# What are the order of the reaction under the following conditions?

The experimental rate of a reaction

$$\ce{A + 2B -> C + D}$$

$$\frac{\mathrm dx}{\mathrm dt} = K[\ce{A}][\ce{B}]^2$$

Calculate the order of the reaction when (i) $$[\ce{A}]\gg[\ce{B}]$$ (ii) $$[\ce{B}]\gg[\ce{A}]$$

My efforts :

Should we ignore the concentration of the reactant that is present negligibly? And thus the orders will be:

(i) Order = 1

(ii) Order =2

• When you have something in large amount the effect of that species becomes negligible. You are limited by your limiting reactant. Jul 7, 2016 at 21:14
• @mamun So it should be the reverse (i) In case I since B is the limiting reactant the order of the reaction will depend on it and the order will be 2 (ii) In the 2nd case it is A that acts as the limiting reactant and the order of the reaction is 1 Jul 7, 2016 at 21:19
• Your latest comment is spot on. Nice job. Jul 8, 2016 at 0:57

Let's compare the rate when $A$ and $B$ are present at their initial amounts, say $A_0$ and $B_0$, to when the reaction is 99% complete.

When $[\ce{A}]\gg[\ce{B}]$, the reaction will be 99% complete when 99% of $B$ is gone, i.e. when $B = 0.01B_0$.

At that point, $$\frac{dx}{dt} = k[A][B]^2 = k(A_0 - \frac{0.99}{2}B_0)(0.01B_0)^2$$

Now since $A\gg B$ also means $A_0\gg B_0$, then it follows that $(A_0 - \frac{0.99}{2}B_0) \approx A_0$. Thus, when the reaction is 99% complete and $B$ is limiting,

$$\frac{dx}{dt} \approx kA_0(0.01B_0)^2 = kA_0 (0.0001)B_0^2$$

If we lump $k$ and $A_0$ into a single parameter, $k_{app}$, then we can write $\frac{dx}{dt} = k_{app}B^2$, which shows that when $B$ is limiting, the reaction is 2nd order.

The case for $A$ being limiting shows the reverse.

The order overall is 3 , first in A and second in B.

When [A]>>[B], A can be considered as constant, because all of B can be consumed but [A] is hardly changed. In this situation the concentration of A ([A]) can be lumped in with the 3rd order rate constant $k_3$ to give a pseudo-second order reaction, $k_{psuedo}=k_3[A]$. This has 2nd order units $dm^3mol^{-1}s^{-1}$. A similar argument applies when [B]>>[A] but with a different pseudo order rate constant.