J d y , 1950
APPLICATION OF
[CONTRIBUTION FROM
THE
THE
GIBBS-DUHEMEQUATION TO MULTICOMPONENT SYSTEMS
2909
RESEARCHLABORATORY, UNITEDSTATES STEELCORPORATION OF DELAWARE, KEARNY, NEWJERSEY]
Application of the Gibbs-Duhem Equation to Ternary and Multicomponent Systems BY L. S. DARKEN The literature contains very few data on the thermodynamics of isotherms of ternary or multicomponent solutions. Likewise, little is to be found on the theoretical treatment of such solutions, except for regions limited to the vicinity of a pure solvent. This paper deals with a n isothermal isobaric section of a ternary system in which section there is but one phase regionthat of the single phase solution. With the aid of no extra-thermodynamic assumption other than the assumption of Henry's law as a limiting law a t infinite dilution, i t is demonstrated that i t is possible to calculate the molal free energy, hence the partial molal free energies of the other two components, from an experimental knowledge of the partial molal free energy of one component a t all compositions. The same method may readily be extended to systems of more than three components and to functions other than the free energy. As a preliminary step let us consider the applicability to a ternary system of the well known relation, G; = G (1 - Ni)(dG/dNJ, valid under isothermal isobaric conditions for binary solutions; this relation follows from the generalized Gibbs-Duhem equation. Here, G represents the molal value of any extensive property] g; the corresponding partial molal quantity for the i t h component and N; the mole fraction of the ith component. Since any solution, no matter how many components i t contains, may be regarded a s binary if we arbitrarily limit composition changes to those corresponding to addition or removal of only one component, i t follows that the foregoing equation applies to a multicomponent solution, provided that the derivative dG/dNj be interpreted as a partial derivative, all ratios of mole fractions except those involving Ni being constant. For a ternary system, focusing our attention upon component 2 which is here regarded as the component whose partial molal quantity is experimentally known, we may then write'
+
(1) A more detailed derivation of Eq. 1 for a ternary system is:
the relation
G
=
NIGI
-
Gz = G
+ (1 - n'z)
(1)
By rearranging terms and dividing by (1 - Nz)2, this expression may also be written
Integration from NZ= 1 to N z gives G
-
(1
-
G N2) lim ___ N~-+I 1 - NZ
the integration and limit to be taken a t constant value of N1/N3.la Let us now apply this relation to the excess molal free energy of mixing Fx5which is defined as Fxs= F - F' where Fi is th_e ideal contribution to the free energy - Fzxs= FZ- Fzi = RTln yz where y2 is the activity coefficient of component 2. Equation 3 now becomes
i t being understood that the integration and limit are to be carried out a t constant Nl/N,; in the usual triangular method of representing composition, the integral is taken along a straight line connecting the point of interest to the corner corresponding to pure component 2, as in Fig. 1. The limit in Eq. 3a is an indeterminate forin which may be evaluated by aid of 1'Hopital's theorem whereby
which by Eq. 1becomes
+
and by virtue of the relations F"5 = N1FIx5 1 - Nz 1 - Nz, N~i72~S f N3F3"] N 1 = and N3 = N3 Ni l+iV, l+N3 ~
+ NzGz + NaEa
is differentiated partially with respect t o Na a t constant Nl/Na, giving
Gz
-
~
NlCI
1
- Nz??i - A72
Rearrangement of terms gives Eq. 1. ( l a ) Integration of Eq. 2 from Nz = 0 to
+
zz - G
AT3??a =-
1
-
Nz
h7gleads to
+
By virtue of the Gibbs-Duhem equation (Nld& Nnd& h'id(la = 0) the sum of the first, third and fifth terms is zero, whence
thus providing a method of determining G for a ternary composition from a knowledge of (1) Gz for ternary compositions of constant Nl/Ns and (2) G for the binary system with the same ratio N I / N ~ .
