Characterization of polychlorinated alkane mixtures—a Monte Carlo modeling approach

Abstract

A Monte Carlo model was developed to characterize the molecular composition of polychlorinated alkane mixtures. The model is based upon a simulation of the free-radical chlorination process by which polychlorinated alkane mixtures are produced industrially from n-alkanes. In the model, the free-radical chlorination reaction was simulated by randomly selecting a position on a partially converted alkane molecule for target by chlorine free-radical attack. The relative reactivities of the hydrogen atoms on the alkane chain towards chlorine free-radical substitution were either determined experimentally or extrapolated from experimental results and incorporated into the model. The result of the simulation is the prediction of the detailed molecular composition of any PCA mixture. Good agreement was found when comparing the distribution of molecules predicted by the model to analytically determined distributions of real PCA mixtures. Results from the model were then coupled with rules describing the action of biological enzymes to estimate the upper limit possible for the aerobic biodegradation of PCA mixtures.

Appendix: derivation of analytical solution

For the chlorination process governed by equation (Tomy et al. 1998), each isomer i containing β chlorine atoms can be identified by the set of β elements ${{\bf S}_{i}^\beta}$, as:

$$ {\bf S}_i^\beta =\left\{ {p_1,p_2,p_3,\ldots p_\beta } \right\} $$

(4)

If the hydrogen atoms are numbered in sequence from the end of the molecule, 1 ≤ p _j ≤  2n + 2 represents the position of the hydrogen atom in the alkane chain that has been replaced by a chlorine atom. It follows that each element of the set is unique, repeats are not permitted, and the total number of elements in the set cannot exceed 2n + 2. The molar concentration of an isomer containing n carbon and β chlorine atoms distributed according to the set ${{\bf S}_{i}^\beta}$ is given by ${C_{n,\beta}^{{\bf S}_i^\beta}}$. The total concentration of all unique isomers containing β chlorine atoms, C _n,β, is given by:

$$ C_{n,\beta} =\sum_{p_1 =1}^{2n+3-\beta } \;\; \sum_{p_2 =p_1 +1}^{2n+4-\beta } \;\; \sum_{p_3 =p_2 +1}^{2n+5-\beta } \;\cdots \sum_{p_{\beta -1} =p_{\beta -2}+1}^{2n+1} \;\; \sum_{p_\beta =p_{\beta -1} +1}^{2n+2} C_{n,\beta}^{{\bf S}_i^\beta } $$

The molar ratio of chloride atoms to alkane molecules in the hydrocarbon phase is given by:

$$ X=\frac{C_{\rm Cl} }{\sum\limits_{\beta =0}^{2n+2} {C_{n,\beta}}} $$

(6)

Choosing a basis of one mole of alkane chains, ${\sum_{\beta =0}^{2n+2} {C_{n,\beta}}=1}$ and the total amount of chlorine contained in the hydrocarbon phase is equal to:

$$ X=\left( 0 \right)C_{n,0} +\left( 1 \right)C_{n,1} +\left( 2 \right)C_{n,2} +\cdots+\left( {2n+2} \right)C_{n,2n+2} $$

(7)

In a closed batch process, the change in the chlorine content in the hydrocarbon phase as the extent of the reaction proceeds is given by:

$$ \frac{\hbox{d}X}{\hbox{d}t}=\left( 1 \right)\frac{\hbox{d}C_{n,1} }{\hbox{d}t}+\left( 2 \right)\frac{\hbox{d}C_{n,2} }{\hbox{d}t}+\cdots+\left( {2n+2} \right)\frac{\hbox{d}C_{n,2n+2}}{\hbox{d}t} $$

(8)

The rate of formation of an isomer ${{\bf S}_{i}^\beta}$ by the replacement of a hydrogen atom at any position p _j in a reactant ${\bf S}_i^{\beta -1} =\left\{ {p_1,p_2,p_3,\ldots p_{\beta -1} } \right\}$, where ${\bf S}_i^{\beta -1} \subset {\bf S}_i^\beta$ and $p_j \notin \left\{ {p_1,p_2,p_3,\ldots,p_{\beta -1} } \right\}$ is defined in the usual way in terms of the reacting species ${{\bf S}_i^{\beta -1}}$ , represented as $r_{n,\beta }^{{\bf S}_i^{\beta -1} \rightarrow{\bf S}_i^\beta}$. The total number of independent reactants that can form the isomer ${{\bf S}_i^\beta }$ is given by the binomial coefficient formula $\left( {{\begin{array}{c} \beta \\ {\beta -\hbox{1}} \\ \end{array}}} \right)$, which yields β as a result. The set of β reactants includes all independent subsets of the set ${{\bf S}_i^\beta}$ having β −1 elements. In the general case where β < 2n + 2, the isomer ${{\bf S}_i^\beta }$ can be lost from the system if one of the hydrogen atoms is displaced by a chlorine atom to yield an isomer with β + 1 chlorine atoms. For each isomer ${{\bf S}_i^\beta }$, there are 2n + 2−β possible products. The net rate of formation of the isomer ${{\bf S}_i^\beta }$ can be determined by considering the sum of all β reactions involving the complete set of isomers containing β −1 chlorine atoms, less the 2n + 2−β possible rates of destruction. Finally, the net rate of removal considering all isomers containing β chlorine atoms can be derived by summing the expressions associated with all isomers containing this degree of chlorination, and is equal to dC _n,β/dt in Eq. (8). The complete expression is cumbersome, but is omitted here for clarity.

