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KineticsNational 2018

An irreversible reaction A + B → products is studied under conditions where the initial concentrations of A and B are equal. Under these circumstances, a graph of $\ln([A])$ as a function of time is linear. What is the order in A?

7% of national qualifiers got this right

Explanation

draft

Since $[A]_0 = [B]_0$ and the stoichiometry is 1:1, we have $[A] = [B]$ at all times during the reaction. This means the rate law

$\text{rate} = k[A]^m[B]^n$

can be rewritten as

$\text{rate} = k[A]^{m+n}$

because $[B]^n = [A]^n$ at every instant. A plot of $\ln[A]$ vs. time being linear tells us only that the overall kinetics observed for A behaves like a first-order decay, i.e., $m+n = 1$ overall (pseudo-first-order behavior). But this does not tell us the individual value of $m$ (the order in A alone).

For example, $m=1, n=0$ (first order in A, zero in B) gives $\text{rate}=k[A]$ , linear $\ln[A]$ vs t. But also $m=0.5, n=0.5$ gives rate $=k[A]^{0.5}[B]^{0.5}=k[A]$ , since $[A]=[B]$ , also producing a linear $\ln[A]$ plot. Many combinations of $m$ and $n$ summing to 1 are indistinguishable under the condition $[A]_0=[B]_0$ .

Therefore, from this experiment alone, the individual order in A cannot be determined — only the sum $m+n$ can be inferred.

Incorrectly assumes that because the plot is linear in $\ln[A]$ , the rate is somehow independent of concentration; but zero order would give a linear plot of $[A]$ itself vs t, not $\ln[A]$ , so this is inconsistent even superficially.

Assumes that a linear $\ln[A]$ vs t plot always implies first order in A alone, forgetting that because $[A]=[B]$ throughout, the coupling of A and B means the overall exponent (m+n) is what's actually being measured, not just m.

Confuses second order behavior (which requires a linear $1/[A]$ vs t plot, not $\ln[A]$ ) with the given linear $\ln[A]$ data, misapplying the integrated rate law form.

Correct: because $[A]=[B]$ at all times, the data only reveal the sum of orders $m+n=1$ ; the individual order in A cannot be isolated without varying the concentrations independently.

Concept · Rate law order determination with coupled initial concentrations