Quasi-Steady Hypothesis

Many reactions involve short-lived intermediates that are so reactive that they never accumulate in large quantities and are difficult to detect. Their presence is important in the reaction mechanism and may dictate the functional form of the rate equation. We begin with a reaction for which a full analytical solution is possible  

The governing ODEs for a constant-volume batch reactor are 

Now suppose that B is highly reactive. When formed, it rapidly reverts back to A or transforms into C. This implies kr ork k/. To apply the hypothesis to this batch reaction, set db/dt = 0. The ODE for B then becomes an algebraic equation and predicts that the concentration of B will be proportional to that of A  

This is the quasi-steady result. The actual concentration of B is initially zero but rises quickly so that Equation 2.25 becomes a good approximation. Erom this point on, B is formed at nearly the same rate as it is consumed. It is actually consumed at a somewhat higher rate so that, after its initial rise, b will gradually decrease, maintaining near proportionality to a. 

Substituting Equation 2.25 gives a simplified ODE for a that has the solution 

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The quasi-steady hypothesis is used when short-lived intermediates are formed as part of a relatively slow overall reaction. The short-lived molecules are hypothesized to achieve an approximate steady state in which they are created at nearly the same rate that they are consumed. Their concentration in this quasi-steady state is necessarily small. A typical use of the quasi-steady... 

FIGURE 2.3 True solution versus approximation using the quasi-steady hypothesis. 

Note that the quasi-steady hypothesis is applied to each free-radical species. This will generate as many algebraic equations as there are types of free radicals. The resulting set of equations is solved to express the free-radical concentrations in terms of the (presumably measurable) concentrations of the long-lived species. For the current example, the solutions for the free radicals are... 

The quasi-steady hypothesis allows the diflficult-to-measure free-radical concentrations to be replaced by the more easily measured concentrations of the long-lived species. The result is... 

Example 2.6 Apply the quasi-steady hypothesis to the monochlorination of a hydrocarbon. The initiation step is... 

Our treatment of chain reactions has been confined to relatively simple situations where the number of participating species and their possible reactions have been sharply bounded. Most free-radical reactions of industrial importance involve many more species. The set of possible reactions is unbounded in polymerizations, and it is perhaps bounded but very large in processes such as naptha cracking and combustion. Perhaps the elementary reactions can be postulated, but the rate constants are generally unknown. The quasi-steady hypothesis provides a functional form for the rate equations that can be used to fit experimental data. 

Compare this maximum value for b with the value for b obtained using the quasi-steady hypothesis. Try several cases (a) kf = ks = lOfe/, (b) kr = ks = 20kf, (c) kf = 2ks = lOkf. 

Example 12.6 Formulate the governing equations for an enz5Tne-catalyzed reaction of the form S P in a CSTR. The enz5mie is pristine when it enters the reactor. Do not invoke the quasi-steady hypothesis. 

Chain lifetimes are small and the concentration of free radicals is low. To a reasonable approximation, the system consists of unreacted monomer, unreacted initiator, and dead polymer. The quasi-steady hypothesis gives... 

Solution The equal reactivity assumption says that kp and kc are independent of chain length. The quasi-steady hypothesis gives d R /dt = 0. Applying these to a material balance for growing chains of length / gives... 

This approximation is an example of the quasi-steady hypothesis discussed in Section 2.5.3. 

The general procedure for applying the quasi-steady hypothesis is the following ... 

This agrees with experimental findings (Boyer et al., 1952) on the decomposition of acetaldehyde. The appearance of the power is a wondrous result of the quasi-steady hypothesis. Half-integer kinetics are typical of free-radical systems. The next example describes a free-radical reaction with an apparent order of 1, or depending on the termination mechanism. 

If the enzyme charged to a batch reactor is pristine, some time will be required before equilibrium is reached. This time is usually short compared to the batch reaction time and can be ignored. Furthermore, so > Eo is usually true so that the depletion of substrate to establish the equilibrium is negligible. This means that Michaelis-Menten kinetics can be applied throughout the reaction cycle and the kinetic behavior of a batch reactor will be similar to that of a packed-bed PER, as illustrated in Example 12.4. Simply replace t with tbatch to obtain the approximate result for a batch reactor. This approximation is an example of the quasi-steady hypothesis discussed in Section 2.5.3. 

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