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# What is the rate law for the following mechanism in terms of the overall rate constant k? Step 1: ##A + B leftrightarrow C## (fast) Step 2: ## B + C -> D## (slow)

The is ##"rate" = k["A"]["B"]^2##.

The reaction is

##"A + B"color(white)(l) ⇌ "C"## (fast) ##"B + C" color(white)(l)stackrelcolor(blue)(k_2color(white)(m))(→) "D"## (slow)

The rate of the reaction is governed by the slow step.

This gives the rate equation

##"rate" = k["B"]["C"]##

The problem is that ##"C"## isn't one of the reactants.

We must express ##"[C]"## in terms of ##"[A]"## and ##"[B]"##.

We can do that because the first step is an equilibrium.

##K = "[C]"/"[A][B]"##

∴ ##["C"] = K["A"]["B"]##

Substituting into the original rate expression, we get

##"rate" = k_2["B"] × K["A"]["B"] = kK["A"]["B"]^2##

Let ##k = k_2K##, and the rate law becomes

##"rate" = k["A]["B"]^2##