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Show 1.6.5 y = A exp(-T) <8) where A = [NH3]0/[NO]0 is the nondimensionalized initial NH3-concentration. Substituting this result into Eq. (7) leaves two equations for x and z, with the initial conditions x = 1 and z = 0 at T = 0. In order to solve these equations, both x and z are tentatively expanded in powers of e: x = x0 + exj • ..., z » z0 • tzi • .„ C9I Equating terms of order e° in Eqs. (5) and (7) yields -z(x0 + xc) = 0, -z(x0 - xc) • 0, with the solution z0 = 0 <10> Equating terms of order e yields 0 = -zi(x0+ xc) • A exp(-r) dx0/dT = -zi<x0 - x^. These equations could also have been obtained by applying the quasi-steady-state approximation to [NH21 For the case xc = constant, corresponding to [OH] • constant, k2/k^ = constant, there results the solution 1 - x 0 - 2xc lnC(x0 - xc)/(l - xc)] = ACi - exp(-T)] = A* (11) zj = (A - 1 + x 0 • 2xcln[(x0 - XjJ/U - X(j)]/<x0 + XjJ (12) The solution for y can be written in the form y = A - A' (13) Equation (12) does not satisfy the initial condition z^ • 0 at T = 0. In order to satisfy this condition, account must be taken of the existence of an "inner solution" valid at small T. It is easily shown that this inner solution is given by xo = i,zo = 0 (14) z = ezj * ... = (1 - exp[-(l • xc)Ti]}eA/(l * xc> (15) where TJ = r/e is the stretched independent variable for the inner region. Matching this solution asymptotically to the solution for the outer region does no affect the results (11) and (12). Equation (11) is a transcendental equation with a number of interesting properties. As xc -» 0 it yields X Q -» 1 - A* for 0 < A* < 1, and x0 -> 0 for A* > 1. As xc -» 1 it yields x0 -» 1. Furthermore, x 0 - > x c a s A'-»» For finite values of A', II - x0l < II - xcl. A plot of x0 as function of A' for various values of xc is shown in Fig. 1, while Fig. 2 shows a plot of x0/xc as a function of xc for various values of A*. It should be noted that A' = A for T = m, while A* < A for T < A For A' * 1, the value of x0/xc equals 2.3 at xc = 0.2. It can be concluded that in order to achieve values of X Q / X C close to unity for 0.1 i x i 0.8, |