ASM Metals HandBook Volume 15 Casting

Prévia do material em texto

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Scand. J. Metall., Vol 11, 1982, p 143-149 
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80-89 
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Th er m od y n am ic Pr op er t ies o f Alu m in u m - Base an d Co p p er - Base A l loy s 
Basant L. Tiwari, General Motors Research Laboratories 
 
I n t r od u ct ion 
THIS ARTICLE will provide accessible information on the thermodynamic properties of liquid aluminum-base and 
copper-base alloys. Three alternative means have been used to report the thermodynamic data: 
· Activities and activity coefficients 
· Partial and integral molar thermal properties 
· Interaction coefficients 
In addition, the utility of phase diagrams will be briefly discussed. Knowledge of activities and activity coefficients is 
necessary in describing solution behavior and in solving problems that involve chemical equilibria. The thermal 
properties are useful in understanding the liquid state and in correlating data on solution behavior. The interaction 
coefficients provide a simple means of calculating activity coefficients in dilute solutions and are also used to correlate 
experimental data on dilute solutions. 
Act iv i t i es an d Th er m al Pr op er t ies 
The activity a of constituent i in a solution is given by the relationship: 
G i = H i – T Si = RT ln ai 
(Eq 1) 
where G i, H i, and Si are the partial molar free energy, enthalpy, and entropy, respectively, of constituent i in 
solution relative to the pure constituent i at the solution temperature as the standard state; T is the temperature (in degrees 
kelvin), and R is the universal gas constant. Thus, from the knowledge of H i and Si as functions of temperature and 
solution composition, one may calculate G i, ai, and i (activity coefficient of component i relative to pure material 
standard state) for all conditions of solution. It is important to realize, however, that thermal property values vary with 
solution composition, unlike the properties of pure substances. 
Replacing ai in Eq 1 with Xi i yields: 
G i = RT ln Xi + RT ln i 
(Eq 2) 
where Xi is the mole fraction of component i. In Eq 2, the term RT ln i is called partial molar excess free energy and is 
expressed by the symbol 
XS
ïGD . It represents the contribution to G i resulting from the departure of solution from ideal 
behavior. Thus, one may write: 
XS
ïGD = RT ln i 
(Eq 3) 
Corresponding to Eq 1, 
XS
ïGD can also be expressed as: 
XS
ïGD = 
XS
ïHD - T 
XS
ïSD = RT ln i 
(Eq 4) 
However, because H i for the ideal solution is 0 by definition, 
XS
ïHD and H i are identical. Therefore, Eq 4 can be 
rewritten as: 
XS
ïGD = H i - T 
XS
ïSD = RT ln i 
(Eq 5) 
Equations 1 and 5 are commonly used to calculate activities and activity coefficients from the thermal properties of the 
solution. 
Although the partial molar thermal properties of individual constituents in a solution are most useful in solving chemical 
process problems, the integral properties are valuable in the discussion of the nature of solutions and also serve in some 
cases as the source of data on the partial properties. The integral molar free energy is related to the partial molar free 
energy by the following relationship: 
Gi = Xi G i + X j G j + . . .. Xn G n 
(Eq 6) 
Similar equations can also be written to express other integral thermal properties. 
In Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14, thermodynamic data in the form of activities, activity coefficients, 
partial molar thermal properties, and integral molar properties have been compiled for selected aluminum-base and 
copper-base alloys. The alloy systems have been chosen based on their commercial importance and on the concentration 