L. S. DARKEN
2910
If the standar? state be so chosen for each component that Fixs = 0 (;.e., so that yi = 1) a t N2 = 1 then the above limit is zero, and Eq. 3a becomes
a prime being used to indicate this particular choice of standard state. If, however, the standard state be_ chosen a s pure component in each case (i.e., Fixs = 0 a t Ni = 1) then the limit becomes
Vol. 72
These two expressions may also be obtained from Eq. 5a by setting N3 = 0 and NI = 0 respectively, N Zbeing set zero in both cases. Substituting for these two constants in Eq. 5a
I
All further treatment in this paper is based on this second choice of standard state. Eq. 3a becomes, on noting again that and
-
NI
[
~
I
1
- Nz
N3 = NI 1+N,
~
+ N3 ~
This is the final equation expressing the excess molal free energy of a ternary solution in terms of the partial molal excess free energy of only one component. As noted previously, the standard state for each component is chosen as that pure component. It will be noted that in general the integrations may be performed, and Fxs thus evaluated, only if there are no miscibility gaps intersecting the lines along which the integrations are to be performed. If, for a particular system, _Fxs be so determined a t all compositions then FlXsand EBXs may be determined, if desired, by the usual methods. For example Eq. 2 may be put in the following forms
~ [A ~ 3 2 3T. d 1~~ 2 =~. ~(sa)
The two constants [ F 1 x S ] ~and z S 1[ F 3 X S ] ~ z = ~ are determinable from measurements on the two binary systems 1-2 and 2-3, respectively. If
Fig. 1.-Illustration on triangular coordinates of compositions t o be investigated in ternary system for proposed method of obtaining p sfor binary system 1-3.
FzXs is regarded as the experimentally determined quantity, these two constants may be expressed in terms thereof by application of the GibbsDuhem equation to each of these two binary systems.
F,-
= (1
- Nl)2 ( ___ b e , ) Nz/.Vs
Thus Fxs,PIxsand FSXsmay be evaluated a t all compositions provided PzXsis known a t all compositions. A Method of Determining the Free Energy of Binary Systems.-Let us suppose that i t is desired to determine experimentally the excess free energy of mixing of a binary solution (components designated 1 and 3) but that the usual methods do not seem practical for this particular case. For example, the vapor pressures may be too small for application of the vapor pressure method. However, i t would frequently be possible to find another substance (designated component 2 ) which is volatile and is miscible in all proportions with components 1 and 3 and all mixtures thereof. In such a case i t would be relatively easy to determine the vapor pressure of component 2 and hence FzXs in the ternary system. Similarly, i t may be difficult to apply the e. m. f . method t o a series of binary liquid metallic solutions ; but perhaps another metal miscible with the other two, could be found. If, further, this other metal behaves reversibly (usually by virtue of lower electronegativity) in the cell, then its
July, 1950
APPLICATIONOF
THE
GIBBS-DUHEM EQUATION TO MULTICOMPONENT SYSTEMS 2911
excess partial molal free energy may readily be determined. In such cases Equation (6) may be applied to determine F x s for the binary system. The binary system 1-3 is obviously a limiting case of the ternary system 1-2-3 in which NZ = 0. Setting N z = 0 in Eq. 6 we find
N.
