DETERMINATION OF ACTIVITIES BY MASSSPECTROMETRY
1403
The Determination of Activities by Mass Spectrometry. I.
The Liquid
Metallic Systems Iron-Nickel and Iron-Cobalt
by G. R. Belton and R. J. Fruehan Department of Metallurgy, University of Pennsylvania, Philadelphia, Pennsylvania
(Received October $8,1966)
The Knudsen cell-mass spectrometer combination has been used to study activities in liquid metallic solutions. By application of the Gibbs-Duhem equation, a simple integration technique has been developed which allows activities and heats of solution to be determined from a series of measurements of the ratio of the ion currents of the solution components, thus overcoming problems caused by changes in the absolute sensitivity of the mass spectrometer. Activities in the Fe-Ni system, determined by this technique, are found to be in excellent agreement with previous work. Heats of solution for the Fe-Ni system and activities in the Fe-Co system at 1590” are presented.
The difficulties involved in the determination of the thermodynamic properties of high-temperature metallic solutions are such that additional methods would be useful. One possible approach, which has received only limited attention, is the use of the mass spectrometer-Knudsen cell combination, a tool which has proved to be so valuable in high-temperature chemistry. The partial pressure of a species within the Knudsen cell is given in such mass spectrometric studies by the expression
Pi
=
Ii+TK/aiPi
(1)
where I t + is the ion current, T is the temperature of the cell, ui is the relative ionization cross section, 8, is the relative detector sensitivity, and K is a constant (usually determined by calibration), which includes geometric factors for the cell and ion source arrangement. Accordingly, it might be expected that the ratio of ion currents obtained in successive experiments with a solution and a pure component would suffice to determine the activity of that component in the solution. Unfortunately, small changes in absolute detector sensitivity between experiments appear to be unavoidable,* thus making the direct approach unworkable. Berkowitz and Chupka3 have pointed out that this problem can be avoided in activity studies as long as an ion current ratio is measured in each experiment
and the activity is derived from these ratios. In a limited study of molten salt systems, they presented and demonstrated two techniques: (a) the use of the monomer-dimer ratio and (b) the use of an inert, congruently vaporizing, internal standard. A variation of technique a was used by Buchler and S t a ~ f f e r , ~ who reported activities of LiF for two compositions in the LiF-BeF2 systems. They measured the BeF2+/ LiBeFz+ ratio over both of the melts and over the solution in equilibrium with solid LiF. By using a double Knudsen cell source they were also able to determine the activity of BeF2 by direct comparison of BeF2+ over the pure compound and the mixture. Lyubimov, et u Z . , ~ had earlier attempted to use a modified Gibbs-Duhem treatment of the ion currents of both components in a study of the solid alloys Fe-Ni and Fe-Co. Unfortunately, only three compositions in each system were studied, thus making the (1) M. G. Inghram and J. Drowart, “Proceedings of the International Symposium on High Temperature Technology,” McGrawHill Book Co., Inc., New York, N. Y., 1960,p 219. (2) See, for example, comments by K. A. Gingerich, J . Phys. C h a . , 68, 768 (1964),and T. A. Milne and H. M. Klein, J. Chem. Phys., 33, 1628 (1960). (3) J. Berkowitz and W. A. Chupka, Trans. N. Y . Acad. Sci., 79, 1073 (1960). (4) A. Bachier and J. L. Stauffer, “Symposium on Thermodynamics, Proceedings Series,” Vol. I, International Atomic Energy Agency, Vienna, 1966, p 271. (5)A. P. Lyubimov, L. Zobers, and V. Rakhovski, Zh. Fiz. Khim., 3 2 , 1804 (1958).
Volume 71,Number 6 April 1067
1404
0. R. BELTON AND R. J. FRUEHAN
Gibbs-Duhem integration somewhat meaningless. In addition, several of their data show the ratio of the ion currents varying inversely with the composition of the alloy. The only other reported and successful study of an alloy system is that of Norman, et U Z . , ~ who determined the Henry's law constant of zinc in very dilute liquid solution with antimony and indium. Essentially, they used the mass spectrometer to measure the rate of change of composition of the alloy due to the volatilization of zinc. From this rate, the orifice area of the cell, and the vapor pressure of zinc, the constant could be computed. In the present paper, it is shown that a simple transformation of the Gibbs-Duhem equation can be used to derive the activities in a solution from measurements of the ratio of ion currents of the components. Measurements are presented for the liquid Fe-Ni system for comparison with existing data from other techniques and for the liquid Fe-Co system.