of the major alloying elements. In general, the concentrations of the alloying elements in the commercial alloys are below 
10 at.%, and under this condition the interaction coefficient method described below provides a convenient means of 
calculating the activities of the constituents in solution. When the concentrations are close to or greater than 10 at.%, the 
interaction coefficient method does not yield accurate values. With this factor in mind, thermodynamic data have been 
selected for the following alloy systems: Al-Mg, Al-Si, Cu-Al, Cu-Ni, Cu-Sn, and Cu-Zn. 
Table 1 Activities, activity coefficients, and partial molar thermal properties for liquid aluminum-magnesium 
alloys at 1073 K 
Al component Al( l ) = Al (ln alloy) (l ) 
XAl aAl Al G Al, 
cal/g-atom 
SAl, 
cal/K-g-atom 
H Al, 
cal/g-atom 
XS
ALGD , 
cal/g-atom 
0.9997 0.9996 0.9999 -0.9 0.0091 9 -0.2 
0.9992 0.9992 1.0000 -1.7 0.0016 0 0.0 
0.9989 0.9989 1.0000 -2.3 0.0105 9 0.0 
0.9945 0.9940 0.9996 -12.8 0.0203 9 -0.9 
0.9886 0.9876 0.9991 -26.6 0.0417 18 -1.9 
0.9785 0.9791 1.0006 -45.0 0.0420 0 1.3 
0.9553 0.9571 1.0020 -93.5 0.0958 9 4.3 
0.8870 0.8887 1.0020 -251.6 0.2157 -20 4.3 
0.8095 0.8008 0.9894 -473.7 0.3790 -67 -22.7 
0.6600 0.6451 0.9775 -934.7 0.7806 -97 -48.5 
0.4175 0.4115 0.9859 -1893.4 1.1548 -654 -30.3 
0.2510 0.2597 1.0353 -2874.9 1.2240 -1561 74.0 
0.1792 0.1652 0.9219 -3839.5 -0.2920 -4153 -173.4 
0.0450 0.236 0.5251 -7988.8 -3.8840 -12157 -1373.6 
Mg component Mg( l ) = Mg (in alloy) ( l ) 
XMg aMg Mg G Mg, 
cal/g-atom 
SMg, 
cal/K-g-atom 
H Mg, 
cal/g-atom 
XS
MgGD , 
cal/g-atom 
0.00027 0.00022 0.8060 -18017 15.4180 -1470.8 -459.73 
0.00079 0.000630.7974 -15696 13.9463 -729.8 -482.71 
0.00104 0.00081 0.7814 -15171 12.6176 -1630.2 -525.87 
0.0055 0.0048 0.8780 -11358 9.6420 -1010.9 -227.42 
0.0114 0.0107 0.9353 -9683 7.4045 -1736.7 -142.53 
0.0215 0.0183 0.8535 -8530 6.9708 -1049.0 -337.68 
0.0447 0.0366 0.8188 -7051 5.4069 -1248.7 -424.12 
0.1130 0.0929 0.8221 -5069 3.6123 -1192.1 -418.20 
0.1905 0.1679 0.8813 -3806 2.7957 -805.9 -269.73 
0.3400 0.3097 0.9109 -2500 1.6331 -747.6 -198.80 
0.5825 0.5176 0.8885 -1405 1.2733 -38.2 -251.88 
0.7490 0.6367 0.8500 -963 1.3102 443.2 -346.49 
0.8208 0.7184 0.8752 -705 1.7577 1180.8 -284.20 
0.9550 0.9198 0.9631 -178 1.1672 1074.3 -79.98 
Source: Ref 1 
Table 2 Integral thermal properties for liquid aluminum-magnesium alloys at 1073 K 
(1-x)Al( l ) + xMg( l ) = Al(1-x) Mg x( l ) 
XMg G, 
cal/g-atom 
S, 
cal/K-g-atom 
H, 
cal/g-atom 
Gxs, 
cal/g-atom 
0.00027 -5.7 0.01320 8.61 -0.34 
0.00079 -14.2 0.01274 -0.58 -0.39 
0.00104 -18.1 0.02365 7.29 -0.55 
0.0055 -75.7 0.07366 3.34 -2.39 
0.0114 -136.9 0.12577 -2.04 -3.53 
0.0215 -227.3 0.19086 -22.54 -6.00 
0.0447 -404.8 0.33346 -47.28 -14.91 
0.1130 -796.1 0.59962 -152.48 -43.49 
0.1905 -1108.7 0.83951 -207.80 -69.79 
0.3400 -1466.9 1.07043 -318.19 -99.61 
0.5825 -1608.9 1.22383 -295.28 -159.37 
0.7490 -1442.6 1.28856 -59.77 -240.99 
0.8208 -1266.9 1.39046 225.25 -264.35 
0.9550 -529.7 0.93986 478.87 -138.20 
Source: Ref 1 
Table 3 Activities, activity coefficients, and partial molar thermal properties for liquid aluminum-silicon 
alloys at 1100 K 
Al component Al( l ) = Al (in alloy) (l ) 
XAl aAl Al G Al, 
cal/g-atom 
XS
AlGD , 
cal/g-atom 
0.90 0.859 0.954 -333 -102 
0.80 0.698 0.872 -786 -298 
0.70 0.503 0.719 -1500 -720 
0.69(a) 0.473 
(±0.01) 
0.685 
(±0.02) 
-1638 
(±50) 
-826 
(±50) 
Si component Si( l ) = Si (in alloy) (l ) 
XSi aSi Si G Si, 
cal/g-atom 
XS
SiGD , 
cal/g-atom 
H Si, 
cal/g-atom 
SSi, 
cal/K-g-atom 
XS
SiSD , 
cal/K-g-atom 
0.00 0.000 0.040 -¥ -7060 -2502 ¥ 4.144 
0.10 0.016 0.155 -9106 -4073 . . . . . . . . . 
0.20 0.051 0.255 -6510 -2991 . . . . . . . . . 
0.30 0.127 0.424 -4509 -1877 . . . . . . . . . 
0.31(a) 0.147 
(±0.02) 
0.473 
(±0.60) 
-4194 
(±300) 
-1634 
(±300) 
. . . 
(±50) 
. . . . . . 
Source: Ref 2 
(a) Phase 
boundary. 
 
