2733
NOTES
dp/dT = p(b
- cTC - 2cT
+
Table III : Water's Latent Heat of Vaporization from NaCl(aq) at 250"
where Ah is the specific latent heat of vaporization, v, is the specific volume of the substance in the vapor phase, and VI is the specific volume in the liquid phase. Equation 5 was tried out a t nine temperatures for water and the Ah values calculated by use of eq 5 are compared with steam table values in Table 11. The deviation, A, similar to the A in Table I, were calculated by the relation l O O ( A h c a 1 c d - A h t a b l e ) / A h t a b l e .
hi = Concn, m
0 1 2
3
joules
AhgL
h,, joules
1715.1 1950 1967 1981
2800.9 2850 2855 2860
hg
-
AhgL joules
1085.8 900 888 879
I11 shows that the values are shifting in the right diTable 11: Water's Latent Heat of Vaporization at Various Temperatures
T ,'C
Ahcalcd, joules
Ahtables joules
273.16 323.16 373.16 423.16 473.16 523.16 573.16 623.16 643.16
2479.4 2378.1 2256.7 2113.1 1928.8 1716.5 1415,3 914.9 432.2
2500.8 2382.2 2256 7 2114.4 1940.4 1716.1 1404.3 893.0 438.4 I
A
-0.85% -0.17 0.0 -0.06 -0.60 0.08 0.78 2.36 -1.41
To extend eq 5 to aqueous solutions of salts, the partial volume of water in the liquid solution phase presents a problem. To circumvent this difficulty, the suggestion made by Gardner, et u Z . , ~ in their osmotic coefficient calculations, that the specific volumes of water in the pure state could be used in place of the partial specific volumes in solution without much error was adopted after the following test. The total ionic volume for each concentration was calculated from crystallographic data6 and was assumed to remain unchanged with changes of temperature. The molar volumes of the solutions were calculated from the data6 on the specific volumes of NaCl solutions under different conditions of temperature and pressure. We found that up to a temperature of 85" and pressures up to 1000 bars, the partial molar volumes of water in solution were almost the same as the mole volume of water under the same conditions of temperature and pressure. The specific latent heats of vaporization of water from sodium chloride solutions a t the concentrations 1, 2, and 3 m were calculated for the temperature 250". To test whether these values were realistic, they were subtracted from the specific enthalpies of steam under the various pressures a t 250" (steam tables). Table
rection. Acknowledgments. The author wishes to thank Drs. C. F. Baes, Jr., and B. A. Soldano of the Oak Ridge National Laboratory for their encouragement and enlightening criticisms during the preparation of this paper. Thanks are also due to Professor R. M. Fuoss for his constructive criticisms. The research was sponsored by the U. S. Atomic Energy Commission under contract with Union Carbide Corporation. (5) R. E. Gibson and 0. H. Loeffler, Ann. N . Y. Acad. Sci., 51, 727 (1948). (6) L. Pauling, "The Nature of the Chemical Bond," Cornel1 Uni-
versity Press, Ithaca, N. Y.
Surface Areas from Mercury Porosimeter Measurements
by Hillar M. Rootare American Instrument Company, Silver Springs, Maryland
and Carl F. Prenzlow Chemistry Department, California State College at Fullerton, Fullerton, California (Received September 8 , 1966)
The mercury porosimeter, developed by Ritter and Drake,lS2 is commonly used to measure the pore-size distribution of powders. I n these measurements the pores are usually assumed to be cylindrical in shape. At least two groups of workers3s4 have calculated rough surface areas from mercury porosimeter data. In each case surface areas were calculated as a "check" (1) H.L. Ritter and L. C. Drake, Ind. Eng. Chem. Anal. Ed., 17, 782 (1945). (2) L. C. Drake and H. L. Ritter, ibid., 17, 787 (1945).
(3) L. G.Joyner, E. P. Barrett, and R. Skold, J. Am. Chem. SOC., 73, 3155 (1951). (4) P. Zwietering and H. L. T. Koks, Nature, 173, 683 (1954).
