Question

The oxidation of \(\mathrm{NO}\) (released in small amounts in the exhaust of automobiles) produces the brownish-red gas \(\mathrm{NO}_{2},\) which is a component of urban air pollution. $$2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)$$ The rate law for the reaction is rate \(=k[\mathrm{NO}]^{2}\left[\mathrm{O}_{2}\right]\) At \(25^{\circ} \mathrm{C}, k=7.1 \times 10^{9} \mathrm{~L}^{2} \mathrm{~mol}^{-2} \mathrm{~s}^{-1}\). What would be the rate of the reaction if \([\mathrm{NO}]=0.0010 \mathrm{~mol} \mathrm{~L}^{-1}\) and \(\left[\mathrm{O}_{2}\right]=0.034 \mathrm{~mol} \mathrm{I}^{-1}\).

Short answer

The rate of the reaction is 2.414 x 10^4 L mol^-1 s^-1.

Step-by-step solution

Write down the rate law

The reaction rate can be calculated using the provided rate law: rate = k[NO]^2[O2]. The rate law indicates that the rate is dependent on the concentrations of NO and O2 to the second and first power respectively.

Insert the given values

Substitute the given concentrations of NO and O2 into the rate law, along with the rate constant k. In this case, k = 7.1 x 10^9 L^2 mol^-2 s^-1, [NO] = 0.0010 mol L^-1, and [O2] = 0.034 mol L^-1.

Calculate the reaction rate

Using the rate law and the given concentrations, calculate the reaction rate: rate = (7.1 x 10^9 L^2 mol^-2 s^-1) * (0.0010 mol L^-1)^2 * (0.034 mol L^-1).

Perform the mathematical operations

First, square the concentration of NO, then multiply the squared value by the concentration of O2 and the rate constant to find the reaction rate.

Key concepts

Understanding the Reaction Rate Law

Chemical reactions occur at various speeds, and understanding the rate at which they occur is crucial in fields like environmental science, engineering, and medicine. The reaction rate law expresses the speed of a chemical reaction as a function of the concentration of its reactants. For a general reaction where reactants A and B form product C, the rate law can be represented as rate = k[A]^x[B]^y, where k is the rate constant, [A] and [B] represent the concentrations of the reactants, and x and y are the reaction order with respect to each reactant. These orders are generally determined experimentally and indicate how changes in the concentration of a reactant affect the reaction rate.

For example, if we double the concentration of A and the rate quadruples, A's reaction order is 2. This tells us the reaction rate is directly proportional to the square of the concentration of A. It's essential to remember that the rate law is determined empirically, and the reaction order is not necessarily related to the stoichiometric coefficients of the balanced equation.

When facing exercises involving reaction rate laws, it's crucial to identify the correct form of the rate law and then plug in the given concentrations and the rate constant to calculate the rate. Patience and attention to exponents are key, as a mistake here can hugely impact the result.

The Role of the Rate Constant

The rate constant, denoted as k, is a proportionality constant that is specific to a given reaction at a certain temperature. It plays a vital role in the reaction rate law because it gives us insight into the reaction's speed under specific conditions. The value of k changes with temperature and can be calculated through experiments or sometimes estimated using theoretical models.

The dimensions of the rate constant will vary depending on the overall order of the reaction. For instance, a second-order reaction rate constant will have units of L mol-1 s-1, while a zero-order reaction rate constant will have units of mol L-1 s-1. Knowing the correct units is necessary for calculating reaction rates accurately.

When solving textbook problems, students should carefully input the value of the rate constant along with the correct units to avoid calculation errors. It's also good practice to check if the calculated reaction rate matches the expected units from the rate law expression, which can serve as a quick verification of your calculation.

Concentration Dependence in Reactions

The rate of a chemical reaction typically depends on the concentration of its reactants; this relationship is captured in the concentration dependence term of a reaction rate law. Concentration dependence indicates how sensitive the reaction rate is to changes in the concentration of the reactants. It can tell us whether a small change in concentration will have a large or negligible effect on the speed of the reaction.

• For zero-order reactions, the rate is independent of the concentration of reactants.
• For a first-order reaction, the rate is directly proportional to the concentration of one reactant.
• Higher-order reactions show more complex relationships, where the reaction rate may depend on the product of concentrations raised to varying powers.

When performing calculations for concentration dependence, it is important to use the correct exponent for each reactant as defined by the reaction order. An incorrect exponent not only leads to the wrong rate but also misunderstands the nature of the reaction mechanism. By carefully observing these details, students can accurately predict how varying concentrations will affect reaction rates and approach equilibrium adjustments with greater confidence.