T H E R M O D Y N A M I C S OF SOLUTIONS Consistency of Spar.!;e Data.
Activities of Nitric and Perchloric Acids
0 . R E D L I C H , W.
E. G A R G R A V E , A N D W.
D. KROSTEK
Lawrence Radiation Laboratory and Department of Chemical Engineering, University of California, Berkeley, Calif.
Usual consistency checks for activities can be applied only to fairly extensive sets of data. A new variant is proposed that can be used even if data are available only for three concentrations. The activities of nitric and perchloric acids are computed in the whole concentration range from total vapor pressure and partly from other data.
application of the relation of Gibbs and Duhem to tests Tof the consistency of data for solutions has been frequently discussed in the past 20 years. If measurements are available only for a small number of solutions, the usual methods of testing fail. Such a check, however, was desired for a few determinations of the partial pressures of nitric acid solutions, differing from other information. A method useful in such cases was therefore developed for a revision of the activities of nitric and perchloric acids, needed in an investigation of their molecular structures.
0
HE
I
A
n
0
-02
The Differential Test for Sparse Data
0
A
For a binary mixture we define
Q
=
XI
log
YI
+
~2
\
(11
log YZ
O A
and find (Redlich and Kister, 1948), according to the relation of Gibbs and Duhem,
dQ/dxz = log (YZ/YI) An analogous relation holds for the activities the function
Q,
=
XI
log
UI
+
I
I
Davis and d e B r u i n , 25°C Potier, 20°C A Haose. 2 5 O C 0
x2
log a2
(2) a1
A 0
-1.0
V
I
I
I
0.I
0.2
0.3
x2
and a2 and Figure 1.
(3)
x2
Function Q, for nitric acid = mole fraction HNOs
namely,
dQp/dXz = log (az/ai)
(4)
The activities may also be replaced by partial vapor pressures if the vapor is perfect. Davis and de Bruin (1964) reported partial vapor pressures pl (water) andp, (acid) for solutions of nitric acid at three concentrations. Their measurements, including results for p2 at other concentrations, deviate considerably from the results of Potier (1956). This author’s data, by the way, are in sufficient agreement with an older compilation (Redlich and Rosenfeld, 1935). From the partial pressures of Davis and de Bruin for three concentrations we derive Q , and separately dQ,/dxz (Equation 4). The data for Q p are indicated in Figure 1 by the circles, the slopes by the short straight lines through the experimental points. The differential test is now carried out by drawing the best line representing the points as well as the indicated slopes. The example of Figure 1 shows that no smooth curves satisfying the prescribed slopes can be drawn through the experimental points. (In addition, the discrepancies between the data of Davis and de Bruin and other authors are appreciable.) The magnitude of the inconsistency of the original data can be estimated by drawing the “best” smooth curve in a diagram
similar to Figure 1 but drawn on a sufficiently large scale. Then the points a t the experimental concentrations and the tangents are marked. The values of Q p and dQp/dx2 are read off the diagram. Now Equations 3 and 4 are solved for a1 and a2 (orp, andpz, or71and YZ). These graphical operations can be easily performed with sufficient precision. I n the very unlikely case that the required scale did turn out to be inconvenient, one would design a suitable deviation function. I n actual use this application of a differential equation to a few given points has been found to be very efficient. Nitric Acid
The activity coefficients of nitric acid are to be based on determinations of the freezing points (Abel et al., 1928 ; Hartmann and Rosenfeld, 1933), partial pressures (Davis and de Bruin, 1964; Haase et al., 1965; Vandoni and Laudy, 1952), and total pressures (Potier, 1956). Freezing points unquestionably give the best information below x z = 0.06. Results for the whole range at 25’ C. are shown in Figure 2. The abscissa xzos5 has been chosen because here the theory of Debye and Huckel prescribes a finite limiting slope. VOL 7
NO. 2
M A Y 1968
211
Table 1.
Potier's results (1956) (extrapolated from O'and 20°to 25' C.) with the other data gives a difference of 0.957 in the standard state values of log 7 2 . I n other words, log 7 2 for pure HNOs based on Equation 5 is 0.957 and log a2 of HNOa is 1.914, while dz is normalized at unity by Potier. A compilation of the best values of log 72 and log a1 (obtained from freezing point data by straightforward application of the equation of Gibbs and Duhem) is shown in Table I.