Table 4 Integral thermal properties for liquid aluminum-silicon alloys at 1100 K 
(1 - x)Al( l ) + xSi( l ) = Al(1-x)Six( l ) 
XSi G, 
cal/g-atom 
Gxs, 
cal/g-atom 
0.1 -1210 -499 
0.2 -1931 -837 
0.31(a) -2430 
(±100) 
-1076 
(±100) 
Source: Ref 2 
(a) Phase boundary. 
 
Table 5 Activities, activity coefficients, and partial molar thermal properties for liquid copper-aluminum 
alloys at 1373 K 
Al component Al( l ) = Al (in alloy) (l ) 
XAl aAl Al G Al, 
cal/g-atom 
XS
AlGD , 
cal/g-atom 
H Al, 
cal/g-atom 
SAl, 
cal/K-g-atom 
XS
AlSD , 
cal/K-g-atom 
1.0 1.000 1.000 0 0 0 0.000 0.000 
0.9 0.889 0.988 -320 -32 29 0.254 0.044 
0.8 0.759 0.949 -753 -144 44 0.580 0.137 
0.7 0.609 0.870 -1354 -381 -60 0.942 0.234 
0.6 0.441 0.735 -2235 -842 -391 1.343 0.328 
0.266 0.532 -3611 -1720 -1055 1.862 0.484 0.5 
(±0.02) (±0.04) (±150) (±150) (±300) (±0.25) (±0.25) 
0.4 0.116 0.290 -5873 -3373 -2878 2.181 0.361 
0.3 0.028 0.095 -9709 -6424 -5012 3.421 1.028 
0.2 0.006 0.029 -14062 -9670 -5864 5.971 2.772 
0.1 0.001 0.008 -19307 -13025 -7415 8.661 4.086 
0.0 0.000 0.002 -¥ -16640 -8625 ¥ 5.838 
Cu component Cu( l ) = Cu (in alloy) (l ) 
cal/g-atom cal/g-atom cal/g-atom cal/K-g-atom cal/K-g-atom 
0.0 0.000 0.042 -¥ -8671 -4225 3.238 
0.1 0.005 0.052 -14349 -8067 -4849 6.919 2.344 
0.2 0.013 0.065 -11833 -7442 -4975 4.995 1.797 
0.3 0.025 0.084 -10025 -6741 -4675 3.897 1.505 
0.4 0.046 0.115 -8396 -5896 -4071 3.150 1.329 
0.085 0.170 -6728 -4837 -3271 2.518 1.141 0.5 
(±0.005) (±0.01) (±150) (±150) (±300) (±0.25) (±0.25) 
0.6 0.166 0.277 -4900 -3506 -1838 2.230 1.215 
0.7 0.352 0.503 -2848 -1875 -679 1.580 0.871 
0.8 0.600 0.750 -1394 -785 -259 0.827 0.383 
0.9 0.839 0.932 -478 -191 -65 0.301 0.092 
1.0 1.000 1.000 0 0 0 0.000 0.000 
Source: Ref 2 
Table 6 Integral thermal properties for liquid copper-aluminum alloys at 1373 K 
(1 - x)Al( l ) + xCu( l ) = Al(1-x)Cu x( l ) 
XCu G, 
cal/g-atom 
H, 
cal/g-atom 
S, 
cal/K-g-atom 
Gxs, 
cal/g-atom 
Sxs, 
cal/K-g-atom 
0.1 -1723 -459 0.921 -836 0.275 
0.2 -2969 -960 1.463 -1604 0.469 
0.3 -3955 -1445 1.828 -2289 0.615 
0.4 -4700 -1863 2.066 -2863 0.728 
-5170 -2163 2.190 -3278 0.812 0.5 
(±150) (±150) (±0.1) (±150) (±0.1) 
0.6 -5289 -2254 2.210 -3453 0.873 
0.7 -4906 -1979 2.132 -3240 0.918 
0.8 -3927 -1380 1.855 -2562 0.861 
0.9 -2361 -800 1.137 -1474 0.491 
Source: Ref 2 
Table 7 Activities, activity coefficients, and partial molar thermal properties for liquid copper-nickel alloys at 
1823 K 
Cu component Cu( l ) = Cu (in alloy) (l ) 
XNi aCu Cu G Cu, 
cal/g-atom 
XS
CuGD , 
cal/g-atom 
0.0 1.000 1.000 0 0 
0.1 0.902 1.002 -374 8 
0.2 0.814 1.017 -747 61 
0.3 0.740 1.058 -1088 204 
0.4 0.677 1.128 -1414 436 
0.611 1.222 -1786 725 0.5 
(±0.02) (±0.03) (±100) (±100) 
0.6 0.534 1.334 -2275 1044 
0.7 0.444 1.480 -2941 1421 
0.9 0.191 1.912 -5992 2349 
1.0 0.000 2.227 -¥ 2900 
Ni component Ni( l ) = Ni (in alloy) ( l ) 
aNi Ni G Ni, 
cal/g-atom 
XS
NiGD , 
cal/g-atom 
0.000 1.906 -¥ 2336 
0.185 1.864 -6120 2222 
0.341 1.704 -3899 1931 
0.455 1.517 -2851 1510 
0.539 1.347 -2241 1079 
0.611 1.222 -1786 725 
(±0.02) (±0.03) (±100) (±100) 
0.682 1.136 -1337 464 
0.752 1.075 -1031 261 
0.826 1.032 -692 116 
0.907 1.008 -353 29 
1.000 1.000 0 0 
Source: Ref 2 
Table 8 Integral thermal properties for liquid copper-nickel alloys at 1823 K 
(1 - x)Cu( l ) + xNi( l ) = Cu(1-x)Ni x( l ) 
XNi G, 
cal/g-atom 
H, 
cal/g-atom 
S, 
cal/K-g-atom 
Gxs, 
cal/g-atom 
Sxs, 
cal/K-g-atom 
0.1 -948 240 0.652 229 -0.006 
0.15 -1195 341 0.842 336 -0.003 
0.2 -1377 435 . . . 
0.3 -1617 . . . . . . 596 . . . 
0.4 -1745 . . . . . . 693 . . . 
0.5 -1786 
(±100) 
. . . . . . 725 
(±100) 
. . . 
0.6 -1742 . . . . . . 696 . . . 
0.7 -1604 . . . . . . 609 . . . 
0.8 -1349 . . . . . . 464 . . . 
0.9 -917 . . . . . . 261 . . . 
Source: Ref 2 
Table 9 Activities, activity coefficients, and partial molar thermal properties for liquid copper-tin alloys at 
1400 K 
Cu component Cu( l ) = Cu (in alloy) ( l ) 
XCu aCu Cu G Cu, 
cal/g-atom 
XS
CuGD , 
cal/g-atom 
H Cu, 
cal/g-atom 
SCu, 
cal/K-g-atom 
XS
CuSD , 
cal/K-g-atom 
1.0 1.000 1.000 0 0 0 0.000 0.000 
0.9 0.802 0.891 -613 -320 -159 0.325 0.115 
0.8 0.539 0.674 -1717 -1096 -749 0.691 0.248 
0.7 0.389 0.556 -2626 -1633 -1443 0.845 0.136 
0.6 0.284 0.474 -3497 -2076 -1662 1.311 0.295 
 
