To determine the order of the reaction with respect to A and B, we use the rate law expression: rate = k [A]^x [B]^y, where x and y are the orders with respect to A and B, respectively.

Step 1: Analyze the effect when only A is doubled.
Initial rate: 0.3 = k [A]^x [B]^y ...(1)
When [A] is doubled (and [B] unchanged), new rate = 0.6 = k (2[A])^x [B]^y ...(2)
Divide (2) by (1):
$\frac{0.6}{0.3}=\frac{k\left(2\right[A\left]{)}^x\right[B]{)}^y}{k\left[A\right]{)}^x\left[B\right]{)}^y}\Rightarrow 2=2^x\Rightarrow x=1$
So, order with respect to A is 1.

Step 2: Analyze the effect when both A and B are doubled.
Initial rate: 0.3 = k [A]¹ [B]^y ...(3)
When both doubled, new rate = 2.4 = k (2[A])¹ (2[B])^y = k * 2[A] * 2^y[B]^y = 2 * 2^y * k [A] [B]^y ...(4)
Divide (4) by (3):
$\frac{2.4}{0.3}=\frac{2\cdot 2^y\cdot k\left[A\right]\left[B\right]{)}^y}{k\left[A\right]\left[B\right]{)}^y}\Rightarrow 8=2\cdot 2^y\Rightarrow 2^y=4\Rightarrow 2^y=2^2\Rightarrow y=2$
So, order with respect to B is 2.

Conclusion:
Order w.r.t A is 1, order w.r.t B is 2. Total order = 1 + 2 = 3.
Therefore, the correct statement is: "Order of the reaction with respect to B is 2."

Related Topics

Rate Law and Order of Reaction: The rate law expresses the relationship between the rate of a reaction and the concentration of reactants. The order of reaction with respect to a reactant is the exponent to which its concentration is raised in the rate law. It is determined experimentally, often by the method of initial rates, as used here.

Method of Initial Rates: This method involves measuring the initial rate of reaction for different initial concentrations of reactants. By comparing how the rate changes when concentrations are changed, the order with respect to each reactant can be found.

Formulae

General rate law: $rate=k\left[A{]}^x\right[B{]}^y$

To find order when concentration is changed: $\frac{r_2}{r_1}={\left(\frac{{\left[A\right]}_2}{{\left[A\right]}_1}\right)}^x{\left(\frac{{\left[B\right]}_2}{{\left[B\right]}_1}\right)}^y$