z
h
I 3
r
k
0
Thus the free energy of formation of the binary solution 1-3, may be obtained by this indirect method. Although many more experimental measurements are required, this indirect method may be of occasional use in cases such as that discussed above in which direct measurements for the binary system are impractical. The general scheme of the procedure is illustrated schematically in Figs. 1 , 2 and 3. Figure 1 shows the compositions of the solutions along lines of constant ratio NI/N3. Figure 2 illustrates schematically the plot of the experimental data. The total area under each curve is then evaluated and plotted (with reversed sign) as indicated by Fig. 3 (top). Fxs at each value of N3 for the binary system 1-3 then corresponds to the departure of the corresponding point from the straight line connecting the termini of the curve; these values of Fxs are indicated by the arrows in the top portion of Fig. 3 and are plotted a t the bottom thereof. Obvious modification of this procedure is used if i t is desired to obtain Fx for ternary compositions. It seems worth while a t this point to call attention to the fact that Equation 7 lends thermodynamic support to the chemists intuition that it is possible to gain some knowledge of the chemistry of a series of binary solutions by investigating the solubility of another substance slightly soluble therein. For example, the solubility of sulfur dioxide gas at 1 atm., in sulfuric acid-water mixtures is a minimum a t a composition (86yo sulfuric acid) corresponding approximately to the one to one ratio of sulfuric acid to water (84.5%sulfuric acid). This behavior is commonly interpreted as evidence of the existence of a hydrate of sulfuric acid in aqueous solution. Although Eq. 7 cannot be fully applied with this limited amount of information, it would not seem unreasonable to infer that such minimum solubility would be found a t higher concentrations of sulfur dioxide also, and hence that the first integral in Eq. 7 would have a minimum value a t or near this ratio of sulfuric acid to water. Granting this inference, it follows then from Eq. 7 that the excess free energy Fxsof the binary system H2S04-H20 has a minimum value a t this ratio. Thus in a sense, the intuitive reasoning is justified
1
0.5
KZ. Fzxs
Fig. 2.--Schematic representation of ___ as func(1 - N2)' tions of Nz along the lines of Fig. 1.
I
I I
I
I
0
0.5 1 N3 (binary system 1-3). Fig. 3.-Top-the integrals of the curves in Fig. 2 plotted as a function of N3 for the binary system 1-3. Bottom-replot of the distances indicated by arrows above, which are values of Fzs for the binary system 1-3.
-although of course thermodynamics gives no direct support to the idea of hydration or compound formation in solution. One further example will be given. Liang, Bever and Floelb have measured the solubility of hydrogen gas a t 1 atm. pressure in molten iron-silicon alloys. The logarithm of the solubility (= FzXs/ 2.303RT Const) is shown in Fig. 4. This is perhaps a rather unexpected type of curve exhibiting a near discontinuity in slope near the one to one %tomratio. If we may be permitted to infer that FzXswould behave similarly over part of the ternary system, then the first integral of Eq. 7, and hence the excess free energy of Che binary system ironsilicon, would exhibit a similar near-discontinuity in slope near the one to one ratio of iron to silicon. The relation may be found more explicitly if we make, for the purpose of illustration, the crude
+
(Ib) H. Liang, M. B. Bever and C. F. Floe, Tvans. A m . Zest. Min Met. Efigrs., 167, 395 (1946).
L. S. DARKEN
2912
Vol. 72
approximation that FzXs/(l- N Z )is~ a function of the ratio NJN3 only, hence from Eq. 7 q i e - s i SyBtern)
=a
- R7i-0, + NlRB1-1)
+ NsP~$-I)
Denoting the low solubility of Hz as NzOthis becomes (approximately)
+
(~lZ1)N1-Ol [ (fj3")Nrl
- (~8")N~-OI