Volume 71, Numbm 8 July 1967
NOTES
2734
on a particular pore-size distribution and in each case cylindrical pores were assumed in the area calculations. T o our knowledge no serious effort has been made to use the mercury porosimeter as a reliable surface area measuring device. We wish to demonstrate that surface areas can be calculated directly from standard porosimeter pressurevolume curves without assuming any particular pore geometry. Furthermore, we wish to show that areas obtained from porosimeter data compare favorably with areas obtained from Brunauer, Emmett, and Teller (BET) measurement^,^ provided mercury does not wet the powder being measured, provided porosimeter pressures are high enough to force mercury into the smallest powder pores, and provided no ink bottle pores are present.
Theory If one neglects gravity effects such as buoyancy of individual powder particles in mercury, the reversible work W required to immerse a mass of powder with an area of 1 cm2is defined6by the relation
w=
71
-
712
(2)
which has heen advanced by Harkins6J in order to estimate the reversible work of immersion W from measurable quantities. In this expression y~ is the surface free energy of liquid mercury in vacuo (480 ergs/cm2), 8 is the contact angle between mercury and the solid surface, and R is the spreading pressure for any mercury vapor adsorbed on the otherwise uncovered solid surface. In the case of mercury intrusion on low-energy (nonwetting) surfaces, ?r is negligible; hence, the work required to immerse an area da of powder surface is equal to dW
= y L cos
8 da
(3)
In the mercury porosimeter this work is supplied when an external pressure P’ forces a volume of mercury dVt into the pores of the powder. If we assume no ink bottle pores are present, spontaneous intrusion cannot occur; hence, we may equate work terms to obtain the expression y~ cos 8
da
=
If one further assumes that The Journal of Physical Chemistry
-P’dVt
y~ and 8 do not
a =
-1P‘dVt y L COS
(5)
e
which may be used as a basis for calculating surface areas from porosimeter measurements. To utilize eq 5, the quantity ,fP’dVt must be evaluated by graphically integrating a porosimeter curve between appropriate integration limits. Strictly speaking, the upper limit should be Vt = V,, a t P‘ = P,,, (if V,, is defined to be the total pore volume and P,,, the pressure required to fill the narrowest pore), whereas the lower limit should be Vt = Oat P’ = 0. In practice, the measured volume of intruded mercury in a porosimeter experiment, V , differs from Vt by an experimental error Vo, which is the unmeasurable volume of mercury intruded during sample preparations. Fortunately, this error has a negligible effect on JP’dVt because pressures are small when the volume V ois intruded. As a consequence, we may write
(1)
In this relat>ion y1 is the surface tension of a clean, outgassed solid surface in vacuo and y12 is the interfacial tension between the solid surface and mercury. Since neither y1 nor ylz can be measured directly, one must utilize the expression
w = y1 - y12= yL COS e +
pressure: this equation may be integrated to furnish eq 5
(4) vary with
-’
m cos 8yL
Jvm“
PdV
(6)
in which we have further defined A to be the area per gram of powder, m to be the mass of powder used in the porosimeter experiment, V,,, to be the apparent total pore volume (ie., Vmax = V,, - V,), and P to be the measured porosimeter pressure. If porosimeter measurements have been made a t 25” so that yL = 480 dynes/cm and if we take 8 = 130°,the right-hand term of the equation becomes
A - 0.02253 S,’m*x PdV m
(7)
This area will be calculated in square meters per gram if P is measured in psia and m is in grams. In practice, the upper integration limit (I‘m,,, V,,,) may be readily recognized as the beginning of a high-pressure plateau. This plateau occurs after all the pores of the powder have been filled with mercury and further small-volume changes are caused by the ( 5 ) S. Brunauer, P. H. Emmett, and E. Teller, J. Am. Chem. Soc., 60, 309 (1938). ( 8 ) H. B. Weiser, “A Textbook of Colloid Chemistry,” 2nd ed, John Wiley and Sons, Inc., New York, N. Y., 1949, p 79. (7) W. D. Harkins and H. K. Livingston, J. Chem. Phys., 10, 342
(1942). (8) L. C. Drake, I d . Eng. Chem., 41, 785 (1949).
