C O M M U N I C A T I ON
ALG E B RAIC R E PR ESE NT A T 10 N 0 F EQU I LI BR I U M PROCESSES IN MULTICOMPONENT S Y S T E M S Reacting chemical systems in which the total number of compounds exceeds that possible in a single equilibrium a r e called multisystems b y Korzhinskii. By algebraically specifying the number of molecules of each compound and the composition of each compound by element, an exact statement of possible phase composition can b e made.
THE phase rule of Gibbs ( 7 ) refers to a reversible equilibrium in a closed system, where the energy of the system is defined by such intensive factors as temperature and pressure. As pointed out by Korzhinskii ( J ) , this totally ignores such extensive factors as composition and relative proportions of the existing phases in any particular chemical system. Conversely, if the composition and relative proportions of many of the possible phases in a chemical system are defined, the energy of the system as measured by the temperature and pressure becomes indeterminate. As will be showm, an exact statement of possible phase composition will also define the multiple equilibria defining the phases. The subject of irreversible changes in chemical reactions has been approached by Korzhinskii (5) from the molecular point of vieiv. In his theory of processes, a system which is not in equilibrium as a \?hole can be considered as being in equilibrium in each of its small elementary parts. I n such a system, application of the phase rule considers only the degree of freedom of such intensive factors as temperature, pressure, and concentration. This totally ignores the extensive characteristics of the system such as the formulas or mass of the individual components. This concept of ‘‘mosaic’’ equilibrium is equivalent to the multivariant equilibrium discussed by Thompson (72), involving several mobile components. As shown by Korzhinskii ( 5 ) , equations of reactions (among minerals of known composition) can be set u p lvith coefficients chosen for the formulas of the minerals in such a way that the inert components on the right and left sides of the equation will be the same, with the system of linear equations being evaluated by determinants. Application of the Gibbs phase rule to systems in which the total number of compounds exceeds that which is possible in an invariant system results in negative degrees of freedom. These systems Lvith negative degrees of freedom are called multisystems by Korzhinskii (7). For ease of solution, the bundle matrix used by Korzhinskii 1) matrix. As showm beloiv, ( 6 ) is limited to a n n X (n linear analysis of simultaneous linear equations is not subject to this limitation. and can readily handle a 3 X 10 matrix, where the degree of indeterminacy is 7. In a chemical system: linear equations can be set u p describing the composition of each component, coupled with a general statement of their interaction, where the coefficients are positive or negative integers, or zero. ,4n example, involving the complex reactions of copper and nitric acid, as given by Sfellor (a),follows. The multisystem of which copper and nitric acid are components can also involve cupric nitrate, ivater, nitrous acid, nitric oxide, nitrous oxide, am-
+
monium nitrate, nitrogen, hydroxylamine, or ammonia. general equation for this multisystem is
+
+ +
+
The
+
Cu b HNOJ = a Cu(NO3) c H20 d HSO2 e NO + f X , O g NHdNG3 h S2 z S H P O H + j NH3 Q
+
+
where the coefficients are positive or negative integers, or zero. T h e material balance by element is
(1)
H. b = 2 c + d + 4 g + 3 i + 3 j N. b = 2a d + e 2f+ 2g
+
+
+ 2h + i + j
0. 3 b = 6 ~ + ~ + 2 d + e + f + 3 g + i
Cu. a
(2) (3)
(4)
= a
As will be noted, the nitrogenous compounds in the general equation are arranged from left to right in order of the valence or average valence in the individual compound. Since the valence of ammonium nitrogen is - 3 , and the valence of nitrate nitrogen is +5, the average valence of nitrogen in S H 4 i Y G 3 is + l . Equations 1, 2 , and 3 are examples of simultaneous linear equations, and involve a total of ten real variables. Equation IV, Q = Q , is not a linear equation, but may be considered the coordinate of an invariant point in the four-dimensional system H * N * 0 * Cu. As discussed by Shilov (701, a system of linear equations with all its constant terms equal to zero UliXl
UziXi
+ +
.... QtiXi
+
Qlflw:!
Q22X2
+ ... . . + + ... . + ,
....
.....
QkOz
+ .... . +
QtnXn
=
0
U2nXn
=
0
QknXn
=
0
is called a homogeneouy linear system. If such a system has a t least one solution, it is called compatible. If a compatible system has a unique solution, the system is determinate; if a compatible system has a t least t\vo different solutions, it is indeterminate ( 9 ) . Homogeneous linear systems are al\vay compatible, since they a1wa);s have the trivial solution x 1 = xq=
.....
=x,=o.
