Leonard C. Grotz
University of Wisconsin-Milwaukee Milwaukee
A
relatively recent trend in general chemistry textbooks is the development of the topics of chemical kinetics and chemical equilibria as a direct consequence of the kinetic-molecular behavior of matter instead of presenting them as additional, apparently unrelated, empirical facts. A simple diagrammatic probability analysis is employed to develop the dependence of reaction rates on concentration rather than to rely on the law of mass action ( I ) . Mechanisms are introduced to show how the law of chemical equilibrium can be derived when the reaction involves more than a single bimolecular collision (8). These developments amplify and enforce the understanding by the student of the fundamental behavior of matter and give him an appreciation of the fact that a science is more than just a collection of observations. A topic in this area that has steadfastly resisted modernization is the treatment of heterogeneous equilibria. The time-honored treatment o f heterogeneous equilibria is to apply the law of chemical equilibrium, which was derived for homogeneous systems, to the heterogeneous cases and then take the concentrations of the substances in the pure solid and liquid phases as constant. The limitations of this procedure are threefold. F i s t , the usual kmetic derivations of the homogeneous equilibrium constant do not allow its application to heterogeneous systems. Second, the application of the term "concentration" to a solid immobile substance is of doubtful significance. Finally, the concentration of a substance in the solid or liquid phase is not the pertinent concentration, but rather the concentration of the substance a t the site of the reaction (at the solid surface or in the vapor phase). The second limitation has sometimes been overcome by introducing a non-mathematical definition of activity, but it is doubtful if this leads to any real enlightenment of the student. The third limitation has sometimes been partially overcome by pointing out the constancy of the vapor pressure of. a pure liquid or pure solid at constant temperature, but this does not afford an explanation when the reaction site is the surface of the solid. To date, no method of overcoming the first, and probably the major, limitation in the treatment of heterogeneous equilibria in general chemistry has been devised. Perhaps there has been a lack of incentive because of the ease with which the law of chemical equilibrium follows from the mathematical definition of activity. The only solution to the difficulties encountered in presenting the topic of heterogeneous equilibria (short of giving half of a course in thermodynamics) seems to be a kinetic derivation of the equilibrium constants for the various types of heterogeneous equilibria. That is, the procedure used to derive the law of chemical
Heterogeneous Equilibria in General Chemistry equilibrium for the homogeneous cases by equating reaction rates a t equilibrium (8) can be extended to the derivation of equilibrium constants for heterogeneous equilibria. This procedure, based on chemical kinetics, is generally referred to as a kinetic derivation to distinguish it from a derivation based on thermodynamics, which is referred to as a thermodynamic derivation. The kinetic derivation is facilitated for the more complex cases of heterogeneous equilibria by employing mechanisms with activated complexes. The activated complexes of the absolute-reaction rate theory are now generally mentioned in general chemistry textbooks (3), but no attempt seems to have been made to integrate more fully this idea with the remainder of the discussion of reaction kinetics. To illustrate how the activated complexes are incorporated into the kmetic derivation, consider first the simple homogeneous gas reaction, -.
'l'hisreactioln can he written as the sum of two steps, v
Hdg) Step %.
+ L(g) 2 S(Y)
"f
~ ( g ) 2HI(g)
In these steps, +(g) is the activated complex a t the top of the energy barrier which can decompose to form either the reactants, HZand IS,or the products, 2HI. When the system is in equilibrium, the reactions for both steps must be in equilibrium, and,
Then equation (a) is solved for [+(g)] and this value is substituted into equation (b). After rearranging to group the specific reaction rate constants together, one obtains,
where K,, is the equilibrium constant, and equation (c) is the law of chemical equilibrium expression for the reaction of HI and Iz. Homogeneous reactions which involve more steps can be treated similarly. When more than two steps are involved, it is easier to arrive a t the law of chemical equilibrium from the rate equations, (a) and (b) above, by dividing them in a way that will cause the concentration of the activated complex to cancel out and also give the concentrations of the products in the numerator of the resulting expression. In the above case, equation (b) is reversed and divided by equation (a), [+(g)l cancels out, and the constants are grouped together to give (c). Volume 40, Number 9, September 1963
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Heterogeneous Equilibria
.
