MATHEMATICAL MODEL
6.7 Reversible Reactions
All chemical reactions are reversible in principle, and most are reversible in practice as well. Hence, we must now return to the reaction in Equation6.1and consider the more general case of reversibility. As before, we take α = 1 without loss of generality, and all other stoichiometric coefficients other thanβ and μ to be zero.
We thus have two reactions to consider:
A+ βB → μD μD → A + βB
We will therefore have to account for both the forward and reverse reactions in the species conservation equations. In the batch reactor, with the usual assumptions that lead to constant volume, the species equations then become
dcA
dt = rA+− rA−, (6.14a)
dcB
dt = rB+− rB−, (6.14b)
dcM
dt = rM+− rM−. (6.14c)
All of the stoichiometric arguments used previously still apply. Clearly,β moles of B still vanish for every mole of A that is reacted, and, similarly,β moles of B are formed for every mole of A that is former by the reverse reaction.μ moles of M are formed for every mole of A that is reacted, and one mole of A andβ moles of B are formed for everyμ moles of M that are lost in the reverse reaction. Hence,
rB−= βrA−, rM+= μrA−, rB+= βrA+, and rM− = μrA+. We can then again define a single reaction rate, denoted r, as follows:
r ≡ rA−− rA+=rB−− rB+
β =rM+− rM−
μ . (6.15)
We thus recover Equations6.6a, b, and c, which clearly depend only on the fact that there is a single reaction, and not on the assumption if irreversibility. This is also true of Equations6.7a and b, which relate the concentrations of the various reactants and products in the batch reactor at any time. Note that r is the difference between two positive quantities and may be positive or negative.
We can gain considerable insight by considering the specific example ofβ = 1 andμ = 2, together with mass action kinetics. We thus write rA−(the forward rate) as k1cAcBand rA+(the reverse rate) as k2cM2, with r= k1cAcB− k2cM2. The batch reactor that is, the system is at equilibrium, which simply means that forward rates exactly equal reverse rates. We denote the equilibrium concentrations of the species with a subscript e; we therefore obtain
Keqis, of course, the equilibrium constant, which is familiar from the general chem-istry course.
We can make one observation that is of experimental significance even before we complete the mathematical steps to determine the time evolution of the concen-trations in a batch reactor. Suppose that we begin the experiment with no M present.
In that case, cMwill be very small, and the second term on the right of Equation6.16, which is quadratic in cM, will be negligible for a finite period of time. The equation will then be approximately the same as Equation6.9,and the system will appear to be irreversible. This is another example of the significance of the time scale when evaluating system response.
Table 6.5. Concentration of H2SO4versus time for the reaction of sulfuric acid with diethyl sulfate in aqueous solution at 22.9◦C, full data set. Data of Hellin and Jungers, Bull. Soc. Chim.
France, No. 2, pp. 386–400 (1957).
concentration of H2SO4,
Time, t (min) cA(t) (g-mol/L)
Equations6.18and6.19can be combined to give dcA
or, formally separating the concentration- and time-dependent terms, KeqdcA
4(cA0− cA)2− KeqcA2 = k1dt.
The left-hand side is a form that is readily found in tables of integrals. Upon inte-gration of the left-hand side from cA0to the current value cA(t), and the right-hand side from t= 0 to the present time, we obtain
ln
EXAMPLE 6.4 The reaction between sulfuric acid and diethyl sulfate studied in Example 6.1 is, in fact, reversible, although the assumption of irreversibility gave a good fit to the data up to a time of 194 minutes. The full data set is shown in Table 6.5.cA0 = cB0. Find the rate expression, assuming that both the forward and reverse reactions may be described by mass action kinetics.
From the data given in Table 6.5, Keq= 4+5.50
2.60− 1,2
≈ 5. The data in Table 6.5are plotted according to Equation6.21inFigure 6.6, with cA0= 5.50 and Keq= 5. The data do follow a straight line and are consistent with the
00 100 200 300 400 0.5
1.5
1.0 2.0 2.5 3.0
IncA(2 – √Keq) – 2cA0 cA(−2 – √Keq) + 2cA0
Slope =4cA0 k1= 6.6 × 103 min1 Keq
√
t, min
Figure 6.6. Computation of the forward rate constant for the reaction between sulfuric acid and diethyl sulfate in aqueous solution.
assumption of mass action kinetics. Calculation of k1from the slope yields k1= 6.7× 10−4L/(g-mol min). This value differs by only 10 percent from the value obtained in Example 6.1 by assuming irreversibility over the first 194 minutes.
6.8 Concluding Remarks
The important concept here is the rate of reaction, which addresses the fact that the mass of a component species is not conserved in the balance equations. The particular rate constitutive equations used in this chapter are quite elementary; mass action kinetics may be followed in systems of interest, but the kinetics may also be far more complex because of chemical steps that are not obvious from the overall stoichiometry. The basic principles developed here are sufficient to enable us to address meaningful design problems in the next chapter despite the elementary forms of the rate equations. More complex reactions will typically be considered in a subsequent course in the chemical engineering core.
Bibliographical Notes
Basic textbooks on physical chemistry contain at least one chapter on the rates of chemical reactions. The classic textbook treatments of reaction kinetics are by Laidler and by Frost and Pearson, the latter now in a third edition as Moore and Pearson:
Laidler, K. J., Chemical Kinetics, 3rd Ed, Prentice-Hall, Englewood Cliffs, NJ, 1987.
Moore, J. W., and R. G. Pearson, Kinetics and Mechanism, 3rd Ed., Wiley, New York, 1981.
More complex reaction schemes and reactor geometries than those covered in this chapter are an essential part of the chemical engineering core course that covers the subjects of kinetics and reactor design.
PROBLEMS
6.1. Suppose that the data inTable 6.1for the reaction of sulfuric acid with diethyl sulfate in aqueous solution are described by a rate of the form r= kcAn. Estimate n using the methodology of Appendix 2A. Investigate the consequences of an error of 1 percent in determining cA.
6.2. Data for the decomposition of dibromosuccinic acid (2,3-dibromo-butanedioic