0. J. KLEPPA
354
Vol. 59
A THERMODYNAMIC STUDY OF LIQUID METALLIC SOLUTIONS. VI. CALORIMETRIC INVESTIGATIONS OF THE SYSTEMS BISMUTH-LEAD, CADMIUM-LEAD, CADMIUM-TIN AND TIN-ZINC BY 0. J. KLEPPA Institute for the Study of Metals, The University of Chicago, Chicago, Illinois Received October 9,1964
A high temperature calorimeter recently developed by the author has been used for determination of the heats of mixin! in the three liquid systems bismuth-lead, cadmium-lead, cadmium-tin at 350 and 450' and in tin-zinc at 430 and 525 . The results have been compared with data obtained in earlier calorimetric work and with integral and differential heat data calculated from e.m.f. equilibrium studies. I n some cases the calorimetric heat data for moderately dilute solutions have been combined with equilibrium information derived from the binary phase diagram, and used for calculation of entropy deviations. These deviation8 agree well with data obtained from e.m.f. studies alone. In the temperature ran e covered the heat of mixing was found to be essentially independent of temperature for bismuth-lead and for cadmium-kad, while negative deviations from the Kopp-Neumann rule were found in cadmium-tin and tin-zinc. It is suggested that these deviations may be explained by the volume expansion believed to accompany the process of mixing.
Introduction The available quantitative informationeon the heats of mixing and heats of formation of alloys has been derived from calorimetric and from equilibrium investigations. By the calorimetric method we measure integral heat data directly, while equilibrium investigations similarly yield differential quantities. A comparison of the two sets of data is possible only after differentiation of the calorimetrically determined data with respect to composition, or after integration of the differential quantities. Up to the present time the calorimetric data have generally been much too inaccurate to permit a reasonably reliable evaluation of differential quantities. The calculation of integral data from equilibrium information, on the other hand, also has been associated with considerable uncertainties, particularly in the extrapolations of data for more concentrated solutions into the unexplored dilute range. It is therefore not surprising that the agreement between the two sets of integral heat data often has been far from satisfactory, although in most cases the results have agreed to within 20-30%. In an earlier communication' the author has reported the development of a new high temperature calorimeter, which permits accurate determinations of heats of mixing and heats of solution in alloy systems. The objective of the present series of investigations is to use this apparatus for a systematic exploration of the thermochemical properties of solid and liquid alloys, and in particular of the alloys formed by group B metals. In the present communication special attention has been given to a comparison with data obtained in earlier equilibrium studies and to the dependence of the heat of mixing on temperature. For this purpose the following liquid systems were selected : bismuth-lead, cadmium-lead, cadmium-tin, and tin-zinc. These systems have all been well explored by the e.m.f. method, and it was believed that the comparison therefore would be of special interest. Among the mentioned systems bismuth-lead was known to have a relatively small negative AH of mixing, while all the others have a positive AH with maxima ranging from about 2000 to 3000 j./g. atom. (1) 0.J. Kleppa, THISJOURNAL, 67, 175 (1954).
Experimental As the details of operation of the high temperature reaction calorimeter have been reported elsewhere, it is here sufficient to state that the apparatus is designed to operate a t temperatures up to about 500°, and that it is particularly suitable for measurement of heat effects of the order of 500 joules. Under favorable conditions an accuracy of about = k l % can normally be achieved. Unfortunately the present work on tin-zinc was performed during the summer months when the rather considerable temperature fluctuations in the laboratory had an unfavorable influence on the experimental precision. This accounts for the somewhat larger spread of the experimental results for this system. In this investigation stock metals. of 99.9 % purity were used. The experiments on tin-zinc were performed a t 430 f 2' and at 525 f 3", while the other systems were studied at 350 =k 2' and 450 f 2". The over-all temperature was measured by a chromel-alumel thermocouple, calibrated a t the melting point of zinc. The fluctuations in experimental temperature of f 2 (3) degrees over extended periods of time had no detectable effect on the determined heats of mixing. No particular difficulties resulted from the high volatility and reactivity of cadmium and zinc. The actual mixing experiments were carried out as reported for the introductory investigation of the system lead-tin.' As in this case, a second sample of pure metal frequently was added to the alloy formed in the first experiment. The molar heat of mixing for the final alloy was then calculated by adding the two measured heat terms. As the results derived from these double experiments are liable to be more uncertain than the results of single experiments, they are identified in the figures by a short vertical bar through the experimental point.