Derivation of the Equations At constant temperatures and pressure, the GibbsDuhem equation may be written as
ENi d In ai i
=0
(2)
where N , and ai are, respectively, the mole fraction and activity of component i and where the sunknation extends over all components. Adding d In a, to each side of eq 2 and rearranging, we obtain d In a,
=
-ENi
d In (ai/a5)
a
(3)
As the activity of a component is directly proportional to the partial pressure (assuming ideal gas behavior) of the corresponding monomeric gas species and this, in turn, is directly proportional to the measured ion current of an isotope of that species (eq l), we obtain d In a5 = --ENi d In ( I i + / I j + ) i
(4)
For a binary system and with the pure component as standard state we obtain on integration
A form which is more suitable for graphical integration is obtained by deducting from eq 5 the following expression which is obtained from the relationship between the mole fractions /.NI=NI
This yields The Journal of P h y a h l Chemistry
where
is the activity coefficient, defined as y1 = al/N1. An expression for the partial molar heat of mixing is readily obtained by combination of eq 5 with a suitable form of the Gibbs-Helmholtz equation to yield y1
As the necessary ion current ratios are measured in single experiments, changes in absolute sensitivity between experiments are not important and should not affect the results.
Experimental Section A Bendix T.O.F. mass spectrometer, Model 12, fitted with a 107 ion source and a M-105-G-6 electron multiplier was used. A detailed description of the instrument has been given elsewhere.' The commercial Knudsen cell assembly (Model 1030), based upon that used by White, et aL18was considerably modified for this investigation. In an attempt to reduce temperature gradients within the cell, a cylindrical tantalum mesh heater, 1-in. diameter X 1 in., was used. The outer cell or susceptor was also of tantalum and had dimensions of 0.625411. diameter X 0.75 in. A blackbody hole was drilled into the side of the cell approximately 0.25 in. from the bottom. Sighting for optical pyrometry was through a horizontal tube which passed through the surrounding water jacket to the inner vacuum chamber. The optical flat on this tube was protected between observations by means of an internal mechanical shutter. Heating was by means of electron bombardment between the tantalum mesh and the cell. Control of the bombardment current was achieved in the following way. The voltage drop developed across a resistor (1 ohm) incorporated in the dc bombardment circuit was applied, after passing through a simple resistancecapacitance delay circuit, to a proportional-acting controller (Wheelco 407). The signal from the controller, after magnetic amplification, was then fed to the control windings of a saturable core reactor incorporated into the ac heating circuit of the tantalum mesh. Control of the bombardment current was such that after (6) J. H. Norman, P. Winchell. and H. G. Staley, J . Chem. Phys., 41, 60 (1964). (7) W.C. Wiley and I. H. McLaren, Rev. Sci. Imtr., 26, 1150 (1955); W. C. Wiley, Science, 124, 217 (1956). (8) D. White, P. N. Walsh, H. W. Goldstein, and D. F. Dever, J . Phys. Chem., 65, 1405 (1961).
DETERMINATION OF ACTIVITIES BY MASSSPECTROMETRY
a thermal equilibration period (usually 3-5 min) temperature fluctuations could not be detected by optical pyrometry. In situ calibrs tion of the disappearing-filament-type optical pyrometer (Leeds and Northrup) was carried out by continuously monitoring the temperature and ion current from a 6-g sample of pure iron while cooling through the melting point at a rate of about lO"/min. A distinct arrest in the ion current of 8-12-sec duration was observed. The Knudsen cells used in the experiments were approximately 0.375-in. 0.d. X 0.5 in. with wall thickness of about 0.077 in. and lid thickness of approximately 0.06 in. Essentially knife-edged orifices were made by high-velocity air abrasion using S i c powder. The alumina cells used for the Fe-Ni alloys had orifice diameters of 0.038 and 0.052 cm and the thoria cells used for the Fe-Co alloys had orifice diameters of 0.027 and 0.043 cm. Alloys were prepared in situ by melting together a total of 0.5-1 g of the component metals, the purities of which were: Fe, 99.9% (principal impurity carbon) ; Ni, 99.998%; and Co, 99.95%. Procedure. Prior to each experiment the electrometer output units of the mass spectrometer were balanced and zeroed and the gate pulse generators were balanced t o prevent drifting off the mass peaks during the experiment. The temperature of the cell was raised to between 1550 and 1600" and held for 1015 min to allow melting and homogenization of the alloy. The ion currents at the desired mass peaks were recorded simu1t:tneously on a calibrated strip chart recorder, background ion currents being determined by the shutter technique.' The principal mass peaks were used for iron and cobalt but the peak at 60 amu was used for nickel to avoid interference with iron at mass 58. After a change in temperature, a period of 3-6 min was usually sufficient to obtain constancy of ion currents. Neasurements were made for both increasing and decreasing temperature sequences to a maximum of about 1680". A number of preliminary experiments were carried out before studying each alloy system to ascertain the effects of the machine variables on the measured ratio of ion currents. No change in the derived ion current ratio was found (Al%) when changes in individual ion currents by a factor of more than 10 were deliberately caused by making the flight of ions off center, by changing the multiplier gain, or by changing the electron bombardment current. Deliberate misalignment of the Knudsen cell and ion source region had no effect on the ion current ratio for a reduction in ion currents by a factor of 2-3; however, gross misa-
1405
I
I
I
I
1
1
5.1
52
83
5.4
5.5
5.6
,091 -q+-
-I I 5.0
T I
(OK)
I
x 10'
Figure 1. Experimental values of the ion current ratio for the Fe-Ni system: 0, measurements with an orifice diameter of 0.038 em; 0,orifice diameter, 0.052 cm.
