To determine the overall order of the reaction, we need to analyze the given reaction mechanism.$\newline$The mechanism consists of three steps:$\newline$(i) $X_2 \rightarrow X + X \, ($fast$) \newline$(ii) $X + Y_2 \rightleftharpoons XY + Y \, ($slow$) \newline$(iii) $X + Y \rightarrow XY \, ($fast$) \newline$The rate-determining step is the slowest step, which is step (ii).$\newline$The rate law for the rate-determining step is determined by the reactants involved in that step.$\newline$In step (ii), the reactants are $X$ and $Y_2. \newline$Therefore, the rate law can be written as:$\newline$Rate $= k[X][Y_2] \newline$To find the overall order of the reaction, we need to sum the exponents of the concentration terms in the rate law.$\newline$The order with respect to $X$ is 1, and the order with respect to $Y_2$ is 1.$\newline$Thus, the overall order of the reaction is:$\newline 1 + 1 = 2 \newline$However, we need to consider the equilibrium in step (ii).$\newline$The equilibrium constant for the reversible step (ii) can affect the concentration of $X. \newline$If we assume that the concentration of $X$ is in equilibrium with $X_2,$ then:$\newline [X] = K[X_2]^{1/2} \newline$Substituting this into the rate law gives:$\newline$Rate $= k(K[X_2]^{1/2})[Y_2] \newline$This simplifies to:$\newline$Rate $= k'[X_2]^{1/2}[Y_2] \newline$where $k' = kK. \newline$Now, the order with respect to $X_2$ is $\frac{1}{2},$ and the order with respect to $Y_2$ is 1.$\newline$Therefore, the overall order of the reaction is:$\newline \frac{1}{2} + 1 = 1.5 \newline$Thus, the correct option is Option 3.