+
.
Results The experimentally determined molar heats of mixing, AH", are tabulated in Tables I-IV and are plotted versus composition in Figs. 1-4. The tables and figures also give values for the calculated quantity, AHM/x1x2, where x1 and xz are the mole (atomic) fractions of the two components. The figures further contain integral heat data from Kawakami's calorimetric study2 as well as data given in the most recent e.m.f. study. Bismuth-Lead.-For this system the results of 27 calorimetric experiments are reported, 15 a t 350" and 12 a t 450". In this temperature range the heat of mixing is independent of temperature within experimental error. The results differ radically from the calorimetric data given by KawakamiJ2 but agree reasonably well with the e.m.f. study by Strickler and Seltz.a Additional data for this and other systems are given by Weibke and Kubas(2) M. Kawakami, Sci. Rep. T6holcu. Imp. Univ., 119, 915 (1927). (3) II. S. Striokler and H. Selts, J . A m . Chem. SOC.,68, 2084 (1936).
THERMODYNAMIC STUDYOF LIQUIDMETALLIC SOLUTIONS
April, 1955
chewskim4These are on the whole in less satisfactory agreement with the present results. Cadmium-Lead.-A total of 29 experiments are reported for this system, 21 a t 350" and 8 at 450". TABLE I MOLARINTEGRAL HEATSOF MIXING IN Bi(1)-Pb(1) Total atoms
g.
Composition XB i
AHM, joule/g. atom
A HM!XB iZPb ,
aoule
(a) At350 i 2'
-
1.3705 1.4276 0.8060 ,8925 .6594 .4823 .7741 .6077 .3112 ,5484 .4175 .7286 .5897 1.4028 1.3199
0.0442 .0824 .09195 .BOO .1847 .2638 .3055 .4157 * 4575 .5191 .6819 .6917 .8546 .8844 .9400
154 279 312 589 - 597 - 804 - 905 1063 1087 1100 - 922 897 - 494 - 390 - 214
-3640 -3690 -3740 -3990 -3960 -4140 -4270 -4380 -4380 -4410 -4250 -4210 -3980 -3810 -3790
1.4812 1.1401 1.5597 1.2567 0.6075 0.7283 0.6334 0.5086 1.1204 1,4508 0.9762 1.3508
(b) At450 i 2' 0.0540 - 188 ,0928 320 .lo16 351 579 .1770 .1930 - 612 .3269 930 .5796 1073 .7331 818 .7571 - 763 .8654 - 465 .8689 - 454 .9243 - 253
-3680 -3800 -3850 -3970 -3930 -4230 -4400 -4180 -4150 -3990 -3990 -3620
-
-
-
TABLB I1 MOLARINTEGRAL HEATSOF MIXINGIN Cd(1)-Pb(1) Total g. atoms
1,1617 1.1749 1.1781 1.2305 0,9296 1.2694 0.7098 0.7142 1.0806 0.8188 .8257 .4984 .6670 .6962 ,8416 .6040 .9314 .9862 .7579 .8667 .9222
Composition XCd
AHM, joule/@;.atom
(a) At350 i 2' 0.0493 435 .0576 514 .0740 647 .lo24 882 .1362 1123 .1406 1177 .1479 1208 .1929 1566 .2571 1888 .2613 1924 .3240 2305 .3391 2286 .5061 2648 .7329 2326 .8185 1885 .8448 1712 .8658 1538 .8844 1398 .go89 1137 .9303 942 .9458 754
AHM/XCdXPb, joule
9300 9480 9440 9600 9540 9740 9590 10060 9880 9970 10530 10200 10590 11880 12690 13060 13240 13680 13730 14530 14710
(4) F. Weibke and 0. Kubaschemrski, "Thermochemie der Legierungen," Berlin, 1943.