After this procedure has been established for homogeneous systems, it can then be applied to the various types of heterogeneous systems. The simplest case to start with would be where the reaction site is the vapor phase as in the reaction, &(g)
+ S(1) * HaS(g)
This reaction is written as the sum of three steps,
To write the rate equation for step 1 it is necessary to point out that the rate of escape of molecules from a liquid a t constant temperature is directly proportional to the amount of surface through which the molecules can escape, that is, the surface area of the liquid. Further, the rate of condensation of the vapor molecules a t constant temperature is directly proportional to the area of liquid which they can strike, that is, again the surface area of the liquid. These considerations are presented in some general chemistry textbooks (4) to give a verbal derivation of the constancy of vapor pressure a t constant temperature, but no one seems to have expressed this mathematically as is done in equation (a) below. The argument for equation (a) is thus similar to that employed by Langmuir in the kinetic derivation of his monolayer adsorption equation (5). The principal difference is that in this case the surface area for both the condensation and evaporation steps is the same. The number of molecules in the liquid phase which strike the surface per unit time is a function only of the temperature, but the number of molecules in the gas phase which strike the surface per unit time will be a function of the concentration of the molecules in the gas phase. With these considerations, and, letting x be the surface area of the liquid sulfur, the following rate equations can be written for the three steps. k d z ) = k,,(z)[S(g)l
(a)
k d W g ) l [S(g)l = ka[+(g)l
(a)
ks1[+(~)1 = kav[H2S(g)l
(c)
Equation (c) is reversed, divided by equation (b) and by equation (a). Common terms are canceled and the constants collected together to give,
I n this example $(s) and *(s) are activated complexes on the surface of the solid carbon. To write rate equations for this case it is necessary to have some measure of the concentrations of $(s) and %(s) on the surface of the carbon. A concentration on a solid surface would have significance if the molecules were mobile on the surface and if the surface was energetically homogeneous. Under these conditions all equal units of surface area would contain the same time average number of molecules. Many adsorbed molecules and surface molecules have been shown to be mobile (6),but an energetically homogeneous solid surface is probably the exception rather than the rule (7). Therefore, it is customary to employ as a measure of the surface concentration of a species the fraction of the total surface area which is covered by that species. This is the procedure that was employed by Langmuir in the kinetic derivation of his monolayer adsorption equation (5), and is the procedure employed in the discussion of heterogeneous reaction rates in physical chemistry (8). In the previous example only one surface area was involved, the surface area of the liquid sulfur. I n this example there are three different surface areas to consider: the free carbon surface, the surface covered by $(s), and the surface covered by @(s). Taking for convenience the total surface area as unity, and letting x be the area of the solid covered by $(s) and y be the area covered by @(s), then (1 - x - y) is the free carbon surface. The rates of the first order reactions of the activated complexes will depend only on the amounts of the activated complexes and this is given by the amounts of surface that they cover. With these considerations the following rate equations can be written.
