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Chapter 16: Problem 31

Give the individual reaction orders for all substances and the overall reaction order from this rate law: $$ \text { Rate }=k \frac{\left[\mathrm{HNO}_{2}\right]^{4}}{[\mathrm{NO}]^{2}} $$

Short Answer

Expert verified
The reaction order for \([\text{HNO}_{2}]\) is 4, for \([\text{NO}]\) is -2, and the overall reaction order is 2.

Step by step solution

01

Identify Reaction Order for [HNO2]

Look at the exponent of \([\text{HNO}_{2}]\) in the rate law \(\text { Rate }=k \frac{\big[\text{HNO}_{2}\big]^{4}}{\big[\text{NO}\big]^{2}}\). The exponent is 4, so the reaction order with respect to \([\text{HNO}_{2}]\) is 4.
02

Identify Reaction Order for [NO]

Look at the exponent of \([\text{NO}]\) in the rate law. The exponent in the denominator counts as negative, so it is -2, making the reaction order with respect to \([\text{NO}]\) equal to -2.
03

Calculate Overall Reaction Order

Sum the individual reaction orders for all species involved. For this rate law, the sum is 4 (for \([\text{HNO}_{2}]\)) + (-2) (for \([\text{NO}]\)) = 2. Therefore, the overall reaction order is 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate Law
The rate law is an equation that links the rate of a chemical reaction to the concentration of the reactants. It provides insight into how the speed of a reaction changes when you alter the concentrations of the substances involved. The general form of a rate law can be written as: \( \text{Rate} = k [A]^m [B]^n \).
In this equation:
  • Rate: The speed of the reaction.
  • k: The rate constant.
  • [A] and [B]: The concentrations of reactants A and B.
  • m and n: The reaction orders for A and B.
Understanding the rate law is crucial for comprehending how different factors such as concentration affect the reaction rate.
Individual Reaction Orders
Individual reaction orders indicate the power to which the concentration of each reactant is raised in the rate law. They show how the rate depends on the concentration of specific reactants. In our rate law, \( \text{Rate} = k \frac{[\text{HNO}_2]^4}{[\text{NO}]^2} \), we can determine the individual reaction orders as follows:
The reaction order for \( \text{HNO}_2 \) is 4 because the concentration of \( \text{HNO}_2 \) is raised to the fourth power in the numerator. This means that the rate of reaction increases significantly with an increase in the concentration of \( \text{HNO}_2 \).
Next, the reaction order for \( \text{NO} \) is -2, as the concentration of \( \text{NO} \) is in the denominator and therefore has a negative exponent. This implies that an increase in the concentration of \( \text{NO} \) will decrease the reaction rate. Hence, each reactant's contribution to the overall reaction rate depends on these individual orders.
Overall Reaction Order
The overall reaction order is the sum of the individual reaction orders of all the reactants in the rate law. It gives a broad sense of the reaction's dependence on the concentrations of the reactants.
For our given rate law, \( \text{Rate} = k \frac{[\text{HNO}_2]^4}{[\text{NO}]^2} \):
  • The reaction order for \( \text{HNO}_2 \) is 4.
  • The reaction order for \( \text{NO} \) is -2.
To find the overall reaction order, we simply add these individual orders:
\( 4 + (-2) = 2 \).
Therefore, the overall reaction order of this reaction is 2. This tells us how the rate of the reaction as a whole responds to changes in the concentrations of the reactants.

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Most popular questions from this chapter

The proposed mechanism for a reaction is (1) \(\mathrm{A}(g)+\mathrm{B}(g) \rightleftharpoons \mathrm{X}(g) \quad\) [fast \(]\) (2) \(\mathrm{X}(g)+\mathrm{C}(g) \longrightarrow \mathrm{Y}(g) \quad\) [slow] (3) \(\mathrm{Y}(g) \longrightarrow \mathrm{D}(g) \quad\) [fast (a) What is the overall equation? (b) Identify the intermediate(s), if any. (c) What are the molecularity and the rate law for each step? (d) Is the mechanism consistent with the actual rate law: Rate \(=k[\mathrm{~A}][\mathrm{B}][\mathrm{C}] ?\) (e) Is the following one-step mechanism equally valid? \(\mathrm{A}(g)+\mathrm{B}(g)+\mathrm{C}(g) \longrightarrow \mathrm{D}(g) ?\)

At body temperature \(\left(37^{\circ} \mathrm{C}\right),\) the rate constant of an enzyme-catalyzed decomposition is \(2.3 \times 10^{14}\) times that of the uncatalyzed reaction. If the frequency factor, \(A,\) is the same for both processes, by how much does the enzyme lower the \(E_{\mathrm{a}}\) ?

Archaeologists can determine the age of an artifact made of wood or bone by measuring the amount of the radioactive isotope \({ }^{14} \mathrm{C}\) present in the object. The amount of this isotope decreases in a first-order process. If \(15.5 \%\) of the original amount of \({ }^{14} \mathrm{C}\) is present in a wooden tool at the time of analysis, what is the age of the tool? The half- life of \({ }^{14} \mathrm{C}\) is \(5730 \mathrm{yr}\).

Many drugs decompose in blood by a first-order process. (a) Two tablets of aspirin supply \(0.60 \mathrm{~g}\) of the active compound. After 30 min, this compound reaches a maximum concentration of \(2 \mathrm{mg} / 100 \mathrm{~mL}\) of blood. If the half-life for its breakdown is \(90 \mathrm{~min},\) what is its concentration (in \(\mathrm{mg} / 100 \mathrm{~mL}\) ) \(2.5 \mathrm{~h}\) after it reaches its maximum concentration? (b) For the decomposition of an antibiotic in a person with a normal temperature \(\left(98.6^{\circ} \mathrm{F}\right)\), \(k=3.1 \times 10^{-5} \mathrm{~s}^{-1} ;\) for a person with a fever (temperature of \(\left.101.9^{\circ} \mathrm{F}\right), k=3.9 \times 10^{-5} \mathrm{~s}^{-1}\). If the person with the fever must take another pill when \(\frac{2}{3}\) of the first pill has decomposed, how many hours should she wait to take a second pill? A third pill? (Assume that the pill is effective immediately.) (c) Calculate \(E_{\mathrm{a}}\) for decomposition of the antibiotic in part (b).

Is it possible for more than one mechanism to be consistent with the rate law of a given reaction? Explain.
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