As further noted by Shilov ( 7 7 1 , “ T h e basic problem encountered in stud>-ing homogeneous linear sptems is the follo\ving: Under Lvhat conditions is a homogeneous system nontrivially compatible-i.e., under \\.hat conditions does a homogeneous system have other solutions in addition to the trivial solution?” The discussion of Shilov’s theorem 21 ( 7 7 ) states, in particular, that if the number of equations in a homogeneous linear system is less than the number of unVOL. 3
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AUGUST
1964
269
knowns, nontrivial solutions will always exist. .4ccording to Veblen (73), “a set of linear homogeneous equations whose coefficients are integers has a set of B - r linearly independent solutions, each ofwhich is a set of relatively prime integers if B is the number of unknowns and r is the rank of the matrix of coefficient.” As noted by Korzhinskii j3),difficulties may be encountered in determining the number of independent compounds. “ T h e number of independent components is the smallest number of those chemical constituents whose combination (addition or subtraction) can give the composition of all possible phases of the system, including phases of variable composition.” This is equivalent to saying that the dependent variables can be expressed as functions of the independent variables. Returning to operations with Equations 1 , 2 , and 3, b can be eliminated between Equations 1 and 2. d + 2e
G =
+ 5f+
3g
+ 6h + 2i + 3j
(5)
Substituting this value of c into Equation 1 yields
b
=
3d
+ 4e + lOf + l o g + 12h + 7i + 9j
(6)
Equating Equation 1 to Equation 2 and substituting the value of c yields
2a
2d
=
+ 3e + 8f + 8g + 10h + 6i + 8j
(7)
G, 6 , and 2a are expressed as functions of the seven variables d to j . All the terms in the equation involving 2a are even numbers except the scalar ‘quantity, 3e. Since the number of copper atoms ( a ) by definition is a n integer, 2a is always a n even number, and e must always be a n even number. Since there is no restriction in the general equation as to whether e is odd or even, the general equation must be multiplied by 2. Substituting the dependent variables in the doubled general equation:
+ + + + + + + + + + + + + + + + + +
+ + + + + + + + + + + + +
(2d 3e 8f 8g 10h 6i 8j) C u 2(3d 4e lOf log 12h 72 9j) H S 0 3 = (2d 3e 8f 8g 10h+ 6 i 8j) C u ( S 0 3 ) 2 2(d 2e Sf 3g 6h 2i 3j) HzO f 2d “ 0 2 2e NO 2f S20 2g “4x03 2h N2 2i IC”20H 2 j NH3
By successively equating the independent variables to 1, the following invariant equations are obtained.
+6 = 2 C‘u(N0s)z + 2 H20 + 2 HNOB +8 = 3 Cu(S03)~ + 4 H?O + 2 S O 8 CU + 20 8 Cu(SO3)p + 10 + 2 SzO 8 CU + 20 HSO3 = 8 C ~ ( S 0 3 ) + 2 6 H20 + 2 10 CU + 24 H S 0 3 = 10 C U ( S O ~+) ~12 H 2 0 + 2 S 2
2 CU 3 CU
”03 “ 0 3
“ 0 3
H20
1
T\TH4103
6 CU 8 CU
+ 14 + 18
“ 0 3
=
“03
=
6 Cu(N03)2 f 4 HzO 8 Cu(NO3)s 6 H20
+
+ 2 SHiOH +2 “3
Each of the above seven equations is invariant, since each involves five components and four elements-i.e., the degree of indeterminacy is 1. T h e only fully determinate linear equations are those involving n variables and n equations.
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FUNDAMENTALS
T h e fact that balanced chemical equations can be written does not automatically determine that the individual compounds in a system are stable or even exist. A general equation can be written involving copper, nitric acid, \vater, a hypothetical trivalent copper nitrate, and the nonexistent oxide, l-402.
2a C u
+ 2(3a + 4 6 )
“03
=
+
2a Cu(SO3)3 (3a 4 6 ) H,O
+
+ 26 N40g
Employing the same technique of solution. the balanced “reaction” equation is
T h e fact that balanced equations can be written involving nonexistent products clearly implies that the independent existence of each compound in a system must be verified by experiment or previous experience. O n the molecular scale, a single molecule of known composition is a separate phase. even though it coexists in solution. This concept is similar to the microcanonical ensemble of systems of Gibbs ( 2 ) ,except that the summation of the composition of the microphases is a constant rather than the summation of kinetic and potential energy. I t Lvould thus appear that if the energy in a chemical system is specified by temperature and pressure. the composition of the microphases is indeterminate. Conversely, if the composition of all microphases is specified, the energy content is indeterminate. This apparent paradox is a chemical analog of Heisenberg‘s principle of uncertainty, %vhere the position and velocity of a nuclear particle cannot be measured simultaneously Lvith high precision. literature Cited
(1) Gibbs, J. W.,“Elementary Principles of Statistical Mechanics,” “Collected LVorks of J. LVillard Gibbs,” Val. I, pp. 116-17, Longmans, Green, New York. 1931. (2) Ibid.. “Equilibrium of Heterogeneous Substances,” Vol. 11. (3) Korzhinskii. D. S.,“Physiochemical Basis of the Paragenesis of Minerals,” p. 13, Russian translation, Consultants Bureau, Inc., New York, 1959. (4) Ibid.,p. 19. (5) Ibid.,pp. 103-10. (6) Ibid., p..104. 171 Ibid..D. 125. (8j Melldr, J. I$’,, “Comprehensibe Treatise on Inorganic and Theoretical Chemistry,” Val. 111, pp. 91-5, Longmans, Green, New York, 1928. (9) Shilov. G. E.. “Introduction to the Theory of Linear Spaces,” p. 2. Prentice-Hall, Enplewood Cliffs. N. J.. 1961. (16) ‘Ibid.,p. 27. (11) Ibid., p. 53. (12) Thompson, J. B., A m . J . Sci. 253, 80 (1955). (13) \-eblen, O., “Analysis Situs,” 2nd ed., p. 183, American AMathematicalSociety, New York, 1946. .
Y
D. R. SWAYZE 47 Sayre St. Elizabeth, 1%’. J . RECEIVED for review June 18, 1963 ACCEPTED April 21, 1964