Barring dissociation, all the partial molal excess free energies appearing on the right side are non= R T [In lVzQ - NI In Na'(N1-1) infinite2 (by virtue of Henry's law) and hence Na In N Z ~ ( N1 ~ -i~t follows ) that the integral is finite. The utility of Eqs. 6 and 7 would be seriously This gives an explicit relation for the excess free ever energy of the binary system, although a very impaired if the function FzXs/(l - N z ) ~ crude approximation is involved. It is thus seen approaches infinite value even though its integral from this relation that on the basis of this crude does not. The behavior of this function must assumption the excess free energy of iron-silicon therefore be investigated, particularly as N Z apsolutions is proportional t o the lengths of the proaches one. Modern solution theory and arrows in Fig. 4. Crude as this may be it in- .expeiiment3are in accord that for non-electrolytes, t o NI (or to 1 - NZ ~ dicates a rather large departure from ideality in N ~ b F ~ x s / l bisNproportional the system iron-silicon (Fxs = -7000 cal. per under the condition considered) a t sufficiently gram atom) ; a strong possibility of a rather un- low concentration. Using a similar expression for usual behavior in this binary system is indicated N&FP/bNz, i t is readily found by aid of the Gibbs-Duhem_ Equation and d'Hopital's theorem by the near discontinuity in slope. that lim Fzxs is not infinite. This matter N2+0 (1 - N2j2 a s well as the further complications involved in 1.2 the treatment of solutions of electrolytes is discussed by Scatchard and P r e n t i ~ s . ~ 9c Attention should be called to the fact that ale 1.0 though, for non-electrolytes, the function f = A FzxS/(l- N z ) is~ probably finite at all composicc tions, the precise evaluation thereof in the vicinity x + of N Z = 1 requires a very high degree of experi'1 0.8 mental _accuracy. Departures from ideality, and Ba , very small hence FzXs and also (1 - N z ) ~are s1 in this vicinity. For the binary metallic systems 2 0.6 on which data are available in the literature there d is no indication of any anomalous behavior of the function, f, in the vincity of Nz = 1. Hence i t seems reasonable to suppose that the best conin the vicinity struction of the curve (jagainst Nz) 0 0.2 0.4 0.6 0.8 1.0 of Nz = 1 is by smooth extension from the region iVsi (in Fe). in which the experimental precision is good, Fig. 4.-Solubility of hydrogen in molten iron-silicon Le., that discordant points near NZ= 1 should be alloys a t 1650" (data of Liang, Bever and Floe). Solubility disregarded in the construction of the curve is expressed as cc. NTP of Hz per gram atom of alloy. (Fig. 2). It will be noticed that to obtain equal FXS Length of arrows indicates crude approximation of ___ precision in f over the range- of experimental 2.303RT observations, the precision in Fzxs must increase for binary system Fe-Si. markedly with increase in NZ; for example the precision a t N Z = 0.9 should be 100 times as great It is apparent that the method of this section as in the vicinity of Nz = 0. Vapor pressure may be extended to find G for an n-component and e. m. f . methods tend to give greater presolution by the introduction of a n additional cision a t high values of Nz but not this much greater. However, attention should be called miscible component and by measurement of theref ore. (2) If dissociation occurs, the difficulty of an infinite integral may Limitations of the Method.--As illustrated in be avoided by choosing the products of dissociation as components. Fig. 2, Eqs. 5 and 7 call for the integration of In metallic systems the chemical elements are chosen as components possibility is avoided. the function Fzxs/(l - Nz)2over the entire range and(3)this M. Margules, Silebcr. d . Wien A k a d . , [ Z ] 104, 1243 (1895); of Nz from zero to one, If the proposed method A. W. Porter, Trans. Far. SOC.,16, 236 (1921); George Scatchard, is to be fruitful, then Fzxs/(l - N2)2must be a Chcm. Rcu., 8, 321 (1931); J. H. Nildebrand, "Solubility of Nonreasonably well-behaved function. Certainly i t Electrolytes," Reinhold Publ. Corp., New York, N. Y.,1936; and Guggenheim, "Statistical Thermodynamics," Cambridge would be disastrous if the integrals thereof ap- Fowler 40, 320 (1944). University Press, 1939; W. J. C. Orr, Trans. F a ? , SOC., proached infinite value. It may easily be shown (4) George Scatchard and S. S. Prentiss, THISJ O U R N A L , 66, 1486 that the integral may be written and 2314 (1934). h
-
v
July, 1950
APPLICATION OF
THE
GIBBS-DUHEMEQUATION TO h/IULTICOMPONENT SYSTEMS
2913
