Hydrogen reacts with nitrogen monoxide to form dinitrogen monoxide (laughing gas) according to the equation: \(\mathrm{H}_{2}(g)+2 \mathrm{NO}(g) \longrightarrow \mathrm{N}_{2} \mathrm{O}(g)+\mathrm{H}_{2} \mathrm{O}(g)\) Determine the rate law, the rate constant, and the orders with respect to each reactant from the following data: $$\begin{array}{|c|c|c|c|} \hline \text { \([\mathrm{NO}](M)\) }& 0.30 & 0.60 & 0.60 \\ \hline \text { \(\left[\mathrm{H}_{2}\right](M)\) } & 0.35 & 0.35 & 0.70 \\\ \hline \text { Rate \(\left(\mathrm{mol} \mathrm{L}^{-1} \mathrm{s}^{-1}\right)\) } & 2.835 \times 10^{-3} & 1.134 \times 10^{-2} & 2.268 \times 10^{-2} \\ \hline \end{array}$$

Understanding the rate law is critical when studying chemical kinetics, as it quantifies the relationship between the concentration of reactants and the rate of a chemical reaction. The rate law is an equation that expresses how the rate depends on the concentration of each reactant. In a given reaction, if we have two reactants, say A and B, the rate law might look like this: Rate = k[A]^{x}[B]^{y}, where:

• k is the rate constant, a coefficient that incorporates various factors such as temperature and is unique to each reaction.
• [A] and [B] represent the molar concentrations of the reactants.
• x and y are the orders of the reaction with respect to A and B, respectively. These are typically determined experimentally and tell us how the rate changes when the concentration of a reactant is altered.

For a student tackling a problem involving chemical kinetics, identifying each part of the rate law is crucial. By correctly determining the values of the rate constant and the reaction orders, students can accurately calculate reaction rates under various conditions.

Reaction order is essentially a measure of how the rate of a reaction will change with the concentration of one of its reactants. It is determined experimentally and is usually a whole number, although it can also be zero or a fraction. The total order of a reaction is the sum of the orders with respect to each reactant.

Examples of Reaction Orders

• If the rate does not change when the concentration of a reactant is altered, the reaction is zero-order with respect to that reactant.
• A reaction is first-order with respect to a reactant if its rate changes directly with its concentration—doubling the concentration of the reactant doubles the rate.
• For a second-order reaction with respect to a reactant, the rate goes up by a factor of four if the reactant's concentration is doubled.

In the context of our exercise, students can see from the data provided that the reaction order with respect to NO is 2, since the rate quadruples when the concentration of NO doubles. Similarly, the reaction is first-order with respect to H2, as the rate doubles when its concentration is doubled. This knowledge is integral when writing the rate law for a given reaction.

The rate constant, denoted as 'k' in the rate law, is a proportionality constant that incorporates the effects of temperature and other conditions on the reaction rate. It has various units depending on the overall order of the reaction. For a first-order reaction, it's typically in s^{-1}, and for a second-order reaction, in L mol^{-1} s^{-1}.

Significance of the Rate Constant

• The rate constant provides a link between the concentration of reactants and the reaction rate.
• A large rate constant indicates a faster reaction at a given concentration of reactants, while a smaller rate constant indicates a slower reaction.
• The value of the rate constant changes with temperature according to the Arrhenius equation.

Using the data from our hydrogen and nitrogen monoxide reaction exercise, the rate constant can be calculated by rearranging the rate law and solving for 'k'. This calculation is critical because the value of 'k' can then be used to predict reaction rates at various concentrations, helping students understand how different variables can affect the speed of a reaction. By mastering the concept of the rate constant, students can deepen their understanding of reaction dynamics and predict how changes in conditions impact the reaction rate.