1.0836 0.6823 1.1878 0.7717 0.6161 0.5114 1.1980 1.1330
(b) At460 i 2' 0.0924 793 .1239 1030 .1720 1377 .2254 1723 .6579 2565 ,7926 2059 ,8956 1291 .9470 739
355
9460 9490 9670 9870 11400 12520 13810 14720
TABLE I11 MOLARINTEGRAL HEATSO F MIXING I N Cd(l)-Sn( 1) Tots1
g. atoms
Composition XCd
AHM, joule/g. atom
AHM/XCdXfJn, joule
1.1105 1.3572 1.0163 0.7969 1,1866 0.8176 1.1069 1.0638 0.8442 0.9184 1.2224 0.7361 1.0229 0.9485 0,4151 0.8106 0.9073 0.6536 1.1154 0.6941 0.8183 0.9142 1.0390
(a) At350 i 2' 0.0540 369 .0573 387 .0584 394 .1115 711 .1147 727 ,1195 772 .1354 871 .1398 874 .2119 1200 .2162 1265 ,2514 1374 .2914 1534 .3496 1687 .4500 1929 .5996 2028 .7541 1722 .7996 1476 .8134 1464 .8793 1087 .8807 1086 .8866 988 .9167 823 .9440 590
7220 7160 7170 7180 7160 7340 7440 7270 7190 7470 7300 7430 7420 7790 8450 9290 9210 9650 10240 10340 9830 10780 11160
0.9016 0.8739 0.9090 1.4895 0.9114 .9557 .8503 .9356 .5787 .6983 .5706 .4442 .6528 1.1136 0.5607 1,0226
(b) At 450 f 2' 0.0300 193 .0401 256 .0503 322 .0547 353 .0796 497 ,0868 539 * 0918 558 ,1745 987 .2065 1137 .3424 1609 .5619 1879 .7218 1674 .7264 1679 .8413 1221 .8457 1202 .9161 745
6630 6650 6740 6830 6780 6800 6690 6850 6940 7150 7630 8340 8450 9150 9210 9690
,
We again find that the heat of mixing appears to be independent or very nearly independent of temperature. The agreement with the recent e.m.f. study by Elliott and Chipman6 is excellent, while Kawakami's data seem to be too low. Cadmium-Tin.-The results of 39 experiments are reported, 23 at 350" and 16 a t 450'. We here note for the first time a dependence of AH on temperature, corresponding to negative deviations from (6) J. F. Elliott and J. Chipman, Trans. Faraday SOC.,47,138 (1951).
0. J. KLEPPA
356 TABLE IV
MOLAR INTEGRAL HEATSOF MIXINGIN Sn(1)-Zn(1) Total atoms
g.
1.2264 1.2177 1.0176 0.8884 .5351 4846 .4114 .4352 .5451 .4045 .3187 .4519 .5118 .3549 .3501 .3533 .3803 .3455 .3375 .4717 .3776 .5636 0.7104 1.0916 1.1276 1.1714 0.6752 .4739 ,5622 $4206 ,3116 .3104 .4361 .5315 .9616 1.5243
Composition X&n
,
AHM,
joule/g. atom
(a) At 430 f 2' 0.0551 506 .0991 920 .1203 1093 ,1365 1195 .1730 1531 .1749 1515 .2134 1775 .2385 1986 .2393 2073 .2393 1968 .2673 2232 .3674 2838 .3677 2781 .4636 3130 ,5058 3322 .5482 3390 .5632 3412 .6387 3346 ,6848 3145 .7546 2909 .7675 2898 .8498 2328 .8780 2036 .9235 1337 .9430 1147 .9472 1055 (b) At525 f 3" 0.1237 977 .2235 1716 .3456 2411 .3703 2494 .4895 2837 .6795 2897 .7703 2737 ,8029 2494 .9214 1411 .9481 953
A.HM]XE~XZ~,
joule
9720 10300 10330 10140 10700 10500 10570 10940 11390 10810 11400 12210 11960 12590 13290 13690 13870 14500 14570 15710 16240 18240 19010 18930 21340 21100 9010 9890 10660 10700 11350 13300 15470 15760 19480 19370
the Kopp-Neumann rule for the heat capacity of the mixture. The agreement with Elliott and Chipman's e.m.f. data is less perfect than for the system cadmium-lead, while we find once more that our results are higher than those given by Kawakami for the same composition. Tin-Zinc.-For this system a total of 36 experiments are reported, among which 26 were performed at 430 and 10 at 525". In spite of the less satisfactory precision the dependence of AH on temperature is again well illustrated. The agreement of the present data with those reported by Kawakami is here somewhat better than for the other three systems. We also note that our results for 430" (but not those for 525') agree well with Taylor's e.m.f. data for about 500°.6 Discussion The author has previously suggested the use of the quantity AHM/xlx2as an aid in the plotting of integral heat data for binary systems.' This quantity is a particularly convenient measuring (6) N. W. Taylor, J . Am. Chem. Soc.. 46. 2885 (1923).
