Question

The rate law for the reaction \(2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{NO}_{2}(\mathrm{g})\) is first order in \(\mathrm{O}_{2}\) and third order overall. What is the rate law for the reaction?

Short answer

The rate law for the given reaction is: Rate = k [NO]^2 [O₂].

Step-by-step solution

Find the order in NO

Since we are given that the reaction is first-order in O₂ and third-order overall, we can determine the order of the reaction with respect to NO. We can express the order of the reaction in terms of NO (NO) and O₂ (O₂): Overall order = Order in NO + Order in O₂ We are told that the overall order is 3, and the order in O₂ is 1, so: 3 = Order in NO + 1 Now, we can find the order in NO: Order in NO = 3 - 1 = 2

Write the rate law

Now that we know the order of reaction with respect to each reactant, we can write the rate law. Recall the general form of rate law: Rate = k [A]^m [B]^n In this case, A = NO, B = O₂, m = 2 (as we found in step 1), and n = 1 (as given in the problem). So, the rate law for the given reaction will be: Rate = k [NO]^2 [O₂]^1 The rate law for the given reaction is: Rate = k [NO]^2 [O₂].

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with the rate at which chemical reactions occur and the factors affecting those rates. It's crucial for scientists and engineers to understand how different conditions like temperature, pressure, and reactant concentrations impact the speed of a reaction. This knowledge plays a vital role in various industries, from pharmaceuticals to environmental science, as it helps in designing and controlling reactions more efficiently.

In our exercise, chemical kinetics helps us determine how the concentration of the reactants influences the rate of the production of nitrogen dioxide (o2). By understanding the relation between reactant concentration and reaction rate, we established the rate law, which is a mathematical equation that models this relationship. The rate law allows us to predict the rate of a reaction under various conditions, which is key to optimizing processes such as chemical synthesis or pollution control. Using chemical kinetics, we can solve for unknown variables in the rate law, giving us deeper insight into the nature of the reaction.

Reaction Order

The reaction order is a term used in chemical kinetics to describe the power to which the concentration of a reactant is raised in the rate law. It indicates how the rate of the reaction is affected by the concentration of that particular reactant. The value is usually determined experimentally and can be zero, a whole number, or even a fraction in some complex reactions.

In our example, we determined the reaction order for o (nitric oxide) to be 2, as calculated in the first step of the solution. This result indicates that the rate of reaction increases by the square of the concentration of o. Similarly, the reaction is first-order for o2 (oxygen), meaning the rate of reaction is directly proportional to the concentration of o2. As the overall reaction order is the sum of the individual orders with respect to each reactant, we added the orders together to find an overall third-order reaction. Understanding the reaction order is necessary for predicting how changes in concentration affect the rate and for the proper scaling of chemical production.

Reaction Rates

Reaction rates signify how quickly the concentrations of reactants decrease or the concentrations of products increase with time. It is usually expressed in terms of mol/L/s, indicating the change in concentration per unit time. Factors such as temperature, catalysts, reactant concentrations, and surface area can affect the rate of a reaction.

For the reaction we've been examining, the rate law describes the dependence of the reaction rate on the concentrations of o and o2. By knowing the rate law, we can calculate the rate of the reaction under different conditions. For instance, if the concentration of one reactant increases, we can predict how much faster the reaction will proceed. Similarly, if a catalyst is introduced, we can analyze its effect on the speed of the reaction. The ability to calculate and control reaction rates is beneficial not only in academic settings when performing experiments but also in industrial applications where time efficiency and product yield are of great economic importance.