Order Reduction for Complex Mechanisms

Preamble

For many complex mechanisms, the solution to the kinetic differential equations involve some pretty difficult mathematics, whilst others can be intractable in their original form. However, it is sometimes possible that the difficulty with which the differential equations can be solved can be drastically reduced through the technique of a reduction in order. This can be accomplished through the introduction of a new dependent-variable, which somehow involves both the time dependence, and a concentration.

For example, whilst the second-order nucleophilic substitution kinetics can be solved quite easily by direct means, it serves to show the method: for the reaction between A and B, combining A and dt into a new dependent variable, we can reduce the second order kinetics to first order kinetics and analyse it as we would any first-order process, ie., a linear plot of ln [B] vs f(t).

Example I: The Bimolecular Reaction

Scheme:

`A+Bstackrel(k) rarrC`

Which can be represented by the differential equation,

`d/(dt) B=-kAB`

To illustrate the method, we can replaced time, the dependent-variable, with the integral `theta=intAdt`. Using this definition, we can rearrange the differential equation,

`dB =-kBAdt=-kBd theta`

Hence, we have reduced the order of the kinetics from second to first, and we can now solve the system as a first order equation, thus,

`B=B_0exp(-k theta)`

The caveat is that we need to analytically ascertain `theta`, which we do by measuring the time dependence of concentration A and integrate numerically. For example, say for the bimolecular reaction we obtain the following concentration dependence through measurement,

To analyse this data and obtain the rate constant by the method of order reduction, we produce the integral, `theta=int Adt`, by means of numerical integration. This corresponds to calculating the area,

As we expect, our new dependent variable, `theta` is a function of t,

Now, if we replace the time-axis with `theta`, we obtain the dependence below. It looks very similar to the our original result graph, but with one, subtle, but important transformation: the decay of concentration B now takes the form of an exponential decay.

We can observe this more clearly by plotting the natural logarithm, and when we do this something remarkable happens: we get a straight-line, the gradient of which is the bimolecular rate constant!

Whilst the curve for ln[A] appears to be linear, it is not. The curve for ln[B] however, is exactly linear. Taking the natural logarithm of the first-order de solution (from above), we obtain,

`lnB=lnB_0-ktheta`

Then using typical methods of least-squares linear regression, we can access the rate constant. The intrinsic accuracy of the method depends strongly upon the accuracy of the integral `theta`, for example, when integrating numerically, usually some sort of graphical approximation technique is employed. This treats the curve as a series of trapezoids, approximating a curve with simple geometrics and then summing the areas, however, this method is not precise, it tends to overshoot the actual function. The situation is improved as the more frequent measurements are taken, but naturally, there's a limit to how frequently we can take a measurement! This naturally prediposes the technique to reactions that can be followed in-situ, however, it will still produce a value in fairly good agreement with the original, non-simplified solution to the kinetic equations.

These figures can useful as initialisation parameters for a more thorough parameter optimisation, say using the linear-approximation numerical method.

Example II: The Parallel-Consecutive Bimolecular reaction

As described, the exact solution to the PCBM kinetics involves transcendental special functions. However, using the order-reduction method, the complexity of the solution can be made drastically simpler. Consider the set of differential equations - there are too many inter-dependent variables for there to exist a closed form solution - to solve the system, we could use the transcendental solution (ibid), or we could employ the order-reduction technique.

Scheme:

`A+Bstackrel(k_1) rarrC`

`A+Cstackrel(k_2) rarrD`

Differential Equations:

`A'(t)=-k_1AB-k_2AC`

`B'(t)=-k_1AB`

`C'(t)=k_1AB-k_2AC`

`D'(t)=k_2AC`

Mass Balance:

`A_0=A+C+2D`

`B_0=B+C+D`

`A_0/B_0=2`

Hence, `C=A-2B`

The species A is involved in both of the two steps, which are originally bimolecular, second order rates; by order-reduction, combining A and t into `theta`, the mechanisms can instead be described as two consecutive first-order processes. Hence, let us define, `theta=int Adt`. The set of differential equations become,

`A'(theta)=-k_1B-k_2C`

`B'(theta)=-k_1B`

`C'(theta)=k_1B-k_2C`

`D'(theta)=k_2C`

Now, the solution to this set may not be obvious by any stretch, but it is certainly very much more straight-forward than the transcendental solution; with the added advantage that this set of equations is tractable, and its solution can be expressed in closed form (ie., with a finite number of algebraic terms). Since the solution to this set of equations has been discussed here, we will not repeat its derivation and instead we provide only the outline (below). This set of order-reduced differential equation have the solutions,

`A(theta)=(k_1B_0)/(k_2-k_1) (exp(-k_1t)-exp(-k_2t))+2B_0exp(-k_1theta)`

`B(theta)=B_0exp(-k_1theta)`

`C(theta)=(k_1B_0)/(k_2-k_1) (exp(-k_1t)-exp(-k_2t))`

`D(theta)=(k_2k_1B_0)/(k_2-k_1) ( exp(-k_2t)/k_2 - exp(-k_1t)/k_1 ) +B_0`

Using the solution `B(theta)`, we can obtain a linear plot of the natural logarithm, `log_e B` against `theta`, providing us access to k1 from the gradient, which incidentally, is ideal for initialising a parameter optimisation using the transcendental solution. Whilst it does not appear to be possible to solve these equation for k2, we can use typical methods of least-squares, non-linear curve regression to obtain it.

Derivation

We begin by solving `B'(theta)`, which we see is a linear first-order process with the solution, `B(theta)=B_0exp(-k_1theta)`. We substitute the definition into the differential equation `C'(theta)` to obtain,

`C'(theta)=k_1B_0exp(-k_1theta)-k_2C`

We note this differential equation is of the standard form,

`y'=F(x)-G(x)y`

which we can solve using the integrating factor method, and obtain the solution,

`C=(k_1B_0)/(k_2-k_1) (exp(-k_1t)-exp(-k_2t))`

Next we can solve `D'(theta)`; we simply integrate again:

`dD=k_2C=k_2(k_1B_0)/(k_2-k_1) (exp(-k_1t)-exp(-k_2t)) d theta`

To obtain the solution,

`D=(k_2k_1B_0)/(k_2-k_1) ( exp(-k_2t)/k_2 - exp(-k_1t)/k_1 + c )`

Solving the initial value problem we arrive at,

`D=(k_2k_1B_0)/(k_2-k_1) ( exp(-k_2t)/k_2 - exp(-k_1t)/k_1 ) +B_0`

Then using the mass-balance equations, `C=A-2B`, we can find A,

 `A=C+2B=(k_1B_0)/(k_2-k_1) (exp(-k_1t)-exp(-k_2t))+2B_0exp(-k_1theta)`