VOl. 59
stick for the energetic asymmetry in the system, and we note that the value of AHM/x1x2 for x = 0 is equal to the relative partial molal heat content of the minor component in its infinitely dilute solution in the major component. We recall also that the very simplest types of solution theories, such as Hildebrand's "regular" solution theory for components of equal volumes and the zeroth approximation of the "quasi-chemical" theory, predict a completely symmetrical, parabolic heat of mixing curve. I n these cases the quantity of AHM/xlx2 will be constant, independent of composition (and temperature). It is apparent that in the four systems covered in the present investigation AHM/xlx2is far from independent of composition. In fact the systems cadmium-tin, cadmium-lead and tin-zinc exhibit a large degree of energetic asymmetry, increasing in the order mentioned. It may be pointed out that this is also the order of increasing relative difference in atomic volumes of the two components. It is noted that the solution of the component with the larger atomic volume in the component with the smaller volume (Cd, Zn) is always associated with a larger heat absorption than the opposite process. This is in agreement with a rule found to be valid in many binary liquid mixtures. The system bismuth-lead differs radically from the other three systems and has a nearly symmetrical negative heat (AH) of mixing curve. However, the quantity AH'/x1x2 is far from independent of composition and has for equi-atomic mixtures fallen to a value about 25% below the terminal values. It is noted that the plots of AHM/xa2 for the other systems exhibit similar negative deviations from the straight line connecting the two terminal values. We shall show that these deviations may possibly be accounted for by the short range order (or clustering) present in the mixtures. Short Range Order.-According to the quasichemical theory the Helmholtz energy of mixing for one mole of a simple binary mixture is given by the expression'
In this expression x is the coordination number (which is assumed to be the same in the mixture and in the pure components and which we set -10 for simplicity), while the single interaction parameter X is independent of composition and given by N(2Wl2
-
WI1
- W22)
Here w12, w11 and w22 are the potential energies of the 12, 11 and 22 bonds, and N is Avogadro's number. We recall that the first term on the right-hand side in the expression above represents the zeroth approximation where the effect of short range order (or clustering) is neglected, while the second term takes this effect into account. As we are dealing with a process in the condensed phase we may disregard the small difference between H and E , and get
(7) 0.J. Kleppa. dbid.. 73. 3346 (1950).
THERMODYNAMIC STUDY OF LIQUID METALLIC SOLUTIONS
April, 1955
357
- 6000 - 5000 X
2- 3000 2 - 1200
d U
-1000
. -. bi
2
.-B-
-800
tca
- 600
- - - SELTZ, e.m.f.,360-470°C.
-400
-4-THIS STUDY, 350'C.
-4THIS STUDY, 450' C.
-200
0
Bi
0.1
0.4 0.5 0.6 0.7 Atomic fraction, XPb. Fig. l.--Heats of mixing in Bi(1)-Pb(1).
