are explained there for vapor pressure which hold here also for viscosity. Furthermore] i t is much more convenient to plot and to use a graph of viscosity as indicated above than a graph of temperatures a t which viscosities are equal. Others have also considered similar more or less empirical functions (11, 15) which may be related to the thermodynamic derivations given above. Many other empirical formulas have heen particularly related to the viscosities of petroleum fractions. USE OF CRITICAL CONSTANTS
The use of reduced temperature (16) tends to straighten log plots of reduced pressure as compared to those of pressure itself. Figure 2 is a plot of fluidity against the vapor pressure of water a t th'e same reduced temperature (TIT,). A series of ordinates was erected against the vapor pressure log scale on the horizontal axis at values of the corresponding temperatures divided by the critical temperature of water; the fluidity of the several materials was plotted on this grid work against the reduced temperature for that material. Since the lines against temperature, as determined by vapor pressure of reference substance (Figure l), are straight within experimental error of the determination of data, the use of critical temperature cannot improve the presentation substantially. An interesting point, however, is that if these lines are extended to their respective critical temperatures (Le.] where TB = T/T,= l), they tend to converge to a narrow band, except for water and the halogens. The fact that they do not converge to a point (i.e., the extended lines at
the critical points of all substances do not give the same indicated fluidities) is probably due to the divergence of the liquids from ideality, as well as the fact that the isobaric pressure (1 atmosphere) used for all determinations of viscosity represents different reduced pressure for each substance. The slopes of the lines for a given substance on the two plots [Figures 1 (lower graph) and 21 are in each case substantially the same. Most noticeable is the narrowing of the band of the lines of all compounds, both those plotted here and others which fall directly on top of this representative group. Presumably, a general relation might be worked out. If it could include corrections for the fundamental properties of the individual fluid, it might bring even closer together all of the viscosity data for fluids. Even on the present basis these data do not vary more than 50% as a maximum from the values of a representative material such as benzene. Also investigated was a plot of fluidity at the same value for the difference between critical and experimental temperature. This log plot of 9 against T, - T (from calculated values for water which are calibrated on the X-axis,against the log of vapor pressure) also gives straight lines, but there is no other reason to warrant its complicated preparation and use. VISCOSITY FUNCTION AGAINST VISCOSITY FUNCTION
Still other methods of expression which .follow directly from Figures 1 and 2 would be plots of viscosity of the compound against viscosity of a reference substance (not water, since i t forms a broken line) a t the same temperature; or of fluidity against fluidity a t the same temperature; or of viscosity against viscosity or of fluidity against fluidity a t the same reduced temperature. These plots would also give straight lines and could be employed. However, in the usual case i t may be somewhat easier to calibrate the log paper used for plotting on the basis of the vapor pressure scale rather than of the viscosity scale in order to obtain the temperature ordinates used for the actual plotting.
November, 1945
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Proof of the previoua paragraph is self-evident. In the upper graph of Figure 1,for example, the viscosity of both benzene and of luretic acid is a straightcline function aa plotted logarithmically against the vapor pressure of water at the me temperature. Hence, the viscosity of benzene (or of any other material which gives a straight line on this graph) would be a straightline tunction of the viscosity of acetic acid taken at the 8&me temperature. The same would hold for a plot of fluidity a t the same temperature against that of a reference substance, or for either viscosity or fluidity against that of a reference substance a t the same reduced temperature. The last mentioned plot is analogous to the plot of vapor pressure for one substance against vapor pressure of another at the same temperature (14)or critical temperature (16). The elopes of the lines on such a plot are the ratio of activation energies of viscosity of the two liquids (or of reduced activation energy). ACKNOWLEDGMENT
Appreciation is expressed to Alfred F.Schmutzler and Allan P. Colburn for interest and suggestions during the course of this study, and to Samuel Josefowitr for drafting the figures.