Activity Coefficients of Nitric Acid at 25' C.
lim a2 =
0.08
0.058
0.10 0.16 0.22 0.28 0.34 0.40 0.46 0.52
0.126 0.314 0.459 0.579 0.683 0.772 0,838 0.886 0.921 0.945 0.965 0.974 0.979 0.981 0.980 0.977 0.972 0.963 0,957
0.64 0.70 0.74 0.78 0.82 0.86 0.90 0.94 0.98 1 .oo
=
11
- log a1
72
-0.01518 -0,03248 -0.06146 -0.1044 -0.1420 -0.1350 -0.131 -0.076 -0.006
0.58
01
xz=o
1%
X2
1.80 x 10-5 9.00 X 10-6 3.60 x 10-4 1.799 x 10-8 8.930 X 1.770 X 10+ 0.02 0.04 0.06
lim
x22;
x2=0
1.544 X 7.635 X 3.016 X 1.469 X 7.400 X 1.533 X 1.750 x 0.0388 0.0645 0.0928 0.125 0.241 0.373 0.523 0.692 0.879 1.070 1.264 1.46 1.68 1.91 2.08 2.26 2.45 2.66 2.90 3.22
10-6
10-6 10-4
lo-*
Perchloric Acid
The isopiestic measurements of Robinson and Baker (1946) and ofHaase and coworkers (1965) below x2 = 0.22 are in good agreement. Information over the rest of the range has been based on Mascherpa's total pressures (1965). I n addition to the definition (Equation 5) we also define activities 61 and 6 2 based on the pure acid as reference state by lirn (61/x12) = 1; lim
62
= 1
x2=l
x2=1
(8)
The vapor may be considered to be a perfect gaseous mixture. Denoting the total pressure and partial pressures by P , pl, and p 2 , and the vapor pressure of the pure components by ,510 andp+, we have for the azeotropic solution
a1 = p d p P = xlP/plo
(9)
and 62
= p2/pz'
=
x2P/pza
(10)
(The definitions of a1 and 6 2 are conventional for nonelectrolytes, those for a2' and 72' for binary electrolytes.) I n the range 0.417 < x2 < 0.710 perchloric acid solutions a t 25' C. (0.393 < x z < 0.835 at 0') are in equilibrium (Mascherpa, 1965) with the solid hydrate HClO(.HzO. Therefore, the product ala2 (or the product 6162) has the same value in the entire interval. The derivation of activities from total pressures has often been discussed. The peculiar form of the pressure curve of perchloric acid caused considerable difficulties. Finally the differential equation of Lewis and Murphree (1924) was integrated with increasing x z beginning from x2 = 0 for an arbitrary set of values of lirn dy2ldx2, and with decreasing x2 be-
I
Landoit-Bornstein Potier
Haass D a v i s and de Bruin I
x*=O
ginning from
x2
= 1 for a set lirn dy2/dx~. The criteria for x* E 1
0.0
the selection of the final curve from the family of integral curves were :
1.0
0.5 xo.5 2
Figure 2.
The definition of the activity coefficient 7 2 chosen in the present work is different from the usual convention. The activity is expressed by uz = (XzYZy;
lim
a2 =
x22
(5)
X2EO
T h e conventional definition by the molality UZ'
= (myz')'; limaz' = m2 m=O
(6)
leads to the relation YP = (1
+ 0.018016 m) 72'
(7)
Haase's data (1965) are based on a long extrapolation to x2 = 0. They can be tied quite well to the freezing point results by addition of 0.009 to his values of log 72. T h e comparison of 212
l&EC FUNDAMENTALS
Correct results for the azeotropic a1 and 62. (6) T h e computation according to Gibbs and Duhem furnished the activity a2 of perchloric acid for concentrations below the azeotrope. Similarly the water activity was obtained above the azeotrope, with due consideration being given to the constancy of the product 6 1 6 2 in the presence of the solid hydrate. A criterion is the smoothness of the curve of log (yT/y2). (c) Recalculated total pressures must check within experimental errors of about 0.05 mm. (d) The computation must satisfy Equations 3 and 4. (a)
Activity coefficient of nitric acid
In principle, these conditions are trivial, but actually they are not easily satisfied because of the influence of experimental and small computational errors. The results are shown in Figures 3 and 4 and Tables I11 and IV. The activity coefficients in Figure 3 are defined as Y Z = bzo*S/xz
(11)
The limiting value of log 72 = -4.50 for x2 = 0 shows that the difference of the standards for log a2 and log 6 2 is - 9.00.
