NOTES
1825 30
-E
25
m
and all others zero. We notice in this case two nonvanishing nondiagonal matrix elements, which give rise to configuration interaction. The configuration interaction of the (yx) and (22) orbitals is just that needed to mix them into the properly symmewized combinations 1 / 4 3 [ (xx) f (yz) 1 which are written in the text for simplicity (z y)z. Then ((z
((x
+ Y>4II.‘i(X + !Ax)
- Y)ZlVl(X - Y)4
1 = 7(PZW)
=
3 -$Jz(Y))
5 0
I
I
I
I
5
10
15
20
?(298)/(x
t y)
-
cal/”K
gm-atom
22
Figure 1. Entropy correlation for 298 and 1000’K.
102
standard partial molar entropy at one temperature, S o ( T 2 )to , that a t another temperature, TI), for a series of aqueous ions of similar charge and type (Le., simple ions, oxy anions, and acid oxy anions). The relationship between the partial molar entropies at two temperatures is linear and can be written in the form
- 189(P4(r)) - 189(P4(?))
The configuration interaction of the (9)and (xy) orbitals is the usud type between two states of the same overall symmetry. So, by assuming equal mixing of the states, we solve the 2 X 2 secular determinant and denote the orbital with higher energy as (x2, zy)- and the one with lower energy (9,xy)+ (see also ref 5). It may be pointed out that neglecting the presence of Yooterms and putting pn’ (Y) = 0 in eq 2a, 4a, 6a, and 8a, which is equivalent to removing the corresponding ligands to infinity, there result the potentials, respectively, for a four-coordinated complex of square planar geometry (Ddh), a tricoordinate complex of trigonal planar geometry (D3h), a five-coordinated complex of pentagonal planar geometry (DSh),and a fivecoordinated complex of square pyramidal geometry (C4,).
S”(T2) = a(T2, Yl)
+ b(T2, Tds”(T1)
(1)
where the intercept and the slope depend on the two temperatures selected for the correlation and the ionic type. The relationship has been used to estimate partial molar entropies at elevated temperatures from room-temperature values. The absolute entropies of solid metals and compounds at room temperature and above are examined in this note and are found to obey a similar linear relationship.
Table I : Coefficients in the Entropy Correspondence Equation
Estimation of the High-Temperature Entropy of Solids by the Ehtropy Correspondence Principle’
by J. G. Eberhart and E. N. Gruetter Sandia Laboratory, Albuquerque, N e w Mexico 871 16 (Received October 6 , 1 9 6 7 )
In recent years, Cobble and C r i s ~ ~have - ~ developed an entropy correspondence principle, which relates the
T,O
K
298 500 1000 1500
a(T, 298), cal/deg g-atom
0.00
2.92 6.86 9.38
b ( T , 298)
1,000 1.038 1.105 1.141
(1) This work ‘was supported by the L’. S. Atomic Energy Commission. (2) C. M. Criss and J. W. Cobble, J . Am. Chem. Soc., 86,5385 (1964). (3) J. w. cobble, Science, 152,1479 (1966). (4) J. W. Cobble, Ann. Rev.Phys. Chem., 1 7 , (1966). ~ Volume 7St Number 6 M a y 1968
1826
XOTES of high-temperature entropies from a lowtemperature value and an empirically determined intercept and slope characteristic of the two temperatures. From the relationships S ” ( T ) / ( z y) = a ( T , 298) b(T, 298)[5”(298)/(x y)] and C‘, = T(bX/ dT),, it also follows that c,”(T)/(x y) = a ( T , 298) P(T, 298)[9(298)/(x y)], where a ( T , 298) = da(T, 298)/d In T and P(T, 298) = db(T, 298)/d In T. Thus the entropy correspondence equation can also be used to estimate heat capacities at an upper temperature from entropies at a lower temperature.