(±0.025) (±0.05) (±300) (±300)(±250) (±0.3) (±0.3) 
0.4 0.169 0.422 -4949 -2400 -1418 2.522 0.701 
0.3 0.125 0.417 -5781 -2432 -1049 3.380 0.988 
0.2 0.082 0.408 -6971 -2493 -640 4.522 1.324 
0.1 0.038 0.379 -9104 -2698 81 6.561 1.985 
0.0 0.000 0.317 -¥ -3197 1050 ¥ 3.034 
Sn component Sn( l ) = Sn (in alloy) ( l ) 
XSn aSn Sn G Sn, 
cal/g-atom 
XS
SnGD , 
cal/g-atom 
H Sn, 
cal/g-atom 
SSn, 
cal/K-g-atom 
XS
SnSD , 
cal/K-g-atom 
0.0 0.000 0.007 -¥ -13609 -8000 4.006 
0.1 0.007 0.072 -13706 -7301 -5233 6.053 1.477 
0.2 0.072 0.362 -7304 -2827 -1901 3.860 0.661 
0.3 0.197 0.656 -4523 -1173 252 3.411 1.018 
0.4 0.340 0.849 -3003 -454 706 2.649 0.828 
0.467 0.934 -2119 -190 681 2.000 0.622 0.5 
(±0.05) (±0.1) (±300) (±300) (±250) (±0.3) (±0.3) 
0.6 0.580 0.966 -1516 -95 506 1.444 0.429 
0.7 0.681 0.973 -1069 -77 311 0.986 0.277 
0.8 0.784 0.979 -678 -57 176 0.610 0.167 
0.9 0.892 0.991 -317 -24 49 0.261 0.052 
1.0 1.000 1.000 0 0 0 0.000 0.000 
Source: Ref 2 
Table 10 Partial molar thermal properties for liquid copper-tin alloys at 633 K 
Sn( l ) = Sn (in alloy) ( l ) 
XSn aSn Sn G Sn, 
cal/g-atom 
XS
SnGD , 
cal/g-atom 
0.98 0.965 0.985 -47 -20 
0.985 0.995 -13 -7 0.99 
(±0.005) (±0.005) (±5) (±5) 
Table 11 Heats of formation of solid and liquid copper-tin alloys at 723 K 
(1 - x)Cu(s) + xSn(s) = Cu(1 -x)Snx(s) 
(1 - x)Cu( l ) + xSn( l ) = Cu(1-x)Sn x( l ) 
Cu( l ) = Cu (in alloy) ( l ) 
XSn Phase H, 
cal/g-atom 
xSn Phase H, 
cal/g-atom 
Cp
(a), 
cal/K-g-atom 
H Cu, 
cal/g-atom 
( )CpCu l , 
cal/K-g-atom 
0.825(b) l -1105 1.7 . . . . . . 0.091
(b) (Cu) -280 
0.850 . . . -990 . . . . . . . . . 
0.204(b) -1300 0.900 . . . -760 1.2 . . . . . . 
0.209 -1260 0.950 . . . -530 
(±50) 
. . . . . . . . . 
0.244(b) -1686 1.000 l 0 -260 1.9 (±2) 
0.250 -1800 . . . . . . . . . . . . . . . . . . 
0.255(b) -1920 
(±100) 
. . . . . . . . . . . . . . . . . . 
(a) Cp = heat capacity. 
(b) Phase boundary. 
 