NOTES
2735
inherent compressibility of mercury and the powder particles themselves. If a porosimeter curve is corrected for compressibility, the end point (P,,,, V,.,) should occur at the beginning of a truly horizontal plateau. Otherwise the plateau will have a slight slope dV/dP given by the equation
dV/dP =
CHgmH,
-k Crn
(8)
in which C R ~is the compressibility of mercury, C is the compressibility of the solid before it was powdered, mHgis the mass of mercury in the dilatometer, and m is the mass of the powder.
Results The surface areas of about 20 powders with widely different properties were studied by mercury porosimeter (Aminco-Winslow S-7109 Porosimeter, American Instrument Co., Silver Springs, Xld.) and gas adsorption technique^.^ Mercury porosimeter curves for a few selected samples of those powders are shown in Figure 1. To avoid possible confusion in analyzing experimental data, a specific example has been made of the porosimeter curve for Sterling FT shown in Figure 1. The end point (P,,,, V,,,) for this powder has been designated with the letter A and the region to be graphically integrated in order to evaluate LvmaxPdV has been marked by the letter B.
The contact angle 8 was assumed to be 130” for all of the powders studiedlo since no mercury contact angle data for these solids could be found in the literature. Precise measurement of the contact angle might be expected to improve the accuracy of the porosimeter areas. However, for purposes of this paper, further refinement of porosimeter areas seems unnecessary because BET areas are accurate a t best to no better than 10% on an absolute area basis.11p12 Table I shows the comparison between porosimeter areas calculated by eq 7 and BET areas. In view of the uncertainties involved in both methods of the area determination, the agreement between porosimeter and BET surface areas is surprisingly good. Caution must be exercised on three further points, however. First, for the present a t least, the method seems limited to powders with areas below 100 m2/g. Higher area powders have very narrow pores which
Table I: Surface Areas Calculated from Mercury Intrusion Data -Surface No.
1 2
3
PORE DIAMETER,m1emnr STERLING-FT
4 5
6
7
0.90
t
Figure 1.
) I 1
8 9 10
11 12 13 14 15 16 17 18 19 20
Mercury, mVg
Sample
Aluminum, dust Anatase, titanium dioxide Boron nitride (BN) Calcium cyanamide (E) Carbon black, Spheron-6 Carbon Sterling FT (2700’) Copper, powdered Fluorspar Fly ash Glass, alkali borosilicate (porous) Hydroxylapatite (BPI-2610) Iron oxide (A) Iron, powdered Silver iodide Tungsten, powdered Tungsten carbide Vanadium oxide catalyst Zinc, dust Zinc (powder on silver screen) Zinc (powder on silver screen)
1.35 15.1 19.6 2.75 107.8 15.7 0.34 2.48 2.34 11.0 55.2 14.3 0.20 0.48 0.11 0.11 0.40 0.34 1.47 2.16
areaNitrogen, mr/g
1.14 1.25Kr 10.3 20.0 3.17 110.0 12.3 0.49 2.12 2.06 7.9 55.0 13.3 0.30 0.53 0.10 0.14 0.40 0.32 1.60 2.00
(9) 9.J. Gregg, “The Surface Chemistry of Solids,” Reinhold P u b lishing Gorp., New York, N. Y., 1961, p 122. (10) Bulletin 2330-A, July 1964, American Instrument Co., Inc., Silver Springs, Md. (11) S. Brunauer, “Physical Adsorption,” Vol. I, Princeton University Press, Princeton, N. J., 1943. (12) J. R. Sams, Jr., G. Contabaris, and G. P. Halsey, Jr., J . Phys. Chen., 66, 2158 (1962).
