June, 1929
HEAT CAPACITIES AND HEAT FUNCTIONS OF METALS
1711
2. The atomic radius calculated from these data is 2.171 A. a t room temperature. URBANA, ILIJSOIS [CONTRIBUTION FROM THE LABORATORY OF PHYSICS, UNIVERSITY OF ILLINOIS ]
PARTIAL MOLAL HEAT CAPACITIES AND RELATIVE PARTIAL MOLAL HEAT FUNCTIONS IN SOLUTIONS OF MOLTEN METALS BY ALBERTN . GUTHRIEAND RECEIVED FEBRUARY 8, 1929
EARLE. LIBMAN PUBLISHED J U N E 5 , 1929
There exist in the literature no data on the partial molal properties of solutions of molten metals. There is likewise a great dearth of experimental work from which such data may be calculated. It seems worth while, therefore, to put on record the following results for the partial molal heat capacities and relative partial molal heat functions of Pb-Sb and BiCd solutions calculated from the work of Wiist and Durrer.‘ By direct calorimetric measurements these investigators have determined the values of the specific heats of the above solutions a t different concentrations from which the partial molal heat capacities may be determined. They have also obtained the heats absorbed per gram in going from the pure solid components to the liquid solution a t the eutectic temperature, for different concentrations. From these data the relative partial molal heat functions may be calculated.
Relative Partial Molal Heat Functions Consider a solution made up of components a and b. Let H = E PV be the total heat function of the solution. Then the partial molal heat function ha of component a in the solution is defined as (dH/dn,)pm, where the n’s are the number of moles of the respective components in the solution. If hl is the value of ha in some arbitrarily chosen reference state, ha = ha - h: is called the relative partial molal heat function. I‘t is convenient to choose for the reference state the infinitely dilute solution of a in b,2 Consider a system composed of some pure a and some solution. Let some of the a go from the pure phase into the solution. If Q is the heat absorbed by the system, then the molal heat of solution of a is defined as I , = (dQ/dn,)pTnb. Knowing 1, we may calculate ha from the relation
+
-
ha = 1, -
Finally, when
[lalna =
o
Ea is known, Lb can be calculated from
(1)
“Forschungsarbeiten auf dem Gebiete des Ingenieurwesens,” Heft 241 (1922). Lewis and Randall, “Thermodynamics and the Free Energy of Chemical Substances,” McGraw-Hill Book Co., Inc., New York, 1923, pp. 87-95.
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Vol. 51
ALBERT iV. GUTHRIE AND EARL E. LIBMAN
I,et us now fix our attention upon a system of two components completely miscible in the liquid state, completely immiscible in the solid state. Such a system has a defi30 nite eutectic temperature. Let L be the heat absorbed when one n. mole of b and n, moles of a go from I 4 0 the pure solid state a t atmospheric x 20 pressure and eutectic temperature into the liquid state to form a CI solution. Then if X, and Xb are % the molal heats of fusion, a t the eutectic temperature, of a and b 0"
2
;d
+
10
L
i
= Ab
f
A&o
+
r'
na =
ladno
0
(3)
and 0 0
Values of L are given by Wiist and Durrer in the article mentioned, for different concentrations. plotting L against n, and drawing tangents, values of 1, may be determined by (4). As shown in Figs. l and 2, the L-n, curves for the SYStems Pb-Sd and Bi-Cd are straight lines. Thus (dL/'dn,)pT,, is, in each case, a constant, independent of 12 the concentration, and by (4) the heats of solution for these systems are likewise constant, independent 1 . 10 of the concentration. Then by 2 (1) and (2) the relative partial X 8 molal heat functions a t the eutecd tic temperature are strangely $ 6 enough zero for these solutions at all concentrations (Tables I and 5
1 2 3 4 5 Moles of Sb in one mole of Pb. Fig. 1.-Relative partial molal heat functions in Pb-Sb solutions a t eutectic temperature.
d
11).
