$$\begin \textrm_3\underset}}\textrm_2 + \textrm; \qquad \textrm_3+\textrm\overset\ 2\textrm_2. \end$$

(1)

In their study, Bond et al. considered as one of their examples the reactions shown in (1) involving ozone, oxygen and atomic oxygen, and the associated system of nonlinear ordinary differential equations (ODEs), their equations (22)–(24):

$$\begin x'= & -k_1x + k_zy -k_xy \end$$

(2)

$$\begin y'= & \ k_1 x - k_zy - k_2xy \end$$

(3)

$$\begin z'= & \ k_1 x - k_zy +2 k_2xy. \end$$

(4)

Here x, y and z are, respectively, the concentrations of \(\textrm_3\), O and \(\textrm_2\) in arbitrary units, with initial conditions \(x(0)=1\), \(y(0)=z(0)=0\). The prime symbol denotes differentiation with respect to t (time). Effectively it is assumed that each of the three reactions is of first order in the (one or two) species involved.

Equations (2)–(4) imply that \(3x'+y'+2z'=0\) exactly, hence that \(3x+y+2z\) is constant. The QSSA applied to y (i.e. setting \(y'=0\)) implies that \(3x+2z\) is constant (at \(3x(0)+2z(0)=3\)), and that, approximately,

$$\begin y = k_1 x/(k_ z + k_2x). \end$$

(5)

It then follows that (approximately)

$$\begin x'= \fracz+k_2x} \end$$

(6)

and

$$\begin z'=\fracz+k_2x}. \end$$

(7)

These equations are indeed consistent with \(3x'+2z'=0\). If one can solve (6) for x(t), then \(z(t)=\frac (1-x(t))\) gives one z. But first one has to solve (6). The solution proceeds as follows. First, use \(z(t)=\frac (1-x(t))\) to put Eq. (6) in the form

$$\begin x' = \frac, \end$$

(8)

where A and B are functions of the three rate coefficients \(k_1\), \(k_2\), \(k_\) (also known as rate constants): \(A=-2k_1 k_2/C\), \(B=\frac k_/C\), with \(C=k_2 -\frac k_\). Then solve (8), subject to \(x(0)=1\):

$$\log x -B/x = At - B.$$

Given t (and the rate coefficients), one can solve numerically for x(t) and deduce z(t). Equation (5) then provides y(t). Plots of \(\big (x(t),y(t),z(t)\big )\) (as approximated by the QSSA) against t can therefore be produced if useful. The behaviour of y(t) under the QSSA is particularly important. It should (after a possible induction period) be ‘small’ and almost constant. If not, the QSSA has produced a significant contradiction which invalidates its use.

Perhaps it is worthwhile to mention here a minor difference in terminology. In much of the chemical literature, the QSSA is described as equating to zero the derivatives of concentrations of intermediates. Others refer to ‘fast variables’ rather than to concentrations of intermediates.

But, since Bond et al. assume in their treatment of this example that y (the concentration of O), is a fast variable, and that x and z are slow, there is the following possible approach to the behaviour in the early stages. Assume that, in those early stages (the ‘induction period’), the slow variables x and z are approximately constant (at a and c respectively). With these values for x and z, the Eqs. (2)–(4) can then be approximated by a linear system:

$$\begin x'= & -k_1a +( k_c -k_a)y \end$$

(9)

$$\begin y'= & \ k_1a- (k_c + k_2a)y \end$$

(10)

$$\begin z'= & \ k_1a+ ( - k_c +2 k_2a)y. \end$$

(11)

If we are mainly interested in small t and the initial conditions \(x(0)=1, y(0)=z(0)=0\), we can take \(a=1, c=0\) in our Eqs. (9)–(11), getting

$$\beginx'= & -k_1 -k_2y \end$$

(12)

$$\begin y'=& \ k_1- k_2y \end$$

(13)

$$\begin z'= & \ k_1+ 2 k_2y. \end$$

(14)

Equation (13) is just a first-order linear ODE for y, which we solve subject to \(y(0)=0\). The solution is

$$y(t)= \frac\big ( 1-\exp (-k_2t) \big ),$$

which for small t is approximately \(k_1 t\).

Equations (12) and (14) can now be solved for x(t) and z(t), subject to \(x(0)=1\) and \(z(0)=0\):

$$x(t) = 1-2k_1t +\frac(1-\exp (-k_2t)) = 1-2k_1t +y \approx 1 -k_1t$$

and

$$z(t) =-2\frac +3k_1t + \frac\exp (-k_2t) = 3k_1t -2y \approx k_1t.$$

It is clear in the above that x and z are not exactly constant—just as the QSSA, by assuming that \(y'\) is approximately zero, produces a \(y'\) that is not exactly zero. And note that setting the derivatives \(x'\) and \(z'\) equal to (approximately) zero in (12) and (14) would produce as approximate y-values negative multiples of \(k_1/k_2\). This suggests that one would need \(k_1\) much less than \(k_2\) if one wished to use the approximations proposed here.

Since \(x(0)+y(0)+z(0)=1\), one might naively suspect that (with these approximations) the total is always 1. But the above solutions for x, y and z imply that \(x(t)+y(t)+z(t)=1+k_1t.\) Here, however,

$$3x+y+2z= 3( 1-2k_1t +y )+y+2( 3k_1t -2y )= 3,$$

so these approximations keep constant a quantity which is proportional to the total number of oxygen atoms. It has been described as a weakness of the QSSA that it keeps \(3x+2z\) constant; ideally it should keep \(3x+y+2z\) constant. The small-t approximations discussed here do not have the same weakness.

We present in Fig. 1 two plots which demonstrate these approximations to x, y and z, for the case \(k_ 1=1, k_= k_2=10\). In both plots the bottom (red) line represents (the approximation to) the ‘fast’ variable y. This approximation to y appears to change little after time about 0.1. But the approximations to the slow variables x and z, although initially close to constant, deteriorate rapidly. In this case we see therefore that, even before the end of the apparent induction period for the QSSA, the proposed approximations to x and z cannot reasonably be regarded as adequate.

Ozone reactions, plots of small-t approximations to the three concentrations x (black), y, z (blue). The bottom (red) line is the approximation to y, the ‘fast variable’