Table 12 Integral thermal properties for liquid copper-tin alloys at 1400 K 
(1 - x)Cu( l ) + xSn( l ) = Cu(1-x)Sn x( l ) 
XSn G, 
cal/g-atom 
H, 
cal/g-atom 
S, 
cal/K-g-atom 
Gxs, 
cal/g-atom 
Sxs, 
cal/K-g-atom 
0.1 -1922 -666 0.897 -1018 0.251 
0.2 -2834 -979 1.325 -1442 0.331 
0.3 -3195 -934 1.614 -1495 0.400 
0.4 -3300 -715 1.846 -1427 0.509 
-3167 -475 1.923 -1238 0.545 0.5 
(±300) (±150) (±0.24) (±300) (±0.24) 
0.6 -2889 -264 1.875 -1017 0.538 
0.7 -2483 -97 1.705 -784 0.491 
0.8 -1937 13 1.392 -545 0.398 
0.9 -1196 52 0.891 -291 0.245 
Table 13 Activities, activity coefficients, and partial molar thermal properties for liquid copper-zinc alloys at 
1200 K 
Cu component Cu( l ) = Cu (in alloy) ( l ) 
XZn aCu Cu G Cu, 
cal/g-atom 
XS
CuGD , 
cal/g-atom 
0.334(a) 0.432 0.648 -2003 -1034 
0.4 0.334 0.557 -2614 -1396 
0.219 0.438 -3621 -1968 0.5 
(±0.05) (±0.1) (±400) (±400) 
0.6 0.139 0.348 -4705 -2520 
0.7 0.085 0.284 -5869 -2998 
0.8 0.049 0.245 -7187 -3349 
0.9 0.023 0.229 -9010 -3519 
1.0 0.000 0.235 -¥ -3454 
Zn component Zn( l ) = Zn (in alloy) ( l ) 
aZn Zn G Zn, 
cal/g-atom 
XS
ZnGD , 
cal/g-atom 
0.132 0.398 -4814 -2199 
0.207 0.517 -3757 -1572 
0.347 0.695 -2521 -868 
(±0.05) (±0.1) (±400) (±400) 
0.505 0.841 -1630 -412 
0.657 0.939 -1002 -151 
0.789 0.986 -564 -32 
0.900 1.000 -250 1 
1.000 1.000 0 0 
Source: Ref 2 
(a) Phase boundary. 
 
Table 14 Integral thermal properties for liquid copper-zinc alloys at 1200 K 
(1 -x)Cu( l ) + xZn( l ) = Cu(1-x)Zn x( l ) 
XZn G G
xs, 
cal/g-atom 
xZn G, 
cal/g-atom 
Gxs, 
cal/g-atom 
0.334(a) -2942 -1423 0.6 -2860 -1255 
0.4 -3071 -1466 0.7 -2462 -1005 
-3071 -1418 0.8 -1889 -695 0.5 
(±400) (±400) 0.9 -1126 -351 
(a) Phase boundary. 
 
 
Ref er en ces ci t ed in t h i s sect ion 
1. B.L. Tiwari, Metall. Trans. A, Vol 18A, 1987, p 1645-1651 
2. R. Hultgren et al., Selected Values of the Thermodynamic Properties of Binary Alloys, American Society for 
Metals, 1973 
 
I n t er act ion Coef f i c ien t s 
The use of interaction coefficients, first suggested by Wagner (Ref 3), provides a convenient means of organizing 
thermodynamic data on dilute solutions. Wagner proposed a Taylor series expansion for the excess partial molar free 
energy in order to express the logarithm of the activity coefficient of a dilute constituent in a multicomponent solution. 
The values of the activity coefficients calculated from this method are as accurate as the original data from which the 
interaction coefficients are determined, provided it is applied to solutions that are quite dilute (Xi < 0.1). 
Consider a dilute solution of i, j, and k dissolved in a common solvent, s. If all but the first-order terms of the Taylor 
series expansion are neglected, the activity coefficient of solute i is expressed by the relationship: 
ln ln ln
ln ln o i i ii i i j k
i j k
X X X
X X X
g g g
g g
æ öæ ö æ ö¶ ¶ ¶
= = + +ç ÷ç ÷ ç ÷ç ÷¶ ¶ ¶è ø è øè ø
 
 
(Eq 7) 
In Eq 7, the partial derivatives are called interaction coefficients and are expressed by the symbol jie , where the 
superscript denotes the constituent that affects the activity coefficient of the subscript constituent. Thus: 
Pablo-pc
Highlight
 
(Eq 8) 
In Eq 8, oig is the limiting value of i at infinite dilution: 
o
ig = lim i 
Xi ®0 
(Eq 9) 
If the solution of i in s obeys Henry's law over a small range of concentration, then i is a constant over this range, and 
i
ie = 0. 
It can be shown by using the Gibbs-Duhem equation that: 
j
ie = 
i
je 
 
