To solve this problem, we need to understand the rate of reaction in terms of the change in concentration of reactants and products.$\newline$For the given chemical reaction:$\newline N_2 (g) + 3H_2 (g) \rightleftharpoons 2NH_3 (g), \newline$the rate of reaction can be expressed in terms of the rate of change of concentration of each species.$\newline$The general expression for the rate of a reaction is:$\newline$Rate$= -\frac{1}{stoichiometric coefficient} {\frac{d \ [Reactant]}{dt}} = \frac{1}{stoichiometric coefficient} {\frac{d[Product]}{dt}} \newline$For this reaction, the rate expressions are:$\newline -\frac{d[N_2]}{dt} = -\frac{1}{3} \frac{d[H_2]}{dt} = \frac{1}{2} \frac{d[NH_3]}{dt} \newline$Let's analyze the given options:$\newline$Option 1: $-\frac{1}{3} \frac{d[H_2]}{dt} = -\frac{1}{2} \frac{d[NH_3]}{dt} \newline$This option is incorrect because the negative sign on the right side is not appropriate for the product.$\newline$Option 2: $-\frac{d[N_2]}{dt} = 2 \frac{d[NH_3]}{dt} \newline$This option is incorrect because the stoichiometric coefficient for $NH_3$ is 2, so the correct expression should be $\frac{1}{2} \frac{d[NH_3]}{dt}. \newline$Option 3: $-\frac{d[N_2]}{dt} = \frac{1}{2} \frac{d[NH_3]}{dt} \newline$This option is correct because it matches the stoichiometric coefficients in the balanced equation.$\newline$Option 4: $3 \frac{d[H_2]}{dt} = 2 \frac{d[NH_3]}{dt} \newline$This option is incorrect because it does not correctly represent the rate expressions.$\newline$Therefore, the correct relation is given by Option 3.