Volume 71,Number 8 July 1967
NOTES
2736
cannot be filled with mercury, even a t 15,000 psi, the maximum pressure of most commercial porosimeters. Second, the method may be applied only to mercury intrusion data. We found that extrusion data do not agree with BET results. Finally, caution should be exercised in analyzing areas of high-energy surfaces such as metals, which might be wetted by mercury or even amalgamated. Surprisingly, this does not pose as severe a problem as one might suspect, because most high-energy surfaces readily form nonwettable oxide layers on exposure to air. We suspect that the areas of AI, W, Cu, Zn, and Fe powders mentioned in Table I show agreement with B E T areas for this reason and that “clean” surfaces of these metals would furnish incorrect areas by the porosimeter method. In conclusion, we believe that the porosimeter method is a useful and reliable method for measuring the areas of low-energy surfaces. At the very least it can be used as a rapid, independent check on gas adsorption results.
The Kinetics of Light Absorption in Photobleaching Media
by Henry C. Kessler, Jr. Research Laboratories, Hamilton Standard Division, United Aircraft Corporatwn, Windsor Locks, Connecticut (Received August I , 1966)
It has long been recognized that the classical Lambert-Beer law is not directly applicable to photochemically active solutions for which the change in absorptivity which accompanies the photochemical reaction results in significant time-varying concentration gradients. The procedure usually adopted in such cases is to assume either that the duration of illumination is so short that the variation of intensity with distance is approximately exponential or that the absorption is so low that the incident intensity prevails throughout the entire reaction zone. It is therefore of interest to construct an analytical model of the coupled absorption and reaction processes that makes neither of the previous assumptions. The result is applicable to photobleaching media and reduces to the Lambert-Beer law when the pigment concentration is constant. Phenomena of this type are encountered in the irreversible photochemical conversion of riboflavin to lumichrome in neutral solution and in the photoThe Journal of Physical Chemistry
bleaching of rhodopsin. With natural retinal extracts, however, the direct product of rhodopsin bleaching is unstable and is rapidly converted to an isomeric form of vitamin A. The rate of accumulation of the immediate product of bleaching is therefore only partially determined by the rate of bleaching of rhodopsin. It will be shown that a quantitative relationship between them can be established if the kinetics of conversion to vitamin A is assumed to be first order. Since first-order kinetics is the limiting case of Michaelis-Menten kinetics for low substrate concentration, this assumption corresponds approximately to the conditions of in vitro experiments carried out a t sufficiently high dilution of rhodopsin even though the conversion process is enzymatically catalyzed.
Analysis Consider a semiinfinite cylindrical column of solution containing bleachable pigment. The section of the cylinder is arbitrary and its generators are parallel to the x axis. Let the column extend from z = 0 to x = QO and be illuminated on its 2 = 0 end by a parallel, monochromatic beam of radiation in the x direction having a wavelength a t which the pigment is strongly absorbing. The illumination begins a t t = 0 with intensity I o which is constant thereafter. It will be assumed that the bleached product is completely transparent a t this wavelength and that the duration of illumination is sufficiently brief so that diffusive redistribution of pigment resulting from the concentration gradient produced by the bleaching process can be ignored. If the derivative of beam intensity with respect to depth a t any time is proportional to the product of the intensity I (erg cm-2 sec-’) and the pigment concentration c (mole ~ m - ~then ) bI/bx = -pcI, where p (cm2 mole-’) is the absorption coefficient. The bleaching process will be assumed to be of first order so that the rate of change of pigment concentration with time is proportional to the product of pigment concentration and the absorbed intensity pI. Then dc/bt = -kpcI, where k (mole erg-’) is the photochemical efficiency of the reaction. For a constant initial pigment concentration c(2, 0) = co this can be integrated to get c = coeQwhere Q = - k p
Sot
Idt. As I
=
- (lcp) -%&/at the intensity equation becomes b2Q bQ bxbt - = - p c o ( z ) eo but (bQ/dt)eQ = d/bt(eQ), so this can be integrated with respect to time to get