Partial Molal Heat Capacities
$ 4 Y
d
z
2
Let C, be the total heat capacity of a solution of components a and 0 b. Then the partial molal heat 0 1 2 3 4 5 capacity of a is defined as tpa = Moles of Bi in one mole of Cd. (dcp/dn,)pT,,. The data of Wiist Fig. 2.-Relative partial molal heat funcand Durrer give the specific heats tions in Bi-Cd solutions a t eutectic temfor the solutions Pb-Sb and Bi-Cd perature.
June, 1929
1713
HEAT CAPACITIES AND HEAT FUNCTIONS OF META1,S
TABLEI RELATIVE: PARTIAL
yo
Sb,
7 13 25 50 75
MOLALHEATFUNCTIONS
I N P B S B SOLUTIONS AT EUTECTIC T E M PERATURE
Mole frac. G = gesamte ttsb =moles of S b of Sb Warmeeffekte"
8.05 0.128 9.08 ,254 ,567 13.40 22.76 1.70 30.55 5.10 .836 Wiist and Durrer. L = G X wt. of solution. 0.114 .203 ,362 .630
Wt. of soh
L = heat (dL/absorbed dnsb)PTnpb
222.8 238.2 276.3 414.4 828.8
1794 2165 3705 9430 25300
T A B L E 11 RELATIVE PARTIAL bIOLAL HEATFUKCTIONS I N BI-CD
4760 4760 4760 4760 4760
-
0 0 0 0 0
SOLUTIONS AT
i;~b
0 0 D 0
0
EUTECTIC
TEMPERATURE
Bi,
%
Mole frac.
of Ri
-
G = gesamte ngi moles of Bi Warmeeffekten
0.056 8.44 8.61 .118 .187 8.98 ,263 9.47 9.47 50 ,350 9.69 60 .446 63 .477 9.55 10.02 TO ,556 10.16 80 .682 90 .828 9.92 Wust and Durrer. L = G X wt. of solution. 10 20 30 40
0.054 .134 ,230 ,368 ,538 .807 ,916 1.257 2.145 4.850
L = heat (dL/absorbed dnBi)PTnCd h ~ i
Wt. of soh
123.8 140.5 160.6 187.3 224.8 281.0 304.0 374.7 562.0 1124.1
1046 1210 1443 1775 2130 2695 2900 3755 5705 11150
0 0 0 0 0 0 0 0 0 0
2160 2160 2160 2160 2160 2160 2160 2160 2160 2160
-
Ilcd
0 0 0 0 0 0 0 0 0 0
TABLEI11 COMPARISONS OF Component
Bi
Temperature, "C.
300 0.0344 ,0345
4)
DATA Cd
1000 0.0416 ,0419
300 0.0733 .0733
1000 0.0823 .0823
Pb
Sb
700 0,0337 .0338
700 0.0547 .0550
TABLE IV PARTIAL MOLALHEATCAPACITIES IN PB-SB SOLUTIONS Pb, 96 25 50 75 87 93 Mole fraction of Pb 0.164 0.370 0.638 0.797 0.886 Sp. heat of the soh., 700°C." ,0491 .0443b .0387 ,0367 ,0351 Partial specific heat, 700 all concns. Pb, ,0337 Sb, .0547 Partial molal heat capacity, 700 ", all concns. Pb, 6.98 Sb, 6.66 a Wiist and Durrer. In the original article some error was made in the calculation of this value and i t has been recalculated from the experimental data. O,
for different weight per cents. These are plotted in Figs. 3 and 4. The tangents to these curves at any concentration intersect the axes a t points
1714
ALBERT N . GUTHRIE AND EARL E. LIBMAN
Vol. 51
representing the partial specific heats of the component^.^ For the systems considered the curves are straight lines and so the partial specific heats, and also the partial molal heat capaci0.06 ties, are constant, independent of ci the concentration and equal to the molal heat capacities of the pure > components in the liquid state. 0 This circumstance affords a means d0.05 v of checking the results by coms vi parison with data obtained by 0 Wiist, Meuthen and Durrer4 from c, the pure components (Table 111). 0.04 In making use of the valuable 2 E data obtained by these investigaV tors, it is to be noted that some 3 errors have been made in the cal(0 0.03 culations. Thus Table 29 on the 0 20 4o 6o true specific heats of lead, above % By weight of lead. the melting point, is in error and Fig. 3.