11 1s
LITERATURE CITED
(1) Andrade, E. N. da C., Nature, 125,582 (1930). (2) Andrade, E. N.da C., Phil. Mag., (7117,497,698 (1934). (3) Andrade, E. N. da C., Proc. Phys. SOC.(London),52,748(1940). (4)Creighton, H.J. M., J. Franklin Inst., 193,647 (1922). (6) Doraey, N. E., "Properties of Ordinary Water Substance". New York, Reinhold Pub. Corp., 1940. (6) Ewell, R.H.,J. C h a . Phys., 5,571,967 (1937). (7)Ewell, R. H., and Eyring, H., Zbid., 5, 726 (1937). (8) Eyring,H.,Ibid., 4,283 (1936). (9) Glasatone, S., Text-bok of Physioal Chemistry, New York. D.Van Nostrand Co., 1940 (10) Gusman, 3. de, Andes soc. eapafl.fls. quCn, 11, 353 (1913). (11) Irany, E.P.,J . Am. Chem. SOC.,60,2106,1938; Phil. Mag.. 33. 685 (1942). (12) Magat, M., Trans. Faraday SOC.,33,81 (1937). (13) Nieaan, Phil. Mag., 32,441 (1941). (14) Othmer. D.F.. IND. ENO.CIIEM.,32,841 (1940). t16j Zbid., 34, 1072 (1942). (16)& I& 36,669(1944). (ln Othmer,D.F., and Gilmont, R., Ibid., 36,858 (1944). Othmer, D.F.,and Sawyer, F. G., Ibid., 35,1289 (1943). (19) O t h e r , D.F., and White, R. E., Ibid., 34,952 (1942). (20) Perry,J. H., and Smith, E. R., Zbid., 25, 195 (1933). (21) Porter, A. W., Phil. Mag., [e] 23, 468 (1912).
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Hydrolysis and Catalytic Oxidation of Cellulosic Materials DETERMINATION OF STRUCTURAL COMPONENTS OF COTTON LINTERS 'H.I?.NICKERSON2 AND J. A. HABRLE'
......,..,.....,.
A
BOILING aqueous solution containing 2.6 moles of hydrochloric acid and 0.6 mole of ferric chloride per liter evolves carbon dioxide rapidly from glucose at a nearly constant rate which is proportional to the glucose present in the system (6). This reagent under the same conditions also liberates carbon dioxide from cellulosic materials but at slow, initial rates which increase as hydrolysis is continued (4, 6). These observations have led to the conclusions that some of the cellulose is hydrolyzed to simple sugars which, by oxidation, yield carbon dioxide, and that the course of the hydrolysis can be determined from instantaneous rates of carbon dioxide evolution (6). The technique and apparatus originally employed in this laboratory were modified and improved by Conrad and Scroggie (9),who verified And extended some of the earlier work. Hydrolysis-time curves obtained by this method for a number of different cellulosic materials have in common a shape that indicates rapid, early disintegration of part of the cellulose and a subsequent slower and more constant breakdown of the remainder. As Badgley and collaborators (1) stated in a recent review, such curves are generally regarded as evidence of a structural heterogeneity. The rapid initial hydrolysis represente the easily accessible or disordered fraction; the slower and more constant, subsequent hydrolysis, the denser less accessible or highly nrdered fraction of the material. However, these two parts of a typical hydrolysis-time curve are not sharply differentiable and, For previous papers in this series, see literature citations 8-8.
* Present addrens, A. C. Lawrence Leather Company, Peabody, Mass. 8
Preaent address. Crescent Heights, New Brighton. Pa.
,
Mellon Institute, Pittsburgh, Pa.
consequently, only a rough approximation of the distribution between ordered and disordered states can be obtained from the
tast. The study described in this paper was undertaken to yield deeper insight into the meaning of hydrolysis-time phenomena and, if possible, to develop a practical method of resolving the typical curve. Samples of cotton linters were subjected to acid hydrolysis for varying times under controlled conditions, and the resulting insoluble residues were washed and dried. This series of hydrocelluloses waa then investigated by the hydrolysiq-oxidation method. MATERIALS AND METHODS
A well-blended batch of processed, high-viscosity acetategrade cotton linters was used without additional purification as starting material. Hydrocelluloses, representing 0, 0.07,0.2,0.8, 2, 4, and 7 hours of treatment with boiling hydrochloric acidferric chloride rea ent or its equivalent, were isolated for investigation by the metfiods described below. The intact sample (0 hours) was boiled 4 minutes in water; the 0.07-and 0.2-hour samples were digested singly in boiling 2.5 N hydrochloric acid for 4 and 12 minutes, respectively. The rest of the series was prepared by refluxing a suitab!e batch of linters in h drochloric acid-ferric chloride reagent siphomng off portions o r t h e boiling BUS nsion 5-10 minutes short of the times indicated, and, as quickgas possible, filtering the hot SUIpension on c o m e alundum crucibles. Crucible and residue were then given preliminary washes with cool, dilute hydrochloric acid and water; finally, the crucible containin moist residue was transferred to boiling 2.6 N hydrochloric acid for 5-10 minutes to complete the digestion. A volume of 40 ml. of hydrolyzing solu-
