For the reaction, $$2A + B \rightarrow$$ products, when the concentration of $$A$$ and $$B$$ both were doubled, the rate of the reaction increased from 0.3 mol L$$^{-1}$$ s$$^{-1}$$ to 2.4 mol L$$^{-1}$$ s$$^{-1}$$. When the concentration of $$A$$ alone is doubled, the rate increased from 0.3 mol L$$^{-1}$$ s$$^{-1}$$ to 0.6 mol L$$^{-1}$$ s$$^{-1}$$. Which one of the following statements is correct?

We assume that the rate follows the usual rate law expression $$r = k[A]^m[B]^n$$, where $$m$$ is the order with respect to $$A$$ and $$n$$ is the order with respect to $$B$$. The symbol $$k$$ is the rate constant.

For the initial experiment we let the initial concentrations be $$[A]=a$$ and $$[B]=b$$. The initial rate is given to be

$$r_1 = k\,a^{\,m}\,b^{\,n} = 0.3\;\text{mol L}^{-1}\text{s}^{-1}.$$

Now both $$[A]$$ and $$[B]$$ are doubled, so they become $$2a$$ and $$2b$$ respectively. The new rate therefore is

$$r_2 = k\,(2a)^{\,m}\,(2b)^{\,n}.$$

Using the laws of exponents, we write

$$r_2 = k\,2^{\,m}a^{\,m}\,2^{\,n}b^{\,n} = 2^{\,m+n}\,k\,a^{\,m}b^{\,n} = 2^{\,m+n}\,r_1.$$

We are told that the numerical value of the rate increases from $$0.3$$ to $$2.4$$, so

$$\dfrac{r_2}{r_1}=\dfrac{2.4}{0.3}=8.$$

Comparing with the algebraic factor we obtained, we have

$$2^{\,m+n}=8.$$

Since $$8=2^{3}$$, we immediately deduce

$$m+n = 3.$$

Thus, the total order of the reaction is three, but we still need individual orders $$m$$ and $$n$$.

Next, only $$[A]$$ is doubled while $$[B]$$ is kept unchanged. The concentration of $$A$$ becomes $$2a$$ and that of $$B$$ stays $$b$$. The resulting rate is

$$r_3 = k\,(2a)^{\,m}\,b^{\,n} = 2^{\,m}\,k\,a^{\,m}b^{\,n}=2^{\,m}\,r_1.$$

The problem states that the rate rises from $$0.3$$ to $$0.6$$, so

$$\dfrac{r_3}{r_1}=\dfrac{0.6}{0.3}=2.$$

Equating the two factors gives

$$2^{\,m}=2 \quad\Longrightarrow\quad m = 1.$$

Substituting $$m = 1$$ into the earlier relation $$m+n = 3$$, we obtain

$$1 + n = 3 \quad\Longrightarrow\quad n = 2.$$

Therefore, the order with respect to $$A$$ is $$1$$ and the order with respect to $$B$$ is $$2$$. The only statement among the options that reflects this outcome is “Order of the reaction with respect to B is 2.”

Hence, the correct answer is Option A.