Question

Hydrogen reacts with nitrogen monoxide to form dinitrogen monoxide (laughing gas) according to the equation:\({{\bf{H}}_{\bf{2}}}{\bf{(g) + 2NO(g)}} \to {{\bf{N}}_{\bf{2}}}{\bf{O(g) + }}{{\bf{H}}_{\bf{2}}}{\bf{O}}\).Determine the rate law, the rate constant, and the orders with respect to each reactant from the following data:

Short answer

The rate law represented as\({\bf{rate = k(NO}}{{\bf{)}}^{\bf{2}}}{{\bf{(}}{{\bf{H}}_{\bf{2}}}{\bf{)}}^{\bf{1}}}\)

Reaction order with respect to NO is 2 and with respect to Cl is 1

Value of rate constant \({\bf{9}}{\bf{.0 \times 1}}{{\bf{0}}^{{\bf{ - 2}}}}{{\bf{L}}^{\bf{2}}}{\bf{Mo}}{{\bf{l}}^{{\bf{ - 2}}}}{\bf{s}}\)

Step-by-step solution

Rate law

The rate law for a chemical reaction is an expression that provides a relationship between the rate of the reaction and the concentration of the reactants participating in it.

General rate law for the given reaction is represented as

\({\bf{rate = (NO}}{{\bf{)}}^{\bf{m}}}{{\bf{(}}{{\bf{H}}_{\bf{2}}}{\bf{)}}^{\bf{n}}}\)

From the given table we can make the following equations

\(\begin{align}2.835 \times {10^{ - 3}}mol{L^{ - 1}}{s^{ - 1}} &= k{(0.30)^m}{(0.60)^n}\,\,\,\,\,\,......(1)\left( {0,1} \right)\\1.134 \times {10^{ - 2}}mol{L^{ - 1}}{s^{ - 1}} &= k{(0.35)^m}{(0.35)^n}\,\,\,\,\,\,......(2)\\2.268 \times {10^{ - 2}}mol{L^{ - 1}}{s^{ - 1}} &= k{(0.60)^m}{(0.70)^n}\,\,\,\,\,\,......(3)\end{align}\)

Order of each reactant

The order of reaction refers to the power dependence of the rate on the concentration of each reactant.

On dividing equation (2) by (1), we get

\(\frac{{Rate(R{}_2)}}{{Rate(R{}_1)}} = \frac{{0.01134}}{{0.002835}} = \frac{{k{{(0.60)}^m}{{(0.35)}^n}}}{{k{{(0.30)}^m}{{(0.35)}^n}}}\)

\(\begin{align}{(2)^m} = 4\\m = 2\end{align}\)

On dividing equation (3) by (2), we get

\(\frac{{Rate(R{}_3)}}{{Rate(R{}_2)}} = \frac{{0.02268}}{{0.01134}} = \frac{{k{{(0.60)}^m}{{(0.70)}^n}}}{{k{{(0.60)}^m}{{(0.35)}^n}}}\)

\(\begin{align}{(2)^n} = 2\\n = 1\end{align}\)

Hence the rate of given reaction is

\({\bf{rate = k(NO}}{{\bf{)}}^{\bf{2}}}{{\bf{(}}{{\bf{H}}_{\bf{2}}}{\bf{)}}^{\bf{1}}}\)

The overall order of reaction \({\bf{ = m + n = 2 + 1 = 3}}\)

Rate constant

The rate constant is the proportionality constant in the equation that expresses the relationship between the rate of a chemical reaction and concentration of reacting substances.

\({\bf{Rate = k}}\left( {{\bf{NO}}} \right){{\bf{\;}}^{\bf{2}}}{\bf{(}}{{\bf{H}}_{\bf{2}}}{\bf{)}}\)

\(\begin{align}2.835 \times {10^{ - 3}}mol{L^{ - 1}}{s^{ - 1}} &= K{(0.30M)^2}(0.35M)\\k &= \frac{{2.835 \times {{10}^{ - 3}}mol{L^{ - 1}}{s^{ - 1}}}}{{{{(0.30M)}^2}(0.35M)}}\\ &= 9.0 \times {10^{ - 2}}{L^2}Mo{l^{ - 2}}{s^{ - 1}}\end{align}\)