Nitrogen monoxide reacts with chlorine according to the equation: \(2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{NOCl}(g)\) The following initial rates of reaction have been observed for certain reactant concentrations: $$\begin{array}{|c|c|c|}\hline \text { \([\mathrm{NO}]\left(\operatorname{mol} / L^{1}\right)\) } & \text { \(\left[\mathrm{Cl}_{2}\right](\mathrm{mol} / \mathrm{L})\) } & \text { Rate \(\left(\mathrm{mol} \mathrm{L}^{-1} \mathrm{h}^{-1}\right)\) } \\\\\hline 0.50 & 0.50 & 1.14 \\\\\hline 1.00 & 0.50 & 4.56 \\ \hline 1.00 & 1.00 & 9.12 \\\\\hline\end{array}$$ What is the rate law that describes the rate's dependence on the concentrations of \(\mathrm{NO}\) and \(\mathrm{Cl}_{2}\) ? What is the rate constant? What are the orders with respect to each reactant?

The rate law is a mathematical expression that links the rate of a chemical reaction to the concentration of its reactants. It has the general form \[\text{Rate} = k[\text{A}]^{m}[\text{B}]^{n}...\] where \( k \) is the rate constant, \( [\text{A}], [\text{B}], ... \) represents the concentration of reactants, and \( m \) and \( n \) are the reaction orders for each reactant respectively.

In the exercise provided, the rate law assists in understanding how the rate of production of \( \text{NOCl} \) is affected by the concentration of nitrogen monoxide (\( \text{NO} \)) and chlorine (\( \text{Cl}_2 \)). By examining the changes in concentration and observing the resultant changes in the reaction rate, one can deduce the powers to which each reactant concentration is raised in the rate law—this indicates the order of the reaction with respect to each reactant.

Understanding the rate law helps predict how a rate will change with varying reactant concentrations—an essential concept for controlling chemical reactions in industrial applications, and studying reaction mechanisms in research.

Reaction order is a key term in chemical kinetics that indicates the dependency of the rate of reaction on the concentration of each reactant. It is often determined experimentally and is denoted by an exponent in the rate law equation.

The reaction order can be zero, first, second, or even fractional based on how the rate changes with concentration. A zero-order reaction means the rate is independent of the concentration of that reactant, while a first-order reaction shows that the rate is directly proportional to its concentration. A second-order dependency suggests the rate is proportional to the square of that reactant's concentration.

• For \( \text{NO} \), if the rate increases by a factor of \(k\) when the concentration of \( \text{NO} \) alone is increased by a factor of \(n\), then \( \text{NO} \) has a reaction order of \(n\) in the reaction.
• Similarly, for \( \text{Cl}_2 \), the reaction order is deduced by observing how the rate changes when only its concentration is varied.

By carefully studying the provided data and comparing rates with varying reactant concentrations, the reaction order with respect to each reactant is identified, which in turn assists in the determination of the rate law of the reaction.

The rate constant, denoted as \( k \), is the proportionality factor in the rate law equation. Its value provides insight into the intrinsic speed of a reaction under given conditions and remains constant at a fixed temperature.

The units of the rate constant vary depending on the overall order of the reaction and can help you check if your rate law is consistent with the units of rate (usually \( \text{mol} \cdot \text{L}^{-1} \cdot \text{time}^{-1} \)). For instance, if the overall reaction order is one, the unit for \( k \) might be \( s^{-1} \), while for a second-order reaction, it might be \( M^{-1}s^{-1} \) or \( L\cdot mol^{-1}\cdot s^{-1} \).

In the exercise, once the reaction orders are known, any set of the provided concentration-rate data pairs can be used to calculate the rate constant by rearranging the rate law. The computed \( k \) allows for the prediction of the rate for any given concentrations of the reactants, making it a fundamental aspect of the rate law and indispensable for chemists in various practical applications.