-Partial molal heat capacities in Pb- should be recalculated directly Sb solutions. from the experimental data given in the article. Unfortunately the errors noted have been reproduced in the “Handbook of Chemistry and Physics” published by the Chemical R u b b e r P u b l i s h i n g Co., the 0.09 Smithsonian Physical Tables (7th 9 edition), and perhaps elsewhere. 0,08 We have seen that the partial 2 molal heat capacities of Bi-Cd 6 0.07 are constant and equal to the .e molal heat capacities of the pure 3 liquid components both a t 300 0.06 and 1000°. It is reasonable to 3 assume that this is true for all % + 0.05 temperatures. If we do so we * 2 may readily calculate the relative 3 0.04 partial molal heat functions a t temperatures other than that of 8
2
-
-
.e
Lc
8
2
~
0.03
the eutectic. 0 20 40 60 80 100 We have (dh,/dT), = cpa 2nd yo By weight of bismuth. (dh:/dT)* = c g a ) SO that (dha/- Fig. 4.-~artial molal heat capacities in BidT), = [d(h, - hi)/dT], = Cd solutions. 3
Lewis and Randall, ref. 2, p. 38.
4
Ref. 1, Heft 204 (1918).
June, 1920
Bi,
%
10 20 30 40 50 60 63 70 80 90
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HEAT CAPACITIES AND HEAT FUNCTIONS OF METALS
TABLEV PARTIAL h.IOLAL HEAT CAPACITIES IN BISXCTH-CADMIUM SOLUTIONS Partial specific heat, Bi, mole &. ht. of soln." fraction
3OOOC.
0.056 0.0695 ,118 ,0656 ,187 ,0614 ,263 ,0576 ,350 .0536 ,446 .0499 ,477 .0489 ,556 .0459 .682 .0422 .828 .0382 Wiist and Durrer.
1000o c .
all concentrations
0.0785 ,0743 .0695 .0659 .0614 ,0576 .0569 .0537 .0500 ,0459
300 "C. 300 "C.
Cd Bi
looooc. looooc.
0.0344 0.0416 ,0733 .0823
Partial molal heat capacity, all concentrations
300 "C.
Bi Cd
300 "C. looo"c' 1000°C.
7.20 8.69 8.24 9.25
0 cPc- cpa. But on the assumption just made tpa- c:~ = 0, and we have (dh,/dT) = 0. From this we see that ha is independent of the temperature. Thus not only a t the eutectic temperature but a t all temperatures the relative partial molal heat functions of the components are zero at all concentrations.
Conclusions 1. The relative partial molal heat functions of the components of the systems Pb-Sb and Bi-Cd have been calculated for the eutectic temperatures. They are found to be equal to zero independent of the concentrations. 2. The partial molal heat capacities of the components of the system Pb-Sb have been calculated a t 700" and have been found to be constant and equal to the molal heat capacities of the pure liquid components. The same calculation has been performed for the system Bi-Cd at the temperatures 300 and 1000" and in each case the partial molal heat capacities have been found to be constant, independent of concentration and equal to the molal heat capacities of the pure components. 3 . Assuming that the partial molal heat capacities are the same as the molal heat capacities of the pure liquid components, not only at the temperatures for which this has been verified but a t all others, it is shown that the relative partial molal heat functions are zero not only at the eutectic temperature but a t all others. URBANA, ILLINOIS