lignment did change the ratio. During the subsequent experiments, the cell position and flight of ions were adjusted to give maximum ion currents. Ionizing potentials of 30 and 40 v were used for the Fe-Ni and Fe-Co systems, respectively, although it was subsequently found that the ion current ratios were essentially independent of ionizing potential between about 25 and 60 v.
Results and Discussion The Iron-Nickel System. The experimental values of the ion current ratios for the Fe-Ni alloys are predented in Figure 1 as values of In ( I F ~ + / I N ~ vs.+ ) 1/T"K. Derived values for the slopes and for In ( I F e + / I ~ i +at) 1600" (approximately the mean temperature of the experiments) are given in Table I. For the alloys with six or more experimental points, the average standard deviations in these quantities are 1070 and 0,009, respectively, with maxima of 1690 and 0.015. This latter value represents approximately 1.5% in the arithmetic ratio which is of the order of the readability of the measuring system. The two experiments with different orifice diameters gave values of In ( I F e + / I ~ i +at) 1600°, which agreed within 0.01. Values of In ( I p e + / 1 ~ i +) In (NFe/NNi) derived from the data in Table I are shown plotted vs. composition in Figure 2. Graphical integration of the area under the curve gives, according to eq 7, the activity coefficient of nickel. Activities of nickel and iron, Volume 71,Number 6 April 1967
G. R. BELTONAND R. J. FRUEHAN
1406
Table I : Experimental Values of Ion Current Ratios for Fe-Ni Alloys Mole fraotion of Fe
0.091 0.195 0.244 0.318 0.375 0.450 0.600 0.750 0.900
f*)
d In INi+ I d (
- 5940 -6240 - 2460
-3310 -430
40 3080 4690 2100
k)
Table 11: Activities and Activity Coefficients for Liquid Fe-Ni Alloys at 1600" In
t*) IN^+
a t 1600°
-0.54 0.71 1.02 1.66 1.98 2.47 3.21 3.91 5.03
"i
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
ONi
0.000 0.073 (f0.003) 0.147 0.222 0.300 0.387 (f0.006) 0.493 0.626 0.762 0.889 ( f0.002) 1 .000
YNi
0.716 0.725 (f0.032) 0.733 0.742 0.751 0.774 (f0.011) 0.821 0.894 0.952 0.988 (f 0.002) 1.000
aF8
1.OOO 0.899 (f 0.001) 0.798 0.695 0.593 0.479 ( f0.004)
0.351 0.225 0.124 0.050 (f0.003)
O.OO0
YFs
1.OOO 0.999 (f0.001) 0.997 0.993 0.988 0.958 (f0.008) 0.877 0.750 0.622 0.503 (f0.031) 0.397
?
Figure 2. Integration plot for the Fe-Ni system at 1600'.
determined by this method, are compared with the data from the transpiration study of Zellars, et U Z . , ~ in Figure 3 ; the agreement is excellent. Other work by Speiser, et aZ.,l0 which is in equally good agreement, is not shown for clarity. An estimate of the uncertainty was made by carrying out the integration on several curves, each drawn to be consistent with the experimental points to within twice the average standard deviation. The derived uncertainties together with the activities and activity coefficients are presented in Table 11. The heats of solution of the components may be determined through eq 8 by graphical integration of the slopes given in Table I. Integration of the reasonable curve shown drawn through the data in Figure 4 yields the values presented graphically in Figure 5. Unfortunately, the accuracy of these data cannot be considered high because of the large uncertainties in the slopes. Furthermore, there is an inconsistency in the data, for the reasonably well-established heats of vaporization of Ni and Fell require the value of R[d In ( I ~ e + / l ~ i + ) / d ( l / T to ) ] be approximately 4-5 The Journal of Physdcal Chemistry
Atbm Fraction Ni Figure 3. Comparison of derived activities with the work of Zellarrs, et al.
