Consider the reaction: 2 O 3 (g) → 3 O 2 ( g) The rate law for this reaction is: Rate = k [O 3 ] 2 [O 2 ] Suppose that a 1.0-L reaction vessel initially contains 1.0 mol of O 3 and 1.0 mol of O 2 . What fraction of the O 3 will have reacted when the rate falls to one-half of its initial value?

Hi everyone here we have a question telling us that the rate law for the reaction A to B is ready equals K times concentration of a divided by concentration of B squared. An initial setup involves a 1.0 liter flask with a 0. mol of A and 0.5 mol of B determine the fraction of a that has reacted when the rate of the reaction is a third of the initial rate. So the concentration of A Equals 0.5 Molar. Because my polarity equals moles over leader and we have 0.5 moles per one liter. And the concentration of B Is also 0.5 malware. R right one equals k Times 0.5, Divided by 0.5 Squared Equals three. Rate too right to equals k Times 0. minus x divided by 0.5 Plus two x squared To get the ratio of right to and rate one. We are going to take rate too equals k Times 0.5 -X Divided by 0.5 Plus two x squared three rate too equals k Times 0. Divided by 0.5 squared one third Equals 0.5 More minus x Times 0. Squared Divided by 0. Plus two x squared 0.5 Plus two x Squared equals three times 0.5 - Times 0.5 Squared four X squared Plus 2.75 x -0.125 equals zero x equals zero point 0428. The fraction of a reacted Is 0.042, eight marbles, Divided by 0.5 moles Equals 0.086. And that is our final answer. Thank you for watching. Bye.

Ch.14 - Chemical Kinetics

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Tro 4th Edition

Ch.14 - Chemical Kinetics

Problem 86

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Chapter 14, Problem 86

Consider the reaction: 2 O₃(g) → 3 O₂( g) The rate law for this reaction is: Rate = k [O₃]² [O₂] Suppose that a 1.0-L reaction vessel initially contains 1.0 mol of O₃ and 1.0 mol of O₂. What fraction of the O₃ will have reacted when the rate falls to one-half of its initial value?

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Identify the initial rate of the reaction using the rate law: \( \text{Rate}_{\text{initial}} = k [\text{O}_3]^2 [\text{O}_2] \).

Determine the expression for the rate when it falls to half of its initial value: \( \text{Rate}_{\text{half}} = \frac{1}{2} \times \text{Rate}_{\text{initial}} \).

Set up the equation for the rate at half its initial value: \( \frac{1}{2} \times k [\text{O}_3]^2 [\text{O}_2] = k [\text{O}_3']^2 [\text{O}_2'] \), where \([\text{O}_3']\) and \([\text{O}_2']\) are the concentrations when the rate is halved.

Assume \( x \) moles of \( \text{O}_3 \) have reacted, then \([\text{O}_3'] = 1.0 - x\) and \([\text{O}_2'] = 1.0 + \frac{3}{2}x\) because for every 2 moles of \( \text{O}_3 \) that react, 3 moles of \( \text{O}_2 \) are produced.

Substitute \([\text{O}_3']\) and \([\text{O}_2']\) into the rate equation and solve for \( x \) to find the fraction of \( \text{O}_3 \) that has reacted.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rate Law

The rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. It is typically formulated as Rate = k [A]^m [B]^n, where k is the rate constant, and m and n are the orders of the reaction with respect to reactants A and B. Understanding the rate law is crucial for predicting how changes in concentration affect the reaction rate.

Reaction Order

The reaction order is the sum of the exponents in the rate law and indicates how the rate of reaction depends on the concentration of reactants. In the given reaction, the order with respect to O<sub>3</sub> is 2, meaning that the rate is proportional to the square of its concentration. This concept helps in determining how the concentration of reactants influences the speed of the reaction.

Half-life of a Reaction

The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value. For reactions of different orders, the half-life can vary significantly. In this context, understanding how the rate changes as the concentration of O<sub>3</sub> decreases is essential for calculating the fraction that has reacted when the rate falls to half its initial value.

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First-Order Half-Life

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The tabulated data were collected for this reaction at a certain temperature: X₂Y → 2 X + Y a. Determine the order of the reaction and the value of the rate constant at this temperature.

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The tabulated data were collected for this reaction at a certain temperature: X₂Y → 2 X + Y c. What is the concentration of X after 10.0 hours?

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Open Question

Consider the reaction: A + B + C → D. The rate law for this reaction is: Rate = k [A][C]^2 [B]^1/2. Suppose the rate of the reaction at certain initial concentrations of A, B, and C is 0.0115 M/s. What is the rate of the reaction if the concentrations of A and C are doubled and the concentration of B is tripled?

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At 700 K, acetaldehyde decomposes in the gas phase to methane and carbon monoxide. The reaction is: CH3CHO(g) → CH4(g) + CO(g). A sample of CH3CHO is heated to 700 K and the pressure is measured as 0.22 atm before any reaction takes place. The kinetics of the reaction are followed by measurements of total pressure and these data are obtained: t (s) 0 1000 3000 7000; PTotal (atm) 0.22 0.24 0.27 0.31. Find the rate law, the rate constant, and the total pressure after 2.00 × 10^4 s.

Open Question

At 400 K, oxalic acid decomposes according to the reaction: H2C2O4(g) → CO2(g) + HCOOH(g). In three separate experiments, the initial pressure of oxalic acid and the final total pressure after 20,000 seconds are measured. Experiment: 1) PH2C2O4 at t = 0: 65.8, PTotal at t = 20,000 s: 94.6; 2) PH2C2O4 at t = 0: 92.1, PTotal at t = 20,000 s: 132; 3) PH2C2O4 at t = 0: 111, PTotal at t = 20,000 s: 160. Find the rate law of the reaction and its rate constant.

Textbook Question

Dinitrogen pentoxide decomposes in the gas phase to form nitrogen dioxide and oxygen gas. The reaction is first order in dinitrogen pentoxide and has a half-life of 2.81 h at 25 °C. If a 1.5-L reaction vessel initially contains 745 torr of N₂O₅ at 25 °C, what partial pressure of O₂ is present in the vessel after 215 minutes?

Consider the reaction: 2 O3(g) → 3 O2( g) The rate law for this reaction is: Rate = k [O3]2 [O2] Suppose that a 1.0-L reaction vessel initially contains 1.0 mol of O3 and 1.0 mol of O2. What fraction of the O3 will have reacted when the rate falls to one-half of its initial value?

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Key Concepts

Rate Law

Reaction Order

Half-life of a Reaction

Consider the reaction: 2 O₃(g) → 3 O₂( g) The rate law for this reaction is: Rate = k [O₃]² [O₂] Suppose that a 1.0-L reaction vessel initially contains 1.0 mol of O₃ and 1.0 mol of O₂. What fraction of the O₃ will have reacted when the rate falls to one-half of its initial value?