Problem 91 - Chemistry The Central Science In Si Units

Question

The reaction \(2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)\) is second order in \(\mathrm{NO}\) and first order in \(\mathrm{O}_{2}\). When \((\mathrm{NO}]=0.040 \mathrm{M}\), and \(\left[\mathrm{O}_{2}\right]=0.035 \mathrm{M}\), the observed rate of disappearance of 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 \(1.8 ?\)

Step-by-Step Solution

Answer

(a) Rate of disappearance of \(\text{O}_2\) is \(4.65 \times 10^{-5} \ \text{M/s}\). (b) Rate constant \(k = 1.656 \times 10^{-2} \ \text{M}^{-2} \text{s}^{-1}\). (c) Units of \(k\) are \(\text{M}^{-2} \text{s}^{-1}\). (d) Rate increases by a factor of 3.24.

Step 1: Write the Rate Expression

The rate of the reaction can be expressed using the rate law: \[ \text{Rate} = k [\text{NO}]^2 [\text{O}_2] \]where \(k\) is the rate constant.

Step 2: Use the Reaction Stoichiometry

According to the stoichiometry of the reaction, \[ 2 \text{NO} + \text{O}_2 \rightarrow 2 \text{NO}_2 \]the rate of disappearance of \(\text{O}_2\) is half that of \(\text{NO}\). Therefore, the rate of disappearance of \(\text{O}_2\) is \[ \frac{1}{2} \times 9.3 \times 10^{-5} \ \text{M/s} = 4.65 \times 10^{-5} \ \text{M/s}. \]

Step 3: Calculate the Rate Constant

Substitute the known values into the rate expression: \[ 9.3 \times 10^{-5} = k (0.040)^2 (0.035) \]Solve for \(k\): \[ k = \frac{9.3 \times 10^{-5}}{(0.040)^2 \times (0.035)} \approx 1.656 \times 10^{-2} \ \text{M}^{-2} \text{s}^{-1}. \]

Step 4: Determine the Units of the Rate Constant

The units of \(k\) can be determined as: \[ \text{M/s} \cdot (\text{M})^{-2}\] Thus, the units for \(k\) are \(\text{M}^{-2} \text{s}^{-1}\).

Step 5: Effect of Increasing NO Concentration

According to the rate law, increasing the concentration of \(\text{NO}\) by a factor of 1.8 affects the rate as follows:\[ \text{Rate} \propto [\text{NO}]^2 \] If \([\text{NO}]\) is increased by 1.8, the new rate becomes \(1.8^2 = 3.24\) times the original rate.

Key Concepts

Rate Law

The rate law for a chemical reaction is an equation that links the rate of reaction to the concentration of reactants. For the reaction in our example, it is expressed as:

• Rate = k [NO]^2 [O₂]

The rate law tells us how the concentration of each reactant affects the rate of the reaction. In this case, it reveals that nitric oxide (NO) affects the rate more significantly than oxygen (O₂) because its concentration is squared in the rate equation. The terms in the rate law besides the reaction rate include:

• k: The rate constant, a unique value for each reaction at a given temperature.
• [NO] and [O₂]: The concentrations of the reactants.

Understanding the rate law is critical as it helps explain the speed of the reaction and the influence of reactant concentrations.

Rate Constant

The rate constant, denoted by k, is a fundamental part of the rate law. It quantifies the speed of a reaction at a specific temperature. Each reaction has its own unique rate constant, which means it can be affected by temperature, pressure, and other environmental conditions.
To find the rate constant, we solve for k using known values of rate and concentration, as shown in the formula:

• k = \( \frac{9.3 \times 10^{-5}}{(0.040)^2 \times (0.035)} \)

We calculated that the rate constant is approximately 1.656 \times 10^{-2} \text{ M}^{-2} \text{s}^{-1} for this specific reaction. It is important to always include units with the rate constant, as they can vary based on the reaction order.
In summary, k is a bridge between the concentration of reactants and how fast the reaction proceeds.

Reaction Order

Reaction order describes how the rate is affected by the dimensions of the reactants. In the given reaction:

• The reaction is second order in NO, because the rate law includes [NO]^2.
• It is first order in O₂, as represented by [O₂].

This means that any change in the concentration of NO has a squared effect on the rate (it impacts the rate more significantly), whereas a change in O₂ has a direct effect. If NO is increased by a factor, say 1.8, the rate increases by \((1.8)^2=3.24\), making it over three times faster.
Understanding reaction order is vital, as it indicates which reactants significantly impact the reaction rate, guiding how to control the reaction speed by adjusting concentrations.

Stoichiometry

Stoichiometry in a chemical reaction provides the ratio in which reactants combine or products form. It is derived straight from the balanced chemical equation. For the stated reaction:

• 2 NO + O₂ → 2 NO₂

It tells us that two molecules of NO react with one molecule of O₂ to produce two molecules of NO₂. This stoichiometric ratio helps determine the relationship between the rates of disappearance or formation of substances. In this case, according to the stoichiometric coefficients, the rate of disappearance of O₂ is half that of NO, confirming that the stoichiometry plays a crucial role in understanding and calculating reaction rates.
In essence, stoichiometry acts as the scaffolding that supports understanding the quantitative aspects of chemical reactions.