+
+ +
0
500
1500
1000
T
(6) “JANAF Thermochemical Tables,” The Dow Chemical Co., Midland, Mich., 1967. (6) D. R. Stull and G. C. Sinke, “Thermodynamic Properties of the Elements,” American Chemical Society, Washington, D. C., 1956. (7) C. E. Wicks and F. E. Block, “Thermodynamic Properties of 65 Elements-Their Oxides, Halides, Carbides, and Nitrides,” Bureau of Mines Bulletin 605, U. S. Government Printing Office, Washington, D. C., 1963.
2000
- “K
Figure 2. Intercept and slope for the linear correlation between entropy a t 298’K and t h a t at higher temperatures.
For reasons to be stated shortly, entropies will be given for Avogadro’s number of atoms, ie., on a gramatom basis. If s”0 is the standard molar entropy of the compound A,B,, then s””/ (x y) is the entropy per gram-atom. Using experimentally determined entropies taken from various tabulations of thermodynamic plots were made of P(298)/(x y) vs. P(5OO)/(x y), P(lOOO)/(x y), and go(1500)/(x y). A typical result is shown in Figure 1with entropies at 298 and 1OOO”I~. It was found that solid compounds of all stoichiometries and solid elements obey the same linear entropy correspondence relationship
+
+
+
+
+
where, in the specific case of elementary substances, z = 1 and y = 0. The legend in Figure 1, which lists the values of x and y in the compounds, shows the wide variety of stoichiometries satisfying the relationship. The chemical types included in the plot are elementary metals and nonmetals, intermetallic compounds, and halides, oxides, carbides, and nitrides of metals. As can be inferred from eq 2, if molar entropies are used directly in the correlation, a different intercept, (x y)a, is obtained for stoichiometries of each x y value. Least-squares values of the intercept and slope in eq 2 were calculated for a fixed lower temperature of 298°K and upper temperatures of T = 500, 1000, and 1500°K. Table I shows the values of a ( T , 298) and b(T, 298) at these upper temperatures. Plots of a(T, 298) and b(T, 298) us. T in Figure 2 show that the slope and intercept of the linear correlation increase smoothly with the upper temperature. The relationship shown in eq 2 permits the estimation
+
+
T h e Journal o j Physical Chemistry
+
+ +
On the Reaction Complex of the C3H6+-C3H6 Ion-Molecule Reaction by Fred P. Abramson’ and Jean H. FutrelP Aerospace Research Laboratories, Ofice of Aerospace Research, Wright-Patterson A i r Force Base, Ohio 45433 (Received October 12, 1967)
I n an earlier p ~ b l i c a t i o nwe , ~ discussed ion-molecule reactions in olefin systems and suggested a classification scheme which is based on the concept of different types of intermediate complexes (e.g., “loose” and “tight” complexes) being involved in particle-transfer, condensation, and molecular ion reactions. In particular, a difference between C . . .H * .C and C C bonded complexes was proposed. Very similar ideas were put forward by Lampe, Franklin, and Field in a review article on ion-molecule reaction kinetic^.^ Whenever the reaction complex is stoichiometrically equivalent to a stable ionic species, these authors suggested that an intimately bonded complex would be formed and that its dissociation products would resemble the unimolecular dissociation products of that ionic species. For example, they noted that the products of ion-molecule reactions of acetylene and ethylene molecular ions corresponded to the major mass spectral fragmentation peaks of certain hydrocarbon^.^ Although they
-
-
(1) Consolidated Electrodynamics Corporation, Monrovia, Calif. 91017. (2) Department of Chemistry, University of Utah, Salt Lake City, Utah 84112. (3) F. P. Abramson and J. H. Futrell, J. P h y s . Chem., in press. (4) F. W. Lampe, 3. L. Franklin, and F. H. Field, P r o p . Reaction Kinetics, 1, 67 (1961). ( 5 ) J. L. Franklin, F. H. Field, and F. W. Lampe, Advan. M a s s Spectry., 1, 308 (1959).