to the fact that this demand for greater precision in the vicinity of N2 = 1is inherent in any method utilizing the Gibbs-Duhem equation. Frequently, it is concealed by the method of treatlirn "*' is finite a t all temperatures, then ment. In fact, i t is common in experimental in- N-1 (1 - "2)' Rz -H: vestigations to neglect this region entirely, upon in generals the derivative and hence the assumption that Raoult's law is rapidly approached and measurement is unnecessary. is finite. Similarly, since Although i t is true that Raoult's law is here approached rapidly, nevertheless i t is the small departures therefrom which are important in determining the partial molal free energy of the other constituent(s) by any method involving the Gibbs-Duhem equation. The present method makes clear this requirement, whereas some other methods of applying the Gibbs-Duhem equation i t follows in general that the limit of &xs/(l to binary systems do not; this latter fact has con- N2)zand of ( Vz - Vi)/(l - Nz)as N z + 1 is also tributed to a tendency on the part of some finite. experimental investigators to ignore compositions Extension to Multicomponent Solutions.in the vicinity of the pure component whose par- The same method is readily extended to systems tial pressure or activity is being measured. of more than three components. Equations 1 Application to Other Thermodynamic Func- a n d 2 hold in identically the same form except tions.-The same method developed here may be that the partial derivatives are to be interpreted applied to the molal values of other extensive as partials a t constant ratio of all mole fractions thermodynamic functions which have zero value except for ratios involving Nz. Equation 3 is a t NZ = 1; such functions are the enthalpy, then valid under the condition that the integraenergy and volume of mixing, and the excess tion and limit are to be taken in this same way. entropy. This is apparent from the consideration The final equation (6) or (sa) follows in just the that equation (3) and the evaluation of the limit same way and with the same limitation as for a leading t o Eq. 4a are perfectly general. From ternary system. Thus the excess free energy, or Eq. 4a i t is seen that the general form of the limit other extensive property meeting the requirein Eq. 3 is ments discussed in the preceding section, a t all compositions in a multicomponent solution may be derived from a precise knowledge of the corresponding partial molal quantity of any one component at all compositions. which is never infinite unless [C1]jvtEl or Acknowledgment.-The author wishes to ac[ ? & ] ~ , , l is, providing as mentioned above that knowledge the benefit of discussion with George G = 0 a t N2 = 0. Hence it is apparent that the Scatchard (Mass. Inst. of Tech.) who kindly remethod is applicable in principle to any molal viewed the fundamental idea of this contribution quantity which has the value zero a t NZ = 0 and offered suggestions for the improvement of providing that none of the corresponding partial the presentation. molal quantities become infinite a t infinite diluSummary tion and providing that approaches zero as A general method has been developed for deterN2 approaches unity. The general equation is mining a molal quantity and the other corresimilar to Eq. 6. sponding partial molal quantities for ternary and multicomponent solutions under isothermal isobaric conditions, from an experimental knowledge of one of the corresponding partial molal quantities, but a t all compositions. The general relation is
$zl
cz
It will be noted that for the functions
essentially the same_ considerations as to finite ~ by virtue values apply as for FzxS/(l- N Z )this of the thermodynamic relations between HI SI V and F. For example, since
(5) Exceptions ntay conce5ably exist at particular temperatures
at which the plot of lirn
Nr+l
F ---m us. ( 1 - N2)
1/T may exibit infinite slope.
2914
S. S. TODD Na[lo 1
(1
'
- N2)Z
-
dNz]
...
E ! , % . . .= o N I Na
The only assumptions involved are: (1) that G is the molal value of a n extensive function, and hence that the extended Gibbs-Duhem equation may be applied; G must be so chosen as t o have zero value for each pure component; (2) that Henry's law is valid as a limiting law for all components a t infinite dilution. The restrictions on the usefulness of the relation are discussed; departures from Henry's law a t small finite con-
[CONTRIBUTION FROM
THE
Vol. 72
centrations must be such that is proportional ~ the near vicinity of NZ = 1, in to (1 - N z ) in order that the function to be integrated in the above equation be always finite. It is shown that the above equation may be used as the basis of a new method for determining G for an n-component solution; this method involves the introduction of another miscible component and the experimental determination of therefor in the new system of n 1 components.
