Let us consider the infinitesimal addition of one of the components participating in a reaction in which the temperature and pressure of the system are held constant. As follows from eq 15 of (1), the equilibrium shifts in the direction for which

(1)

where is the infinitesimal addition of species k, is the chemical potential of species k in the mixture and is the infinitesimal change in the chemical potential of species k that occurs upon reaction after the addition of an infinitesimal amount of k. While eq 1 is correct, Torres argues that because a finite addition of k, or , can be decomposed into a sum of infinitesimal changes, and because each is required to have the opposite sign from , one may also conclude that

(2)

in which , as defined in ref 1, is the sum of the infinitely many changes .

Although the argument leading to eq 2 is intuitively appealing, the extension of eq 1 to finite changes does not, however, yield a result that is valid in general. When applied to the ammonia synthesis reaction, for example, the finite addition of more nitrogen, > 0, may cause the equilibrium to shift in the direction for which > 0 (3, 4), which now represents the difference in the chemical potential of k when the reaction proceeds and when no reaction occurs, both after a finite perturbation from the same equilibrium state. In fact, as shown in ref 3, many other reactions may yield > 0 for > 0, changes that are not predicted by eq 2.

For the ammonia synthesis reaction (with nitrogen, hydrogen, and ammonia comprising an ideal gas mixture), the chemical potential of nitrogen is given by

(3)

where (T) is the chemical potential of pure N₂ in an ideal gas state at the same temperature T, is the mole fraction of N₂, and P is the total pressure of the mixture. Starting from an equilibrium state, if a finite amount of nitrogen is added at constant T and P, eq 2 requires

(4)

in which and are the final mole fractions of nitrogen after reaction has occurred or not yet occurred, respectively, upon the addition of more nitrogen. When > 0, eq 4 implies that < , namely, the reaction should “moderate” the perturbation (2). But as follows from the analysis in (4), where we now let represent the number of moles of nitrogen added divided by the total number of moles of all components initially present, if one begins with an equilibrium mole fraction of N₂ equal to 0.3, one finds that > for 0.4 ≤ ≤ 4/3. Specifically, for the same initial mole fraction of N₂ of 0.3, one obtains = 0.597992 and = 0.597701 for = 0.74. Although exceeds only by a small amount of substance, the reaction nevertheless fails to moderate the change in the mole fraction of nitrogen.

To understand why the expression is not necessarily equal to a sum over , we elaborate on the discussion provided in ref 3, which likewise considered the ammonia synthesis reaction for a mixture of ideal gases. At equilibrium, the mixture must satisfy (4)

(5)

where K(T) is the equilibrium constant and K_P(T, P) is a function of T and P. By introducing the following quantities (3)

(6)

where n_i is the number of moles of species i, eq 5 can be rewritten as

(7)

Now begin with the system at equilibrium with an initial mole fraction of nitrogen given by < 0.5. If more nitrogen is added while keeping T and P constant, the right side of eq 7 will not change before the reaction commences because each z_i term does not depend upon the amount of nitrogen. Also, if enough nitrogen is added so that the mole fraction of nitrogen equals 1 − (prior to the start of the reaction), no reaction will in fact take place because eq 7 is still satisfied. In other words, 1 − represents another equilibrium state for this system. (Note that the equilibrium state is double-valued only when the system is open—that is, the number of moles of nitrogen is allowed to change—and does not imply that multiple equilibrium states exist when the system is closed [3].) These two equilibrium states merge at = 0.5. This value is not arbitrary; it is equal to the initial mole fraction of nitrogen above which an infinitesimal addition of more nitrogen causes the reaction to shift to the left, producing more nitrogen (1–5). Consequently, if enough nitrogen is added to cause to exceed 0.5, the system will adjust itself when the reaction commences as if it had originated from the equilibrium state residing at 1 −, “forgetting” that it had started from < 0.5.

For example, consider an initial equilibrium state = 0.2, such that the second equilibrium state beyond 0.5 resides at the value 1 − = 0.8. If the amount of nitrogen added results in = 0.7, the system will behave as if it began at the value = 0.8 and nitrogen had been removed from the system. By “thinking” that < 0, the system moderates this negative perturbation (instead of the actual positive addition), when the reaction is allowed to proceed, which according to eq 4 results in an increase in the mole fraction of nitrogen upon reaction. Yet, from the point of view of the finite positive change initiated at = 0.2 (that is our point of view), the reaction violates eq 4. All finite additions of nitrogen that increase the mole fraction from = 0.2 to 0.5 ≤ ≤ 0.8 also yield an increase in upon reaction (only for this range of mole fractions does the system behave as if it started from = 0.8 with nitrogen being removed). Values of between 0.2 and 0.5, and those that exceed 0.8, correspond to perturbations that satisfy eq 4. In general, a violation of eq 4 occurs for the ammonia synthesis reaction for a final mole fraction of nitrogen (before reaction proceeds) between 0.5 and 1 − . Given that the mole fraction of nitrogen just before the reaction begins is equal to (4)

(8)

eq 4 is therefore not satisfied when

(9)

(Equation 9 is also valid for < 0 if the inequalities are reversed.) For = 0.2, eq 9 reduces to 0.6 ≤ ≤ 3.

And so, the above analysis reveals why the sum of the various infinitesimal changes, , does not necessarily equal , the response that follows from a finite addition. For each infinitesimal perturbation, the system only “knows” of one equilibrium state to which it should attempt to return: that is, when starting from < 0.5, the system does not suddenly find itself at > 0.5. Hence, each infinitesimal perturbation yields a response that satisfies eq 4. On the other hand, a finite change, if large enough, enables the system to jump from < 0.5 to > 0.5. The system therefore “forgets” that it originated from the equilibrium state at < 0.5. In other words, the sum of many infinitesimal changes and a single finite change describe two different pathways, each of which may lead to different responses of the system.

Finally, the violation of eq 4 does not only arise for the ideal gas mixture; it also appears if one analyzes the ammonia synthesis reaction under nonideal conditions (5), with eq 2 now being given by

(10)

Literature Cited

1. Torres, E. M. J. Chem. Educ. 2007, 84, 516–519.
2. de Heer, J. J. Chem. Educ. 1957, 34, 375–380.
3. Liu, Z.-K.; Agren, J.; Hillert, M. Fl. Phase Equil. 1996, 121, 167–177.
4. Corti, D. S.; Franses, E. I. Chem. Eng. Educ. 2003, 37, 290–295.
5. Uline, M. J.; Corti, D. S. J. Chem. Educ.2006, 83, 138–144.