# Why we should concern with A∞?

To determine the order of one reactant at pseudo concentration of another reactant the plot of $$\ln(A_t - A_∞)$$ versus $$t$$ is used. $$A_t$$ is absorbance of the intermediate at time $$t$$. $$A_∞$$ is absorbance of intermediate at the end of the reaction (approximately after 24 hours).

Why $$A_∞$$ is important here?

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## 1 Answer

Let's suppose a hypothetical irreversible reaction: $$\ce{B + H -> D + F}$$ where H is a reactant with constant concentration during the reaction and that don't absorb light in the working frequencies. The absorbance of a species $$i$$ related with the concentration by the Lambert-Beer Law: $$A_{i}=c_{i}\varepsilon_{i}l$$ where $$c_i$$ is the concentration of species $$i$$, $$\varepsilon_{i}$$ is the molar absorptivity and $$l$$ is the length of the cuvette.

At an arbitrary time $$t$$, the total absorbance of the solution is given by:

$$A_{t}=c_{B,t}\varepsilon_{B}l+c_{D,t}\varepsilon_{D}l+c_{F,t}\varepsilon_{F}l \tag{1}$$

by the stoichiometry of the reaction, we have: $$c_{D,t}=c_{F,t}=c_{B,0}-c_{B,t}$$; where $$c_{B,0}$$ is the initial concentration of the reactant B, so replacing this in Eq. 1 and defining a auxiliary constant $$\varepsilon_{i}l=k_{i}$$, we have:

$$A_{t}=c_{B,t}k_{B}+k_{D}(c_{B,0}-c_{B,t})+k_{F}(c_{B,0}-c_{B,t})$$ $$A_{t}=c_{B,t}(k_{B}-k_{D}-k_{F})+c_{B,0}(k_{D}+k_{F})\tag{2}$$

At a time $$t\rightarrow \infty$$ (a time long enough that the reaction goes to completion), $$c_{B,\infty}=0$$ and $$c_{D,\infty}=c_{F,\infty}=c_{B,0}$$, so the absorbance is given by:

$$A_{\infty}=c_{D,\infty}k_{D}+c_{F,\infty}k_{F}$$ $$\boxed{A_{\infty}=c_{B,0}(k_{D}+k_{F})} \tag{3}$$

so,replacing Eq.3 in Eq.2 its follows that:

$$A_{t}=c_{B,t}(k_{B}-k_{D}-k_{F})+A_{\infty}$$ $$\boxed{A_{t}-A_{\infty}=c_{B,t}(k_{B}-k_{D}-k_{F})}$$

In conclusion: $$A_{\infty}$$ is important because is directly proportional to the initial concentration of reactant B. When you subtract $$A_{\infty}$$ from the absorbance at an arbitrary time $$t$$ $$(A_{\infty})$$ it allows you to know (indirectly) the concentration of the reactant at the aforementioned time $$t$$.