+
KEARNY,
RECEIVED JUNE 22, 1949
N. J.
PACIFICEXPERIMENT STATION, BUREAUOF
MINES,
UNITEDSTATESDEPARTMENT OF THE
INTERIOR]
Heat Capacities at Low Temperatures and Entropies of Zirconium, Zirconium Nitride, and Zirconium Tetrachloride BY S. S. TODD'
A recent paper of Coughlin and King2 presents high-temperature heat-content data for zirconium metal and its oxide (ZrOz), nitride (ZrN), silicate (ZrSiO,), and tetrachloride (ZrClr). The present paper gives low-temperature heat-capacity values and entropies a t 298.16' K. for the metal, nitride, and tetrachloride, thus making possible freeenergy calculations for the last two substances. Low-temperature heat-capacity and entropy values for the oxide and silicate have been reported by Kelley.as4 Heat Capacities The materials used in this investigation were identical with those described by Coughlin and King,2 and repetition of the methods of preparation and tests of purity appears unnecessary. Correction was made for the hafnium contents, based upon the assumption that corresponding zirconium and hafnium compounds have the same molal heat capacity. This correction increased the measured heat-capacity values by the following amounts: Zr, 1.0%; ZrN, 0.770:'0; and ZrClr, 0.35%. The measurements were made with previously described apparatus. The results, expressed in thermochemical calories6 (1 cal. = 4.1833 int. joules), are listed in Table I and plotted against (1) Pacific Experiment Station, U. S . Bureau of Mines. Article not copyrighted. (2) 5 . P. Coughlin and E. G . King, THISJOURNAL, 72, 2262 (1950). (3) K . K . Kelley, (a) i b i d . , 63, 2750 (1941); (b) Ind. Eng. Ckcm., 36, 377 (1944). (4) In this connection, it appears worth-while to record that Kelley's entropy values should be increased slightly to account for the now known hafnium contents of the materials. Recalculation gives S$a.ls = 12.12 * 0.08 for ZrOz (monoclinic) and Sbs.ls = 20.2 t 0.2 for ZrSiO' (zircon). ( 5 ) K . R.Kelley, B . F. Naylor, and C. H. Shomate, U . S. Bur. Mines Tech. Paper, 686 (1946). (6) E . F. Mueller and F. D. Rossini, A m . J . P h y s . , l a , 1 (1944).
TABLE I MOLALHEATCAPACITIES T, OK.
CP, cal./deg. T ,
OK.
CP cal./d'eg.
T , OK.
calyieg.
Zr (mol. wt.,91.22) 53.2 56.8 60.8 65.6 70.6 75.4 79.8 84.1 94.9 104.5
2.418 115.0 2.640 124.1 2.873 136.1 3.134 146.2 3.383 156.0 3.609 166.1 3.789 176.0 3.945 186.2 4.296 196.1 4.562 206.3
53.1 57.1 62.0 67.5 72.7 77.6 80.5 85.3 95.2 104.8
1.198 1.426 1.699 2.018 2.308 2.574 2.731 2.978 3.477 3.946
4.796 4.971 5.172 5.293 5.409 5.510 5.606 5.672 5.737 5.802
216.4 226.2 236.4 246.0 256.7 266.4 276.4 286.6 296.8 (298.16)
5.861 5.905 5.948 5.981 6.042 6.083 6.127 6.149 6.168 (6.186)
ZrN (mol. wt.,105.23)
52.6 55.9 60.0 64.9 69.3 74.0 80.0 83.9 94.8 104.6
11.24 11.86 12.63 13.53 14.30 15.06 15.98 16.53 17.96 19.15
114.9 124.8 136.2 145.9 155.9 166.2 176.7 186.2 196.4 206.7
4.416 216.6 4.867 226.5 5.358 236.5 5.747 246.0 6.137 256.4 6.517 266.4 6.876 276.6 7.177 286.8 7.495 296.7 7.784 (298.16)
ZrCla (mol. wt., 233.05) 114.6 20.24 216.6 226.7 124.7 21.21 136.1 22.23 236.4 146.3 22.97 246.3 156.2 23.66 256.5 166.3 24.25 266.4 176.3 24.87 276.5 186.4 25.33 286.8 196.4 25.75 296.7 206.7 26.15 (298.16)
8,048 8.287 8.516 8.714 8.954 9.127 9.325 9.482 9,613 (9.655)
26.53 26.79 27.11 27.37 27.66 27.94 28.17 28.37 28.63 (28.65)
