Effect of changing of the concentration of reactants when concentration of the initiator and the concentration of the initial reactant are very small

A + I---> p


What happens to the rate of the reaction when the concentration of the reactant is changed in a system that its concentration of its initiator and the concentration of the initial reactant is very small? Generally, we add a small amount of initiators with respect to the concentration of reactance.eg: I= 1\*10^-4M, A= 1M


Can we take the order of A zero under these conditions?


For example, styrene polymerization is initiated by benzoyl peroxide.





| styrene used (mmol) | benzoyl peroxide (mmol) |
| --- | --- |
| 0.45 | 1.5 |
| 0.93 | 2.48 |
| 2.27 | 1.5 |


 

As you written the reaction in an irreversible fashion, that will be my starting point. You cannot state that the partial order of $A$ is zero, still the rate law has the form
$$ r = kC\_A^\alpha C\_I^\beta \tag{1} $$
However, if you put an excess of $A$, then its concentration as the reaction takes places, will not deviate much from the initial concentration $C\_{A0}$. Thus, Eq. (1) now is
$$ r = \underbrace{kC\_{A0}^\alpha}\_{k'} C\_I^\beta = k'C\_I^\beta \tag{2} $$
Eq. (2) is helpful because it allows to at least determine $\beta$ in this limiting case. For example, if you have data, you can couple Eq. (2) with a mole balance in a batch reactor at constant volume
\begin{align}
 \frac{dC\_I}{dt} &= r\_I \\
 \frac{dC\_I}{dt} &= -\nu\_Ir \\
 \frac{dC\_I}{dt} &= -k'C\_I^\beta \rightarrow
 \ln\left(-\dfrac{dC\_I}{dt}\right) = \ln(k') + \beta\ln(C\_I) \tag{3}
\end{align}
If you have $C\_I$ as a function of time, you can plot the left-hand sight (by numerical differentiation of your data) vs $\ln(C\_I)$. If it results in a straight line, then the slope and the y-intercept will inform you that
$$ k' = \exp(b) \hspace{1 cm} m = \beta \tag{4,5}$$
Eq. (4) won't give you anything valuable, since $ k' = kC\_{A0}^\alpha $, and we don't know $\alpha$. However, at least you have $\beta$ for this set of experiments.


If the reaction mechanism is more complicated, as Poutnik pointed out, then the algebra changes. However, you can always approximate at every step that $C\_A \approx C\_{A0}$ to simplify matters.


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