0.2
0.3
We now leave the domain of strict applicability of the quasi-chemical theory, and assume arbitrarily that the relations above are still valid if we let X vary linearly with concentration. For any concentration we may then obtain a value for X by interpolating linearly between the two terminal Values of AHM/xIx2. By inserting this value in (2) we may calculate a value for ANM/xlx2 which may be compared to the experimental value. For equiatomic mixtures in the four systems under consideration such a comparison is presented in Table V. When the experimental uncertainties and the arbitrary nature of the various assumptions are taken into account, it seems that the agreement between the calculated and observed values of AHM/ ~1x2 is as good as can be expected. Differential Quantities.-It is recalled that in e.m.f. experiments the relative partial molal heat content of the more electro-positive component is obtained through the temperature dependence of the chemical potential. The corresponding quan-
0.8
0.9
1.0 Pb
TABLE V EFFECT OF
SHORT RANQE ORDER ON THE HEATOF MIXINQ (DATA IN KILOJOULES) Syatem (AHM/xixrh.s (temp., "C.) xo.0 Calad. Obad.
Bi-Pb(400) Cd-Pb (400) C d S n (350) CdSn (450) Sn-Zn (430) Sn-Zn (525)
-3.6 12.4
9.5 8.6 16.3 15.0
-3.8 9.6 7.8 7.4 11.7 11.6
-4.4 10.7 8.0 7.5 13.2 11.5
tities for the other component may be obtained from an integration of the Gibbs-Duhem-Margules relation. However, these values will *normally be associated with greater uncertamty. From the integral heat data reported above, on the other hand, we may derive the differential quantities of either component by means of suitable graphical or analytical methods. We shall here give special attention to the partial mold quantities in moderateIy dilute solutions only.
0.J. KLEPPA
358 16000 15000
I'
I
I
I
I
VOl. 59 I
I
I
I
-
13000
12000
:*
I
\
#
9
11000
10000
g
*3.-
9000
42
Ld
2000
$ 1500 X
KAWAKAMI, CALORIMETRIC,350°C.
----ELLIOT, e.mf., 500 1000
OC.
eTHIS STUW,35O0 C. ct THIS STUDY, 45OOC.
500
0 Pb
0.1
0.4 0.5 0.6 0.7 Atomic fraction, XCd. Fig. 2.-Heats of mixing in Pb(1)-Cd(1).
0.2
0.3
For dilute solutions we may expand the integral molar heat of mixing in powers of the mole fraction of the minor component, x
The first term on the right-hand side is by definition zero, while (bAHM/bx),o = a is equal to the relative partial molal heat content of the minor component in its infinitely dilute solution. The coefficient of 2 2 , which we shall denote by b, for simplicity, similarly pravides a measure for the curvature of AHMa t x = 0. By definition we have AHM = xZ1f (1 - x)E2. If the two first non-zero terms only are used in the expansion, the expressions for the two partial molal heat contents, 1,relative to the pure components, will be particularly simple z1
+
-
(minor component) = a Zbx bx2 Zz (major component) = bz*
-
(4)
(5)
0.8
0.9
1.0
Cd
We note that a plot of AH'/x versus x for moderate values of x should yield a straight line, with slope b and intercept a a t x = 0. In general it is of course impossible to predict how far into the non-dilute concentration range the simple linear relationship will hold. This introduces certain ambiguities, particularly in the determined value of b. In the case of the systems considered it was found that the plots indicated a linear relationship up to x -0.2 or more. In Table VI are given values for a and b from the present investigation along with data from earlier e.m.f. studies. An approximate value of b may sometimes be obtained from the binary equilibrium phase diagram in the vicinity of the melting point of the pure component. We recall that if solid solubility is negligible, and if the difference in heat capacity between the pure solid and the pure undercooled liquid may be neglected, the chemical potential a t the liquidus(of the component present in both phases,referred to the pure undercooled liquid) will be given by p - I r o = (T - Tm)AHi/Tm (6)
359
THERMODYNAMIC STUDY OF LIQUIDMETALLIC SOLUTIONS
April, 1955 12000
11000
10000 ai
3
.E
9000
7000
i
L
T
4