kcal at about N N = ~ 0.55 (the maximum on the heat of mixing curve). The present data indicate a value of about zero. It is not unreasonable, however, to assume that the relative slopes are correct, in which case, and taking twice the average standard deviation as a probable uncertainty in the slopes, the uncertainties in the heats of solution are approximately 3 kcal at infinite dilution and approximately 1 kcal at the equimolar composition. (9) G. R. Zellars, S. L. Payne, J. P. Morris, and R. L. Kipp, Trans. AIME, 215, 181 (1959). (10) R. Speiser, A. J. Jacobs, and J. W. Spretnak, ibid., 215, 185 (1959). (11) R. Hultgren, R. L. Orr, P. D. Anderson, and K. K. Kelley, "Se-
lected Values of Thermodynamic Properties of Metals and Alloys," John Wiley and Sons, Inc., New York, N. Y., 1963.
DETERMINATION O F ACTIVITIES BY MASSSPECTROMETRY
.a
al *o l*
1407
0
0
.
' 0 0
-15
-10
I
I
I
1
-5
0
5
K)
15
-4
Figure 4. Integration plot for the heats of solution in the Fe-Ni system.
c
I
0.058
-0--9-
1
I
I
I
I
I
1.1
52
5.3
5.4
1.6
56
I/T
(OK)
x 10'
Figure 6. Experimental values of the ion current ratio for the Fe-Co system: 0 , measurements with an orifice diameter of 0.027 cm; 0, X, orifice diameter, 0.043 cm.
*
which scattered about essentially zero (- 541 1050) with no apparent trend with composition; consequently, simple averages were taken, The average standard deviation for alloys with six or more experimental points was 0.023 or approximately 2.4% of the arithmetic ratio.
Table 111: Experimental Values of Ion Current Ratios for Fe-Co Alloys at 1590" Atom fraction of Go
-0
.2
.4
.6
.8
Atom fraction of co
1.0
A t o m Frocfion Ni
Figure 5. Heats of solution and heats of mixing in the Fe-Ni system.
The maximum heat of mixing, -2.6 f 1 kcal, is considerably smaller than the value of -7.8 kcal reported by Speiser, et ~ 1 . ; ' ~however, as Hultgren12 has pointed out, their value is almost certainly much too negative for this type of system. The Iron-Cobalt System. The experimental data for the Fe-Co alloys are shown in Figure 6 and values for In (Ico+/INi+) at 1590" (approximately the mean temperature of the experiments) are given in Table 111. A least-mean-square analysis of the data gave slopes
0.943 0.910 0.724 0.602 0.513 0.376
1.17 0.94 -0.27 -0.67 -1.07 -1.34
0.335 0.299 0.197 0.119 0.095 0.058
-1.60 -1.84 -2.41 -3.12 -3.21 -3.91
Curves which were integrated in accordance with eq 7 or in accordance with eq 4, when Raoultian behavior was assumed above N = 0.94, are presented in Figure 7 and derived activities are shown in Figure 8. Uncertainties in the derived activities were again estimated by drawing and integrating several extreme (12) Reference 11, p 728.
Volume 71 Number 6 April 1967 I
1408
G. R. BELTON AND R. J. FRUEHAN
1.0
1
1
1
1
1
Table N: Activities and Activity Coefficients for Liquid Fe-Co Alloys at 1590' Nco
aco
YCO
0.00 0.10
0.000 0.132 (2Z0.003) 0.235 0.373 0.489 0.574 (zkO.009) 0.657 0.732 0.817 0.905 (*0.002) 1.000
1.068 1.132 (2Z0.030) 1.175 1.244 1.223 1,1417 ( 2 Z O . 018) 1.095 1.046 1.021 1.005 ( 2 Z O . 002) 1.000
0
0.20 0.30 0.40 0.50
%
-;-;-1 b I
0.60 0.70 0.80 0.90
2
Figure 7. Integration plots for Fe-Go system a t 1590' (a) in accordance with eq 5, and (b) in accordance with eq 7.
