Read the passage carefully and answer the questions :

The speed at which a chemical reaction takes place is called the rate of reaction. The rate of reaction depends on various factors like concentration of the reactants, temperature, etc. The relation between the rate of reaction and the concentration of reacting species is represented by the equation r = k[A]x[B]Y, where x and y are the order of the reaction with respect to the reactants A and B, respectively.

The overall order of the reaction is x ++ y. The rate of reaction can also be increased by the use of a catalyst which provides an alternate pathway of lower activation energy. It increases the rate of forward and backward reaction to an equal extent. It does not alter the Gibbs energy of the reaction.

Q) If the rate of reaction becomes twenty-seven times upon increasing the concentration of reactant by three times, the order of this reaction is

The correct option is 3.

Let us understand the step-by-step reasoning behind this answer.

According to the rate law for a single reactant, the rate of reaction ($r$) is related to the concentration of the reactant ($\left[A\right]$) raised to the power of its order ($n$). This can be written mathematically as:

$$r=k{\left[A\right]}^n$$

Here, $k$ is the rate constant and $n$ is the order of the reaction.

Let the initial rate of the reaction be $r_1$ when the initial concentration is $\left[A\right]$:

$$r_1=k{\left[A\right]}^n$$

According to the given problem, when the concentration of the reactant is increased by $3$ times, the new concentration becomes $3\left[A\right]$. At this concentration, the new rate of reaction ($r_2$) becomes $27$ times the initial rate ($r_1$).

Thus, we can write the new rate equation as:

$$r_2=27r_1=k{\left(3\right[A\left]\right)}^n$$

Substituting the expression for $r_1$ into the equation, we get:

$$27·k{\left[A\right]}^n=k·3^n{\left[A\right]}^n$$

Dividing both sides by $k{\left[A\right]}^n$, we simplify the equation to:

$$27=3^n$$

Since we know that $27$ can be expressed as a power of $3$ (as $3^3=3\times 3\times 3=27$):

$$3^3=3^n$$

By comparing the exponents on both sides of the equation, we find:

$$n=3$$

Therefore, the overall order of the reaction is 3.