Question

The reaction \(2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)\) is second order in NO and first order in \(\mathrm{O}_{2}\). When [NO] \(=0.040 \mathrm{M}\) and \(\left[\mathrm{O}_{2}\right]=0.035 \mathrm{M},\) the observed rate of disappearance of \(\mathrm{NO}\) is \(9.3 \times 10^{-5} \mathrm{M} / \mathrm{s}\). (a) What is the rate of disappearance of \(\mathrm{O}_{2}\) at this moment? (b) What is the value of the rate constant? (c) What are the units of the rate constant? (d) What would happen to the rate if the concentration of NO were increased by a factor of \(1.8 ?\)

Short answer

(a) The rate of disappearance of O2 at this moment is: \[Rate_{O_2} = \frac{1}{2} \times 9.3 \times 10^{-5} M/s\] (b) The value of the rate constant (k) is: \[k = \frac{9.3 \times 10^{-5}}{(0.040)^2(0.035)}\] (c) The units of the rate constant are: \[\text{Units of k} = \frac{1}{M^2s}\] (d) The factor by which the reaction rate increases when the concentration of NO is increased by a factor of 1.8 is: \[\text{Rate Increase Factor} = (1.8)^2\]

Step-by-step solution

a) Calculate the rate of disappearance of O2

To find the rate of disappearance of O2, we need to use the stoichiometry of the reaction. From the balanced equation, we see that 2 moles of NO reacts with 1 mole of O2. Given the rate of disappearance of NO, we can calculate the rate of disappearance of O2 by using the ratio of their stoichiometric coefficients: Rate of disappearance of O2 = (1/2) * Rate of disappearance of NO Thus, we can calculate the rate of disappearance of O2 as follows: \[Rate_{O_2} = \frac{1}{2} \times 9.3 \times 10^{-5} M/s\]

b) Calculate the rate constant (k)

The rate law for this reaction can be written as: \(Rate = k[\mathrm{NO}]^2[\mathrm{O}_2]\) Given the concentrations of NO and O2, and the observed rate of disappearance of NO, we can solve for the rate constant (k) using the rate law equation: \(9.3 \times 10^{-5} = k(0.040)^2(0.035)\) Now, we can solve for k: \[k = \frac{9.3 \times 10^{-5}}{(0.040)^2(0.035)}\]

c) Determine the units of the rate constant

The units of the rate constant can be determined from the rate law equation. For this second-order reaction with respect to NO and first-order with respect to O2, we have: \(Rate = k[\mathrm{NO}]^2[\mathrm{O}_2]\) The units of the rate constant can be found by taking the units of Rate and diving them by the units of the concentration terms: \[\text{Units of k} = \frac{\text{Units of Rate}}{(\text{Units of [NO]})^2(\text{Units of [O2]})}\] Since the units of Rate are M/s and the units of concentrations are M (molarity), the units of the rate constant are: \[\text{Units of k} = \frac{M/s}{(M)^2(M)} = \frac{1}{M^2s}\]

d) Predict the change in reaction rate when NO concentration is increased by a factor of 1.8

According to the rate law equation for this reaction, the rate of the reaction depends on the square of the concentration of NO. To find the new reaction rate when the concentration of NO is increased by a factor of 1.8, we can use the equation: \(New\ Rate = k(\text{New [NO]})^2[\mathrm{O}_2]\) New [NO] = 1.8 * Initial [NO] New [NO] = 1.8 * 0.040 M So the new rate equation becomes: \(New\ Rate = k(1.8 \times 0.040)^2(0.035)\) To determine the factor by which the rate has increased, divide the new rate by the initial rate: \[\text{Rate Increase Factor} = \frac{(1.8 \times 0.040)^2}{(0.040)^2} = (1.8)^2\

Key concepts

Rate Law

In chemical kinetics, the rate law defines the relationship between the concentration of reactants and the reaction rate. In essence, it shows how the rate depends on the molarity of the substances involved in the reaction. For a generalized reaction, the rate law can be expressed as:

Rate = k[A]^m[B]^n...

where k is the rate constant, [A] and [B] represent the concentrations of the reactants, and m and n are the reaction orders with respect to each reactant. These orders must be determined empirically—they cannot be inferred from the stoichiometry of the equation.

Practical Application of the Rate Law

To use the rate law, we measure the rate of the reaction under specific conditions, and then use algebra to solve for the rate constant or the concentrations if they are unknown. The rate law is indispensable for predicting how changes in concentration affect the speed of the reaction.

Reaction Rate

Reaction rate refers to the speed at which reactants are converted into products in a chemical reaction. It's typically expressed as the change in concentration of a reactant or product per unit time, such as moles per liter per second (M/s). For a simple reaction where A → B , the average rate of consumption of A over a time interval Δt is:

Rate = -Δ[A]/Δt

Here, the negative sign indicates that the concentration of A decreases over time. Reaction rate can be affected by factors like temperature, concentration of reactants, surface area, and the presence of a catalyst.

The Relation Between Rate and Stoichiometry

In a balanced chemical equation, stoichiometry provides the ratio of reactants consumed and products formed. It is necessary to understand these proportions to correctly interpret the reaction rates for different substances in the reaction.

Rate Constant

The rate constant, often denoted as k, is a proportionality factor in the rate law of a chemical reaction. It's a direct indicator of the reactivity of the reaction—it varies with temperature and the presence of a catalyst but is independent of reactant concentration. A large rate constant indicates a faster reaction, assuming the concentrations remain constant. Units of k depend on the overall order of the reaction and can range from s^-1, M^-1s^-1, to M^-2s^-1, and so on. Calculating the rate constant involves substituting the measured concentration values and the reaction rate into the rate law.

Dissecting the Units of the Rate Constant

For a reaction where the sum of the orders with respect to each reactant is n, the units for the rate constant k would be:

Units of k = M^(1-n)s^-1

When we do this, we're ensuring that the rate (with units of M/s) is consistent, regardless of the reaction order.

Stoichiometry

Stoichiometry comes from the Greek words for 'element' and 'measure' and is a section of chemistry that pertains to the calculation of the quantities of reactants and products involved in chemical reactions. In essence, stoichiometry is the recipe for a chemical reaction, as it uses the balanced chemical equation to understand the ratio between different molecules involved. Through stoichiometry, scientists can predict the amounts of substances consumed or created in a given reaction.

Stoichiometry and Reaction Rates

Stoichiometry also plays a critical role in determining reaction rates. In a balanced chemical equation, the coefficients indicate the relative rates at which reactants are used and products are formed. So if a reaction states that 2 moles of A react with 1 mole of B to produce 2 moles of C, we can infer that the rate of disappearance of A is twice that of B. This stoichiometric ratio helps us to calculate individual reactant usage and product formation rates from the overall reaction rate.