If chlorine is in excess, the reaction is first order, described by a rate equation of the form:

$$ r_{n,\beta}^{{\bf S}_i^{\beta -1} \rightarrow {\bf S}_i^\beta } =k_{n,\beta }^{{\bf S}_i^{\beta -1} \rightarrow {\bf S}_i^\beta } C_{n,\beta }^{{\bf S}_i^{\beta -1} } $$

(9)

For reactants consisting of pure n-alkane, the initial conditions are given by:

$$ C_n =\sum_{i=1}^{2n+2} {C_{n,i} } =C_{n,0} $$

It is more convenient to work in the chlorination domain. Applying the chain rule to (8) provides the necessary conversion from the time domain.

$$ \frac{\hbox{d}C_{n,\beta}}{\hbox{d}X}={\frac{\hbox{d}C_{n,\beta}}{\hbox{d}t}}\big{/}{\frac{\hbox{d}X}{\hbox{d}t}} $$

(11)

In the simplest case it can be assumed that all hydrogen atoms are replaced at equal rates once encountered by a chlorine free-radical. This is equivalent to assuming that the rate constant of any isomer is proportional to the number of hydrogen atoms on the reacting alkane. From this result, all molecules with the same number of chlorine atoms will have the same reactivity. Thus, from Eq. (9):

$$ r_{n,\beta} =k_{n,\beta } C_{n,\beta} $$

(12)

At any instant during the chlorination process:

$$ \frac{\hbox{d}X}{\hbox{d}t}=\frac{\hbox{d}C_{n,1} }{\hbox{d}t}+\frac{2\hbox{d}C_{n,2} }{\hbox{d}t}+\frac{3\hbox{d}C_{n,3} }{\hbox{d}t}+\frac{\left( {2n+1} \right)\hbox{d}C_{n,(2n+1)} }{\hbox{d}t} +\frac{(2n+2)\hbox{d}C_{n,(2n+2)} }{\hbox{d}t} $$

(13)

Under the simplifying assumption, it follows from a mass balance that:

$$ \frac{\hbox{d}C_{n,\beta}}{\hbox{d}t}=r_{n,\beta -1} -r_{n,\beta} $$

(14)

Combining:

$$ \frac{\hbox{d}X}{\hbox{d}t}=k_{n,(2n+1)} C_{(2n+1)} +\cdots+k_{n,1} C_{n,1} +k_{n,0} C_{n,0} $$

(15)

The values of rate constants are proportional to the number of hydrogen in the carbon chain. Selecting 1 as a basis for the rate constant k _n,(2n+1) it follows:

$$ \frac{\hbox{d}X}{\hbox{d}t}=(1)C_{n,2n+1} +(2)C_{n,2n}+ \cdots +(2n+1)C_{n,1} +(2n+2)C_{n,0} $$

(16)

By inspection, the above expression is equal to the total moles of hydrogen, which can also be written as:

$$ \frac{\hbox{d}X}{\hbox{d}t}=(2n+2)-{\rm X} $$

(17)

These equations can be integrated subject to the condition C _n,β = 0 when X = 0 to yield the solution:

$$ C_{n,\beta } =\frac{X^{\beta }}{(2n+2)^{\beta }}{\kern 1pt} \left( {\frac{(2n+2)-X}{(2n+2)}} \right)^{(2n+2-\beta )}\frac{(2n+2)!}{(2n+2-\beta )!(\beta )!} $$

(18)

An alternative solution to this problem, previously published was used by Colburne and Stern (1965) for comparing experimental chlorination data. This model is of the form:

$$ \frac{C_{n,0} }{C_n }=e^{-X}\quad \hbox{ for }\beta = 0 $$

(19)

$$ \frac{C_{n,\beta } }{C_n }=\frac{X^{\beta }}{\beta !}\cdot e^{-X}\quad\hbox{ for }\beta = 1,\ldots,n-1 $$

(20)

The assumption that allows for this solution is that all alkane molecules react with chlorine radicals at equal rates regardless of the degree of chlorination. However, the model violates the mass balance, or the condition that $\mathop {\lim }\limits_{\beta \rightarrow 2n+2} {C_{n,2n+2}}/{C_n}=1$ and hence does not represent a correct solution.