Again, equation (c) is reversed, divided by equation (b) and by equation (a). Common terms are canceled and the constants collected together to give,
One of the most common examples of a heterogeneous equilibrium that is employed in general chemistry is the thermal decomposition of calcium carbonate (9). This reaction can be treated similarly, but it must be pointed out that the reaction can occur only a t a common interface of some area x between the pure solid phases of CaC03 and CaO if the pure phases are to be maintained (10). The reaction is written as the sum of
A more complicated example is offered by the reaction, C(s)
+ COdg) * ZCO(8)
where the reaction site is the surface of the solid carbon. This reaction can be written as the sum of:
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Journnl of Chemical Education
As before, the rate of reaction of a surface species is taken to be directly proportional to the area covered by that species. Letting y be the interfacial area occupied by $(s), then x - y is the free interfacial area and is a measure of the concentrations of both the CaC03 and CaO which can undergo reaction. Since only the CaCOa and CaO which are a t the common interface can react,
These relations assume that the activated complex can be only one layer in depth, the same assumption that was made in the previous example. Reversing equation (b), dividing by (a),cancelling common terms, and collecting the constants gives
This treatment of the decomposition of CaC03 can serve as a basis for the introduction and the explanation of autocatalysis, where the rate of a reaction increases with the degree of completion of the reaction. Under non-equilibrium conditions the rate of decomposition of CaC03 will be directly proportional to the area of the common interface between the CaC03 and CaO as given by the left side of equation (a)above. Starting out with pure CaCOa the interfacial area will be negligible, but as the reaction proceeds the interfacial area must increase and the rate of the decomposition will therefore increase. Another example of this is given by the phenomenon known as tin disease in which the cubic crystals of tin are reported to be a catalyst for the conversion of the tetragonal crystals to the cubic crystals of tin (11). I n these cases the autocatalysis is brought about because the effective concentration of the reactant as measured by the interfacial area increases as the reaction proceeds. A second type of autocatalysis is observed when a product of the reaction is a catalyst for the reaction. I n the two examples given here the interfacial area cannot be called a catalyst because it is consumed when the reaction goes to completion. A catalyst can be defined as a substance that reacts and is produced in a reaction in a mole ratio of one to one. Solubility Equilibria
Another type of heterogeneous equilibrium constant that is important to general chemistry and qualitative analysis is the solubility product constant. The solubility product constant can be derived in a similar manner if allowance is made for the uncertainty in the extent of the hydration of ions. Taking BaS04 as an example and combining the successive reactions with water molecules into a single step to simplify the derivation gives the following,
Step I .
Ba(s)++
+ ~ H P OI $(s)
Again, #(s) and @(s) are activated complexes. Writing the water molecules in the formnlas for the aqueous ions emphasizes that the solution process is a reaction between the solid ions and water molecules which, afterall, would not occur if no interaction existed. These considerations are usually presented in general chemistry textbooks to explain the process of solution
( I d ) , but they are then taken for granted in further discussions of solutions such as in the consideration of solubility products. Steps 1 and 3 should not in general be interpreted to represent reactions with fixed numbers of water molecules, but rather an activation of the solid ions by the water molecules. Also, the formulas for the aqueous ions should not in general be taken to contain exactly a and molecules of water. but should be taken to represent the interaction between the ions and the water molecules which accounts for the stabilization of the ionic solution. Lewis and Randall, Debye and Hiickel, and others, have shown that in dilute solutions, where the number of water molecules for steps 1 and 3 are in large excess, the properties of electrolyte solutions can be treated without reference to any specific solute-solvent interactions (15). The dilute solution law does not hold for concentrated electrolyte solutions and one of the explanations given is based on the decrease in the amount of water available for the hydration of the ions (14). Taking a total surface area of unity, and letting x be the area of Ba++(s), y the area of SOl-(s), and z the area of #(s), then (1 - x - y - z) is the area of @(s). As before, equating the forward and reverse rates for the steps above gives, kdx)[HsOla = k&)
(a1
k d z ) = 4.1Ba(H~O),++(a~)l(~)( b ) kdy)IHnOlo = k d l - x - y - Z ) (c) kdl
- z - Y - z)
=
k~,ISO~(BO)B(aq)l(z)
(4