(Eq 10) 
and this reciprocal relation is most useful in determining values of from activity data. 
Values of iie are determined from experimental data on the binary system i-s, and values of 
j
ie are determined from 
experimental data on the ternary system i-j-s. Values of o and for aluminum-base and copper-base alloys are given in 
Tables 15, 16, 17, 18, and 19. 
Table 15 Standard Gibbs free energies for solution of elements in liquid aluminum 
Solution 
reaction 
o, 1100 K o
iGD for i (X), 
cal/g-atom 
o
iGD for i (%), 
cal/g-atom 
Be(s)=Be 19.6 9404 - 2.639T 9404 - 9.607T 
Bi(l)=Bi 24.5 5309 + 1.524T 5309 - 11.688T 
Ca(l)=Ca 0.0086 -10,400 -10,400 - 9.93T 
Cd(l)=Cd 19.9 7100 - 0.518T 7100 - 12.498T 
Cu(s)=Cu 0.037 -1135 - 5.538T -1135 - 16.385T 
Fe(s)=Fe 1.6 × 10-4 -27,000 + 7.18T -27,000 - 3.411T 
Ga(l)=Ga 1.1 832 - 0.52T 832 - 11.551T 
Pablo-pc
Highlight
Ge(l)=Ge 0.16 -2761 - 1.176T -2761 - 12.288T 
1
2
H2=H 
. . . . . . 11,664 + 6.523T 
In(l)=In 12.3 6800 - 1.201T 6800 - 13.223T 
Li(l)=Li 0.40 -5800 + 3.435T -5800 - 3.014T 
Mg(l)=Mg 0.18 -3478 - 0.30T -3478 - 9.24T 
Na(l)=Na 293 8230 + 3.798T 8230 - 5.03T 
Ni(s)=Ni 0.7 × 10-6 -28,280 - 2.442T -28,280 - 13.132T 
Pb(l)=Pb 115 9970 + 0.363T 9970 - 12.832T 
Sb(l)=Sb 3.4 13,100 - 9.45T 13,100 - 21.589T 
Si(s)=Si 0.27 9598 - 11.291T 9598 - 20.517T 
Sn(l)=Sn 4.68 5845 - 2.245T 5845 - 14.333T 
Zn(l)=Zn 1.92 2538 - 1.007T 2538 - 11.911T 
Source: Ref 4 
Table 16 Interaction coefficients for elements in liquid aluminum 
i j j
ie 
Temperature,K 
Ag Ag -3.1 1273 
Cu H see 
Cu
He 
973-1273 
H Cu 39.0 973 
H Cu 16.6 1073 
H Cu 20.1 1173 
H Cu 4.3 1273 
H H 0 973-1273 
H Si 11.5 973 
H Si 6.2 1073 
H Si 4.2 1173 
H Si 1.8 1273 
Si H see 
Si
He 
973-1273 
Cd Cd -5.0 1373 
Cu Cu 2.2 1373 
Ga Ga -0.3 1023 
Ge Ge +3.0 1200 
In In -4.5 1173 
Mg Mg 3.0 1073 
Si Si 16.0 1100 
Sn Sn 6.0 1100 
Zn Zn -0.9 1000 
Zn Si 2.2 963-1053 
Li Sn -16.0 949 
Na Si -12 973 
Mg Si -9 973-1073 
Sn Pb -1.5 973-1073 
H Ce -100 800 
H Cu 15 700-800 
H Cr -1 800-900 
H Fe -1 800-900 
H Mg -2 700-800 
H Mn 27 800 
H Ni 19 700-800 
H Th -20 800-900 
H Ti -42 800-900 
H Si 7.1 700-800 
H Sn 0.6 800-900 
Source: Ref 4, 6 
Table 17 Standard Gibbs free energies for solution of elements in liquid copper 
Element(a), i o
ig , 1200 °C 
o
iGD (X), cal/g-atom 
o
iGD (wt %), cal/g-atom 
Temperature, °C 
Ag(l) 3.23 3900 - 0.32T 3900 - 10.52T 1100-1200 
Al(l) 0.0028 -8630 - 5.84T -8630 - 13.84T 1100 
As(v) 4.8 × 10-4 -22,350 -22,350 - 9.44T 1000 
Au(l) 0.14 -4630 - 0.73T -4630 - 12.09T 1175-1325 
Bi(l) 1.25 5960 - 3.6T 5960 - 15.1T 1100-1300 
C(graphite) 1.4 × 105 8550 + 17.8T 8550 + 12.0T 1100-1300 
Ca(l) 5.1 × 10-4 -22,200 -22,200 800-925 
Cd(v) 15.6 -25,700 + 22.9T -25,700 + 12.7T . . . 
Cd(l) 0.53 -1860 -1860 - 9.0T . . . 
Co(s) 15.4 8000 8000 - 9.0T . . . 
Cr(s) 43 11,000 11,000 - 8.72T . . . 
Fe(s) 24.1 12,970 - 2.48T 12,970 - 11.34T 1460-1580 
Fe(l) 19.5 9300 - 0.41T 9300 - 9.27T 1460-1580 
Ga(l) 0.034 -10,800 + 0.61T -10,800 - 8.68T 1100-1280 
Ge(l) 0.009 -16,000 + 1.52T -16,000 - 7.5T 1255-1545 
1
2
H2(g) 
. . . 10,400 + 8.4T 10,400 - 7.5T 1100-1300 
In(l) 0.41 -9550 + 4.71T -9550 - 5.58T 700-1000 
Mg(v) 0.08 -40,200 + 22.3T -40,200 - 15.1T 650-927 
Mg(l) 0.044 -8670 - 0.31T -8670 - 7.53T 650-927 
Mn(l) 0.51 -1950 -1950 - 8.83T 1244 
Mn(s) 0.53 1550 - 2.31T 1550 - 11.14T 1244 
Ni(l) 2.22 2340 2340 - 9.0T . . . 
Ni(s) 2.66 6500 - 2.5T 6550 - 11.5T . . . 
1
2
O2(g) 
. . . -20,400 + 10.8T -20,400 - 4.43T 1100-1300 
Pb(l) 5.27 8620 - 2.55T 8620 - 14.01T 1000-1300 
Pd(s) 1.3 800 800 - 10.1T 1500-1600 
Pt(s) 0.05 -10,200 + 0.87T -10,200 - 10.47T . . . 
1
2
Ss(g) 
. . . -28,600 + 13.79T -28,600 - 6.03T 1050-1250 
Sb(l) 0.014 -12,500 -12,500 - 10.4T 1000-1200 
Se(v) 0.002 -18,200 -18,200 - 9.5T 1200 
Si(l) 0.006 -15,000 -15,000 - 7.5T 1550 
Si(s) 0.01 -2900 - 7.18T -2900 - 14.68T 1550 
Sn(l) 0.048 -8900 -8900 - 10.4T 1100-1300 
Te(v) 0.0328 -10,000 -10,000 - 10.53T 1200 
Ti(l) 8.5 6730 - 0.31T 6730 - 11.74T 1000-1300 
V(s) 130 28,100 - 9.4T 28,100 - 18.1T . . . 
Zn(l) 0.146 -5640 -5640 - 9.1T 1150 
Source: Ref 5 
(a) l, liquid; v, vapor; s, solid; g, gas. 
 
