Many double-displacement reactions are enzyme-catalyzed via the "ping pong" mechanism, so called because the reactants appear to bounce off the enzyme like a ping pong ball. these reactions typically have two reactants and two products. in a third-order reaction involving two reactants and two products, doubling the concentration of the first reactant causes the rate to increase by a factor of 2. what will happen to the rate of this reaction if the concentration of the second reactant is cut in half?
a. it will increase by a factor of 2.b. it will increase by a factor of 4.c. it will decrease by a factor of 2.d. it will decrease by a factor of 4.

D. It will decrease by a factor of 4

Explanation:

According to the question , the equation follows :

Rate law : This states the rate of reaction is directly proportional to concentration of reactants with each reactant raised to some power which may or may not be equal to the stoichiometeric coefficient.

.................(1)

STEP": First, find out the power "a" and "b"

a+b = 3 (because it is given that the reaction follow 3rd order-kinetics)

According to question, doubling the concentration of the first reactant causes the rate to increase by a factor of 2 means,

r' = 2r if [A'] = 2[A]

Here [B] is uneffected means [B']=[B]

hence new rate =

Put the value of [A'] , [B'] and r' in the above equation:

...........(2)

Divide equation (1) by (2) we , get

Here A and A cancel each other

B and B cancel each other

We get,

1^b = 1 ( power of 1 = 1)

This is possible only when a = 1

We know that : a + b = 3

1 + b = 3

b =3 -1  = 2

b = 2

Hence the rate law becomes :

.............(3)

Look in the question now, it is asked to calculate the concentration of [B],if  cut in half

Hence

[B']=1/2[B]

Insert the value of [B'] in equation (3)

............(a)

But

..............(b)

Compare equation (a) and (b) , we get

new rate r' =

r' = 1/4 r