Reversing equation (d), dividing by equations (c), (b), and (a),canceling, and collecting the constants and the water concentration on the right gives,
At low concentrations of electrolyte where the concentration of the free water is essentially constant, [Bd&O),++(aq)l [SO4&O)B(ap)l = K.,
Takiig the water concentration to be constant is the usual simplification procedure employed in general chemistry. This is done in the derivation of ionization constants, hydrolysis constants, and the ion-product constant of water. Because the concentration of water in pure water is 55.5 moles per liter, the simplification is taken to be valid for electrolyte solutions up to 0.1 molar. The loss in validity is no greater than that introduced by takimg the equilibrium constants to be independent of concentration, that is, using concentrations instead of activities. Leaving out the water of hydration gives the solubility product expression in its usual form, I n all of the foregoing examples of heterogeneous equilibria it will be noted that the surfaces of the pure solid and liquid phases make an equal contribution t o both the forward and reverse reactions and, therefore, the surface terms always cancel out. These terms are the x in the first example, the x, y, and (1 - x - y) in the second example, the y and (x - y) in the thud example, and the x,y, z, and (1 - x - y - z) in the fourth example. As a consequence, there is no concentration term in the resulting laws of chemical equilibria Volume 40, Number 9, September 1963
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48 1
for the reactants or products that were present in the form of pure solid or liquid phases. The procedure employed for the four examples given can be used to show that this result will be true for any heterogeneous equilibrium. Also, as in the above examples, it will always be true that those concentration terms which do appear (gases and components of solutions) will be raised to a power given by the coefficient of the suhstance in the chemical equation. Therefore, by induction, the conclusion is reached that the law of chemical equilibrium for a heterogeneous reaction can be obtained by the application of the law of chemical equilibrium for homogeneous reactions to the equation followed by deletion of the concentration terms for all pure solid or liquid substances. The generalization just made serves rts an illustration of the inductive process of the scientific method. The kinetic derivation of equilibrium constants also affords an opportunity to demonstrate to the student in general chemistry how a theory can be checked by observations of deductions made from the theory. This point is often presented as a part of the scientific method (i5),but there are few opportunities in general chemistry to illustrate it. The utility of deductions is limited, however, because while the non-existence of a deduction from a theory will disprove the theory, the existence of a deduction from a theory will only add to the inductive probability that the theory is correct. The existence of equilibrium constants serves as a good example, because as derived above, they are a deduction from the kinetic molecular theory of matter. The experimental verification of the existence of equilibrium constants helps to substantiate the kineticmolecular theory, but does not prove that matter is actually composed of particles in motion. The mech-
482 / Journol of Chemical Education
anisms given for the various examples of heterogeneous equilibria also serve as an illustration of the deductive process. The existence of deductions from the mechanisms, such as autocatalysis, shows only that t,hn mechanisms are consistent with all the observa~ - - --~~~ tions, but does not prove that they constitute the actual processes which occur when the reactants change to the products. ~
~~
Literature Cited (1) QUAGLIANO, J. V., "Chemistry," Prentice-Hsll, Inc., Englewood Cliffs, N. J., 1958, pp. 53942. (2) SIENHO,M. J., A N D PLANE,R. A., "Chemistry," 2nd ed., MeGraw-Hill Book Co., Inc., New York, 1961, pp. 2554. (3) Ibid., pp. 247-9. (4) SISLER,H. H., VANDERWERF, C. A,,
A N D DAYIDSON, A. W., "General Chemistry, A Systematic Approach," 2nd DD. ed.. The Macmillan Co.. New York. 1959.. . . 5840. (5) ADAMSON, A. W., "Physicd Chemistry of Surfaces," Interscience Publishers, Inc., New York, 1960, pp. 464-72. (6) Ibid., pp. 551-5. ( 7 ) MYSELS,K. J., "Introduction to Colloid Chemistry," Interscience Publishers, Inc., New York, 1959, pp.
1924. ( 8 ) PRUTTON,C. F.,
(9) (10) (11) (12) (13)
(14) (1.5)
A N D MARON,S. H., 'Tundamental Principles of Physical Chemistry," The Maomillan Co., New York, 1944,pp. 6 5 4 4 . NEBERGALL, W. H., A N D SCHMIDT, F. C., "College Chemistry," D. C. Heath and Ca., Boston, 1957, pp. 232-3. ADAM,N. K., "The Physies and Chemistry of Surfaces," Oxford University Press, London, 1944, pp. 241-4. NEBERGALL, W. H., AND SCHMIDT, F. C., op. cil., p. 537. SISLER,H. H., ET AL., op. cit., pp. 216-9. BARROW,G. M., "Physical Chemistry," McGrsw-Hill Book Co., Ino., New York, 1961, pp. 564-73. HARNED, H. S., A N D OWEN,B. B., "The Physical Chemistry of Electrolytic Solutiana," 2nd ed., Reinhold Publishing Corp., New York, 1950, pp. 606-9. NEBERGALL, W.H., AND SCHMIDT, F. C., O p . d., p. 11.