C
Table II.
C. 0 (Mascherpa) 25 (Mascherpa) 25 (Lilich) Temp.,
Azeotrope of Perchloric Acid
xz
log a1
log bz
0.31!35 0.3195 0.344
0.082 0.34 0.064
-1.900 -2.009 -2.752
-2.652 -2.523 -3.166
O
Table 111.
A H a a s e O°C 0 H o o s e 25'C
Activity Coefficients of Perchloric Acid
[Belowxz = 0.417: u z =
lim
( ~ 2 ~ 2 ) ~ ;
u2 =
~
2
~
Xz
Table IV.
1% YZ
log a1
-0.017 -0.044 -0.135 -0.278 -0.492 -0.786 -1.175 -1,617 -2.013 -2.286 -2.539
62
=
1; lim 61
- 1 0 ~
0"
c.
bz 25' C.
o:i57 0.139 0.119 0.102 0,084 0.066 0.049 0.024
-
-2
-
m 0 d
-1
-
-4
-
1
HNOI HC1Oh
Vapor Pressures of Vapor Temp., Pressure, OC. M m . Hg 0 14.15
20 25 0 25
47.4 62.5 11.6 36.2
,
,
I
1
0.5
x0.5 2
Figure 3. Activity coefficient of perchloric acid
C
1
1
xlZ)
log
oo c.
(b1/Xi2)
25" C.
-0.146 -0.132 -0.105 -0.086 -0.073 -0.068 -0.068 0.377 0.368 0.329 0.292 0.251 0.207 0.158 0.106 0.055
-0.1 N n
m 0
-0.2
-6:067 -0,065 -0.061 -0.055 -0.044 -0.034 -0.020 -0.009
- 0.3
7
"03
0.8
I .o
0.9 x2
Figure 4. Table V.
1
*2=1
0.278 0.271 0.257 0.241 0.222 0.202 0.180 0.223 0.217 0.189 0.164 0.139 0.117 0.095 0.074 0.044
=
-I
xN
11
Activities bl and bl of Perchloric Acid Solutions X2P1
0.71 0.72 0.74 0.76 0.78 0.80 0.82 0.835 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98
a1 =
-0.051 -0.089 -0.110 -0.119 -0.113 -0.101 -0.075 0.071 0.476 1.004 1.602 2.150 2.756 3.334 3.723 3.959 4.139
(Abovexz = 0.710: lim xz
lim ;
%*SO
x130
0.0004 0.0016 0.0036 0.0064 0.0100 0.0144 0.02 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40
Mascherpa
0
p, Total Pressure, M m . Hg
and HCIO4
Activity bz of perchloric acid lim
zz
-1
bz = 1
Author
Potier, extrapol. Potier Extrapol. Mascherpa Mascherpa
The fact that log ( b ~ / x has ~ ~ a) finite limit at x z = 1 (normed to zero) expresses the complete dissociation of water in nearly anhydrous perchloric acid. Recent measurements of perchloric acid pressure by Lilich and Shalygin (1966) are not in good agreement with Mascherpa's data. An appreciably higher value is given for the azeotropic composition (see Table 11). We have not been able to find any reason for the discrepancies.
T h e vapor pressures of the anhydrous acids are shown in Table V. Acknowledgment
This work was performed under the auspices of the United States Atomic Energy Commission a t the Lawrence Radiation Laboratory of the University of California. literature Cited
Abel, E., Redlich, O., Lengyel, B., Z. Physik. Chem. 132, 189 (1928). Davis, W., de Bruin, A. J., J . Znorg. Nucl. Chem. 26, 1069 (1964). Haase, R., Ducker, K. H., Kuppers, H. A., Ber. Bunsen-Gesellschaft 69, 97 (1965). Hartmann, F., Rosenfeld, P., 2. Physik. Chem. (A)164, 377 (1933). Lewis, W. K.,M.urphree, E. V., J . A m . Chem. SOC.46, 1 (1924). VOL 7
NO. 2
MAY 1968
213
Lilich, L. S., Shalygin, V. M., Vestn. Leningrad Univ. 21 (22), Ser. Fiz. Chim. No. 4, 104 (1966). Mascherpa, G., Rev. Chim. Minirale 2, 379 (1965). Potier, A., Ann. Fac. Sci. Univ. Toulouse 20, 1 (1956). Redlich, O., Rosenfeld, P., "Landolt-Bornstein Tabellen," 3rd Suppl., p. 2145, Springer, Berlin, 1935. Robinson, R. A., Baker, 0. J., Proc. Roy. Sod. New Zealand 76, 260 (1946).