Table 18 Interaction coefficients for elements in liquid copper alloys 
i j i
je 
Temperature, °C 
H Ag -0.5 1225 
H Al 6.2 1225 
H Au -1.9 1225 
H Co -3.1 1150 
H Cr -1.6 1550 
H Fe -2.9 1150-1550 
H Mn -1.1 1150 
H Ni -5.5 1150-1240 
H P 10.0 1150 
H Pb 21.0 1100 
H Pt -8.0 1225 
H S 9.0 1150 
H Sb 13.0 1150 
H Si 4.8 1150 
H Sn 6.0 1100-1300 
H Te -6.6 1150 
H Zn 6.8 1150 
O Ag -0.7 1100-1200 
O Au 8.6 1200-1550 
O Co -68 1200 
O Fe -4.04 × 106/T 2183 1200-1350 
O Ni -36,000/T + 17 1200-1300 
O P -700,000/T + 385 1150-1300 
O Pb -7.4 1100 
O Pt 38 1200 
O S -19 1206 
O Si -6300 1250 
O Sn -4.6 1100 
S Au 6.7 1115-1200 
S Co -4.8 1300-1500 
S Fe -25,400/T + 8.7 1300-1500 
S Ni -29,800/T + 13 1300-1500 
S Pt 11.5 1200-1500 
S Si 6.9 1200 
Ag Ag -2.5 1150 
Al Al 14 1100 
Au Au 3.7 1277 
Bi Bi -6800/T + 1.65 1000-1200 
Ca Ca 20 877 
Fe Fe -5.7 1550 
Ga Ga 7 1280 
Ge Ge 13.4 1255 
H H 1.0 1123 
Mg Mg 9.8 927 
Mn Mn 6 1244 
O O -24,000/T + 7.8 1100-1300 
Pb Pb -2.7 1200 
S S -20,800/T 1050-1250 
Sb Sb 15 1000-1200 
Sn Sn 10 1300-1320 
Tl Tl -4.8 1300 
Zn An 4 1150 
Zn Zn 0.38 902 
Zn Zn 0.72 727 
Zn Zn 1.185 653 
Zn Zn 1.40 604 
Source: Ref 4, 6 
Table 19 Activity coefficients at infinite dilution in liquid metals 
Solvent Solute o Temperature, 
°C 
0.38 700 
0.47 900 
Silver 
0.53 1000 
Aluminum 
Magnesium 0.88 800 
0.14 604 
0.17 653 
Copper Zinc 
0.21 727 
Source: Ref 2, 7 
The standard state for and o described above is the pure material at the temperature of the solution, and the 
concentration is expressed in mole fraction. In dealing with dilute solutions, however, it is common to use a hypothetical 
1 wt% solution as the standard state. Under this condition, the Taylor series expansion corresponding to Eq 8 is: 
log fi = 
i
ie (%i) + 
j
ie (%j) - 
k
ie (%k) 
(Eq 11) 
where fi, which is the activity coefficient for 1 wt% standard state, equals ai/%i and: 
% 0;% % 0
log
%
j i
i
k i j
f
e
j
= ®
æ ö¶
= ç ÷
¶è ø
 
 
(Eq 12) 
In Eq 11, fi is the activity coefficient, and the zeroeth-order term, log 
o
if , disappears because the activity coefficient at 
infinite dilution oif is equal to 1. 
The reciprocal relationship, corresponding to Eq 10, is: 
j ii
i j
j
M
e e
M
= 
 
(Eq 13) 
where Mi is the atomic weight of i. The relationship between the two types of the interaction coefficients is: 
(2.303)(100)
j js
i i
j
M
e
M
e= 
 
(Eq 14) 
where Ms is the atomic weight of the solvent (Al, Cu). 
 
