Initial concentration mol/l[a] initial concentration mol/l[b] initial rate mol/ls 0.20 0.10 20 0.20 0.20 40 0.40 0.20 160 given the data in the accompanying table, what is the reaction order for b?

The generalized rate expression may be written as:
r = k[A]ᵃ[B]ᵇ

We may determine the order with respect to B by observing the change in rate when the concentration of B is changed. This can be done by comparing the first two runs of the experiment, where the concentration of A is constant but the concentration of B is doubled. Upon doubling the concentration of B, we see that the rate also doubles. Therefore, the order with respect to concentration of B is 1.
The same can be done to determine the concentration with respect to A. The rate increases 4 times between the second and third trial in which the concentration of B is constant, but that of A is doubled. We find that the order with respect to is 2. The rate expression is:

r = k[A]²[B]

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It shows that the experiment is designed properly.