Vandoni, M. R., Laudy, M., J . Chim. Phys. 49, 99 (1952). RECEiVED for review August 17, 1967 ACCEPTEDJanuary 11, 1968 Twelfth in a series on Thermodynamics of Solutions. For the convenience of the reader, references to the preceding papers follow.
I to IV. 0. Redlich and A. T. Kister, Znd. Eng. Chem. 40, 341, 345 (1948); J . Chem. Phys. 15,849 (1947); J . Am. Chem. SOC.71,505 (1949).
V. 0. Redlich and J. N. S. Kwong, Chem. Reus. 4 4 , 2 3 3 (1949). VI. 0. Redlich, A. T. Kister, and C. E. Turnquist, Chem. Eng. Prog. Symp. Ser. 48, No. 2, 49 (1952).
VII. 0. Redlich and A. T. Kister, J . Chem. Phys. 36, 2002 (1962). VIII. 0. Redlich and A. K. Dunlop, Chem. Eng. Prog., Symp. Ser. 59, No. 44, 95 (1963). IX. F. J. Ackerman and 0. Redlich, J . Chem. Phys. 38, 2740 (1963).
X. G. W. Lundberg, J . Chem. Eng. Data 9, 193 (1964). XI. 0. Redlich, F. J. Ackerman, R. D. Gunn, Max Jacobson, and S. Lau, IND.ENC.CHEM.FUNDAMENTALS 4,396 (1965).
SURFACE DIFFUSION OF CHEMISORBED HYDROGEN ON NICKEL CHARLES N. SATTERFIELD AND H l R O S H l IINOl Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass.
02139
From steady-state flux measurements through a porous nickel disk, the surface diffusion coefficient for hydrogen on nickel at 37" C. was measured directly and found to be about 3 X lO-'sq. cm. per second. 40 microns The surface diffusion flux was a substantial portion of the total flux only at low pressures-e.g, of Hg. The value for the diffusion coefficient agrees with the results of Gomer et al. obtained by field emission microscopy but is much smaller than that predicted by the indirect thermodynamic calculations of Rideal and Sweett.
catalytic reaction involves at least one species and many investigators postulate that migration or surface diffusion of chemisorbed species on the catalyst plays an important role in their reaction kinetics. Knowledge of the magnitude of the flux of chemisorbed species is essential to an understanding of many aspects of contact catalysis. Surface diffusion of physically adsorbed molecules has been extensively studied-see, for example, the recent reviews by Dacey (1965) and Field et al. (1963). Little has been reported on surface diffusion of chemisorbed species, although they would be expected to be bound more strongly on the surface and hence to be less mobile than physically adsorbed species. The diffusion of hydrogen on nickel was chosen for study here because the chemisorption is reversible and the system is important in heterogeneous catalysis. The only previous direct study was that of Gomer et al. (1957), using field emission microscopy. Rideal and Sweett (1960) report some indirect conclusions based on thermodynamic calculations from adsorption isotherms. The method used here involved measurement of the steady flow of hydrogen through a porous disk of metallic nickel formed by pressing very fine nickel powder under high pressure. The difference between the observed flow and that predicted to occur by Knudsen diffusion (the latter determined from measurements with a nonadsorbing gas, helium) is attributed to surface diffusion. I t is necessary to control the properties of the porous disk and to choose experimental condiHETEROGENEOUS
A chemisorbed
Present address, Matsuyama Plant, Teijin, Ltd., Kita-yoshida, Matsuyama-shi, Ehime-ken, Japan 214
I&EC FUNDAMENTALS
tions carefully, so that the surface diffusion flux is an appreciable fraction of the total. The surface diffusion flux, N,, may be expressed by:
where
T,
accounts for the tortuous path of surface diffusion and
Ds
= aZye-EaIRT = Doe-EaIRT
(2)
The Knudsen diffusion flux, N K , is expressed by:
(3) where
DK
=
re =
9700 r , d T / M 20
PA1
- 0)S,
The ratio of surface to Knudsen flux is then:
From Equation 7 it is seen that surface diffusion is favored by low porosity, high surface area per unit weight, and steep slope of the adsorption isotherm. Low pressure favors surface
