Steady-State Diffusion with Homogeneous Chemical Reaction

3 Steady-State Diffusion with Homogeneous Chemical Reaction 

The following example, taken from Welty et al. ( 1976), illustrates the solution approach to a steady-state, one-dimensional, diffusional or heat conduction problem. 

As shown in Fig. 4.5, an inert gas containing a soluble eomponent, S, stands above the quiescent surface of a liquid, in which the component, S is both soluble and in which it reacts chemically to form an inert product. Assuming the concentration of S at the gas-liquid surface to be constant, it is desired to determine the rate of solution of eomponent S and the subsequent steady-state concentration profile within the liquid. 

Under quiescent conditions, the rate of solution of S within the liquid, is determined by molecular diffusion and is described by Pick s law, where 

At steady-state conditions, the rate of supply of S by diffusion is balanced by the rate of consumption by chemical reaction, where assuming a first-order chemical reaction 

---

Steady-state mass diffusion with homogeneous chemical reaction... 

A to products by considering mass transfer across the external surface of the catalyst. In the presence of multiple chemical reactions, where each iRy depends only on Ca, stoichiometry is not required. Furthermore, the thermal energy balance is not required when = 0 for each chemical reaction. In the presence of multiple chemical reactions where thermal energy effects must be considered becanse each AH j is not insignificant, methodologies beyond those discussed in this chapter must be employed to generate temperature and molar density profiles within catalytic pellets (see Aris, 1975, Chap. 5). In the absence of any complications associated with 0, one manipulates the steady-state mass transfer equation for reactant A with pseudo-homogeneous one-dimensional diffusion and multiple chemical reactions under isothermal conditions (see equation 27-14) ... 

We have restricted our discussion in this section to bistability in well-stirred, homogeneous systems. Multiple steady states may also occur in unstirred systems, where domains of the system in one steady state coexist with domains in the other steady state. In addition to the obvious application to nondhemical systems, chemical systems (in fact the iodate-arsenite system considered here) sometimes exhibit domains that are connected by propagating reaction-diffusion fronts. We will return to this system in our discussion of chemical waves, which will include a description of these fronts. 

In this section, the different behavior of processes with coupled noncatalytic homogeneous reactions (CE and EC mechanisms) is discussed in comparison with a catalytic process. We will consider that the chemical kinetics is fast enough and in the case of CE and EC mechanisms K (- c /cf) fulfills K 1 so that the kinetic steady-state and even diffusive-kinetic steady-state approximation can be applied. 

In these electrode processes, the use of macroelectrodes is recommended when the homogeneous kinetics is slow in order to achieve a commitment between the diffusive and chemical rates. When the chemical kinetics is very fast with respect to the mass transport and macroelectrodes are employed, the electrochemical response is insensitive to the homogeneous kinetics of the chemical reactions—except for first-order catalytic reactions and irreversible chemical reactions follow up the electron transfer—because the reaction layer becomes negligible compared with the diffusion layer. Under the above conditions, the equilibria behave as fully labile and it can be supposed that they are maintained at any point in the solution at any time and at any applied potential pulse. This means an independent of time (stationary) response cannot be obtained at planar electrodes except in the case of a first-order catalytic mechanism. Under these conditions, the use of microelectrodes is recommended to determine large rate constants. However, there is a range of microelectrode radii with which a kinetic-dependent stationary response is obtained beyond the upper limit, a transient response is recorded, whereas beyond the lower limit, the steady-state response is insensitive to the chemical kinetics because the kinetic contribution is masked by the diffusion mass transport. In the case of spherical microelectrodes, the lower limit corresponds to the situation where the reaction layer thickness does not exceed 80 % of the diffusion layer thickness. 

In this form of voltammetry, the concentration distributions of each species in the electrode reaction mechanism are temporally invariant at each applied potential. This condition applies to a good approximation despite various processes still occurring such as mass transport (e.g. diffusion), heterogeneous electron transfer and homogeneous chemical processes. Theoretically it takes an infinite time to reach the steady state. Thus, in a practical sense steady-state voltammetric experiments are conducted under conditions that approach sufficiently close to the true steady state that the experimental uncertainty of the steady-state value of the parameter being probed (e.g. electrode current) is greater than that associated with not fully reaching the steady state. The... 

Thus in the oscillator death state (Mirollo and Strogatz, 1990), the stirring completely inhibits the chemical oscillations, transforming the unstable steady state of the local dynamics into a stable attractor of the reaction-advection-diffusion system. This arises from the competition between the non-uniform frequencies that lead to the dispersion of the phase of the oscillations, while mixing tends to homogenize the system and bring it back to the unstable steady state. Therefore this behavior is only possible in oscillatory media with spatially non-uniform frequencies. The phase diagram of the non-uniform oscillatory system in the plane of the two main control parameters (v, 5) is shown in Fig. 8.4. 

---