The rate of reaction between two reactants A and B can be expressed as:

rate = k[A]^m[B]^n

Where:
- rate is the rate of reaction
- k is the rate constant
- [A] and [B] are the concentrations of reactants A and B, respectively
- m and n are the orders of reaction with respect to reactants A and B, respectively.

We are given that the rate of reaction decreases by a factor of 4 when the concentration of reactant B is doubled. In other words, if [B] is doubled, the rate becomes 1/4 of its original value. Mathematically, this can be expressed as:

(1/4) = k[A]^m(2[B])^n

Simplifying the equation further:

1 = 4k[A]^m[B]^n

Since k, [A], and [B] are constant, the equation above implies that the only way for the rate to remain unchanged is if the powers of [B] are equal to -1. Therefore, we have:

n = -1

Hence, the reaction is first order with respect to reactant B.

Alternatively, we can use the concept of reaction rate constants to determine the order of reaction with respect to reactant B. The rate constant, k, is defined as the proportionality constant that relates the rate of reaction to the concentrations of the reactants. In this case, we can compare the rate constant before and after doubling the concentration of reactant B:

rate1 = k[A]^m[B]^n
rate2 = k'[A]^m(2[B])^n

Since we are given that the rate decreases by a factor of 4, we can write:

rate2 = (1/4)rate1

Substituting the expressions for rate1 and rate2:

k[A]^m[B]^n = (1/4)k'[A]^m(2[B])^n

Cancelling out the common terms:

1 = (1/4)(2^n)k'/k

Simplifying further:

8 = 2^n(k'/k)

Since k' and k are both rate constants, their ratio is also a constant. Therefore, we can write:

8 = 2^n(K)

Simplifying the equation:

2^3 = 2^n

Comparing the powers:

3 = n

Hence, the reaction is second order with respect to reactant B.