Ref er en ces ci t ed in t h i s sect ion 
2. R. Hultgren et al., Selected Values of the Thermodynamic Properties of Binary Alloys, American Society for 
Metals, 1973 
3. C. Wagner, Thermodynamics of Alloys, Addison-Wesley, 1952 
4. G.K. Sigworth and T.A. Engh, Scand. J. Metall., Vol 11, 1982, p 143-149 
5. G.K. Sigworth and J.F. Elliott, Can. Met. Quart., Vol 13, 1974, p 455-461 
6. J.M. Dealy and R.D. Pehlke, Trans. Met. Soc. AIME, Vol 227, Feb 1963, p 88-94 
7. J.M. Dealy and R.D. Pehlke, Trans. Met. Soc. AIME, Vol 227, Aug 1963, p 1030-1032 
 
Th er m al Pr op er t ies f o r Hy p o t h et i ca l St an d ar d St a t e 
It is a common practice in solving problems on chemical equilibria to use the hypothetical pure component (Xi) or 
hypothetical weight percent (%i) as the standard state. For this purpose, thermodynamic properties for the hypothetical 
standard states are needed. The values of o can be used to determine Gibbs free energies of mixing for elements in liquid 
base metal, using Eq 15 and 16: 
o
iGD (Xi) = RT ln 
o
ig 
(Eq 15) 
and 
(% ) ln
100
o
o i s
i
i
M
G i RT
M
g
D = 
 
(Eq 16) 
The values of oiGD (Xi) and 
o
iGD (%i) thus calculated for aluminum-base and copper-base alloys are also given in Tables 
15 and 17. The principal advantage of using the hypothetical standard states is that at low concentrations the activity of i 
can be set equal to its atom fraction or weight percent, depending on the composition scale used to express the standard 
state. 
Ph ase Diag r am s 
A phase diagram is essentially a map that shows the relative stabilities of various phases presentat equilibrium under the 
varying conditions of temperature and composition. Therefore, for a given composition of an alloy, a phase diagram can 
be used to determine the phases, which will be present at equilibrium as the melt solidifies. In addition, a phase diagram 
can also be used to estimate the activity of the components in liquid solution and to understand the behavior of the liquid 
solution (Ref 8). For example, if the diagram is for a simple eutectic system, it is likely that no strong intermetallic 
compound exists. If, however, a strong compound in the solid state appears in the diagram, then association of dissimilar 
atoms is probably occurring in the liquid. Therefore, one can predict whether a liquid solution will exhibit strong negative 
or positive deviations from the ideal behavior from the shapes of the liquids and azeotropes. 
The phase diagrams of the Al-Si and Cu-Zn systems, which are chosen based on their commercial importance, are shown 
in Fig. 1 and 2. The aluminum-silicon diagram (Fig. 1) represents a simple eutectic system, and the alloys containing 8.5 
to 13% Si are widely used to produce automotive parts. Generally, the higher the silicon content, up to the eutectic 
composition (12.6% Si), the greater the fluidity and, consequently, the easier an alloy is to cast. The copper-zinc diagram 
(Fig. 2) however, shows a high solubility of zinc in solid copper-zinc solution and the formation of several other phases 
over a wide range of zinc concentrations. Copper alloys containing up to 35% Zn and significant amounts of other 
alloying elements are quite common. The copper-zinc phase diagram is used to adjust the compositions in order to 
produce alloys with the desired properties. 
 
Fig . 1 The aluminum-silicon phase diagram. Source: Ref 9 
 
Fig . 2 The copper-zinc phase diagram. Source: Ref 9 
 
Ref er en ces ci t ed in t h i s sect ion 
8. L.S. Darken and R.W. Gurry, Physical Chemistry of Metals, McGraw-Hill, 1953 
9. T.B. Massalski et al., Binary Alloy Phase Diagrams, Vol 1 and 2, American Society for Metals, 1986 
 
Ref er en ces 
1. B.L. Tiwari, Metall. Trans. A, Vol 18A, 1987, p 1645-1651 
2. R. Hultgren et al., Selected Values of the Thermodynamic Properties of Binary Alloys, American Society 
for Metals, 1973 
3. C. Wagner, Thermodynamics of Alloys, Addison-Wesley, 1952 
4. G.K. Sigworth and T.A. Engh, Scand. J. Metall., Vol 11, 1982, p 143-149 
5. G.K. Sigworth and J.F. Elliott, Can. Met. Quart., Vol 13, 1974, p 455-461 
6. J.M. Dealy and R.D. Pehlke, Trans. Met. Soc. AIME, Vol 227, Feb 1963, p 88-94 
7. J.M. Dealy and R.D. Pehlke, Trans. Met. Soc. AIME, Vol 227, Aug 1963, p 1030-1032 
8. L.S. Darken and R.W. Gurry, Physical Chemistry of Metals, McGraw-Hill, 1953 
9. T.B. Massalski et al., Binary Alloy Phase Diagrams, Vol 1 and 2, American Society for Metals, 1986