1.00
curves, each drawn to be consistent with the data to within twice the average standard deviation, except that the data for the 91% Co alloy were given little weight, as the average temperature of the determinations was 30" below the median of 1590". The numerical values for the activities and activity coefficients, together with the estimated uncertainties, are presented in Table IV. General Discussion. The excellent agreement between the mass spectrometric and literature values for the Fe-Ni alloys indicates that accurate activities can be determined by this mass spectrometric technique. The Journal of Physical Chemktry
1.000 0.896 (f 0.001) 0,790 0.679 0.587 0.518 (zk0.007) 0.438 0.355 0.256 0.127 (zk0.004) 0.000
YFe
1.000 0.996 (f0.001) 0.988 0.970 0.981 1.038 (2Z0.015) 1.095 1.183 1.282 1.416 (f 0,043) 1.590
In the opinion of the authors, the observed scatter in the data and particularly in the slopes is mostly due to small variable temperature gradients in the Knudsen cell. Fortunately, the ratio of ion currents is not a strong function of temperature ; hence the presence of small gradients should not seriously affect the value of the ratio at the average temperature. It is clear, however, that high precision in derived heats of solution will be achieved only with uniform cell temperatures. An extended tube furnace will probably be necessary, rather than the short cylindrical heater used in the present work. The integration curves shown in Figures 2 and 7b exhibit simple shapes which are of current interest. Darken's has examined many liquid binary metallic systems and has shown that the activity coefficient of the solvent may very often be accurately represented by the equation In
Atom Fraction Co Figure 8. Derived activities for the Fe-Co system a t 1590'.
ape
y1 =
CYN,~
(9)
to surprisingly high solute concentrations. Over the same composition range, the activity coeficient of the solute must be given by In yz = a N j 2
+I
(10)
when I is an integration constant. Similar equations hold for the solvent and solute at the opposite end of the system but with a different value for the constant a. These two terminal regions are then connected by a short transition region where the behavior of the activity coefficients is more complex. It will be readily seen that the functions plotted as abscissas in Figures 2 and 7b are equivalent to (13)
L. 8. Darken, Trans. A I M E . , 239, 80 (1967).
KINETICSOF
THE
HYDROLYSIS OF THE DICHROMATE ION
+
ln(r2/yl) K , where K represents a combination of various mass spectrometer constants. Over the regions where eq 9 and 10 are valid, straight-line behavior of the function will be observed when plotted vs. atom fraction. This behavior is shown by the Fe-Ni system and within experimental error by the Fe-Co system.
Acknowledgments. The authors wish to thank the
1409
American Iron and Steel Institute for their generous support of this work. It was carried out in the Laboratory for Research on the Structure of Matter which is generally supported by the Advanced Research Projects Agency, Office of the Secretary of Defense. M. J. Ginsburg and Dr. R. C . Svedberg made substantial contributions to design of the heating and control equipment.
The Kinetics of the Hydrolysis of the Dichromate Ion.
111. Environmental
Effects on Rate Constants and Activation Energies’
by Berta Perlmutter-Haymanand Yael Weissmann Department of Physical Chemistry, Hebrew University, Jerusalem, Israel
(Received July 18, 1966)
The rate of the hydrolysis of the dichromate ion in the absence of bases and other nucleophiles is accelerated by a number of “inert” salts and is decelerated by tetraethylammonium salts and by certain neutral substances. The acetate-catalyzed hydrolysis is subject to specific cat’ionic effects as would be expected of a reaction between two anions. The activation energy is 7.5 kcal and is constant between 1 and 4 5 O , whereas the activation energy for the uncatalyzed reaction decreases with increasing temperature. The rate of the uncatalyzed reaction is somewhat higher than that calculated from data on isotopic exchange.
Introduction In continuation of our investigation of kinetic salt effects, we considered the acetate-catalyzed hydrolysis of the dichromate ion to be a suitable reaction; it involves two anions and is first order with respect to each of them.2 However, the contribution of the uncatalyzed reaction to the observed rate is not negligible under our experimental conditions. For an accurate determination of the salt effect on the catalyzed reaction, a knowledge of the same effect on the uncatalyzed reaction is therefore desirable. Although for technical reasons this knowledge could not be obtained for all the salts employed, the results obtained for the uncatalyzed reaction seemed interesting in themselves and further environmental effects on this reaction
were therefore investigated. Activation energies for the two reactions were measured as a help toward understanding the mechanism.
Experimental Section The course of the reaction was followed spectrophotometrically, as described previously.2 All experiments were carried out under conditions where the reaction is pseudo-first order and the back reaction can be neglected. The temperature in the cell compartment was kept (1) Taken in part from a thesis t o be submitted t o the Senate of the Hebrew University by Y. Weissmann in partial fulfillment of the requirements for a Ph.D. degree. (2) B. Perlmutter-Hayman, J . Phys. Chsm., 69, 1736 (1965).
Volume 71, Number 6 April 1967
