   # The decomposition of dinitrogen pentaoxide N 2 O 5 (g) → 2 NO 2 (g) + ½ O 2 (g) has the following rate equation: Rate = k [N 2 O 5 ]. It has been found experimentally that the decomposition is 20.5% complete in 13.0 hours at 298 K. Calculate the rate constant and the half-life at 298 K. ### Chemistry & Chemical Reactivity

9th Edition
John C. Kotz + 3 others
Publisher: Cengage Learning
ISBN: 9781133949640

#### Solutions

Chapter
Section ### Chemistry & Chemical Reactivity

9th Edition
John C. Kotz + 3 others
Publisher: Cengage Learning
ISBN: 9781133949640
Chapter 14, Problem 61GQ
Textbook Problem
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## The decomposition of dinitrogen pentaoxideN2O5(g) → 2 NO2(g) + ½ O2(g)has the following rate equation: Rate = k[N2O5]. It has been found experimentally that the decomposition is 20.5% complete in 13.0 hours at 298 K. Calculate the rate constant and the half-life at 298 K.

Interpretation Introduction

Interpretation:

For the given reaction and the rate constant and the half-life at 298K has to be calculated.

Concept introduction:

Arrhenius equation:

• Arrhenius equation is a formula that represents the temperature dependence of reaction rates
• The Arrhenius equation has to be represented as follows

k=AeEa/RTlnk=lnAeEa/RTlnk=(EaR)(1T)+lnA

• Ea represents the activation energy and it’s unit is kJ/mol
• R represents the universal gas constant and it has the value of 8.314 J/K.mol
• T represents the absolute temperature
• A represents the frequency factor or collision frequency
• e represents the base of natural logarithm
•  Arrhenius equation equation was proposed by Svante Arrhenius in 1889.

In order to establish the plausibility of a mechanism, one must compare the rate law of the rate determining step to the experimentally determined rate law.

Rate law: It is generally the rate equation that consists of the reaction rate with the concentration or the pressures of the reactants and constant parameters.

Rate constant: The rate constant for a chemical reaction is the proportionality term in the chemical reaction rate law which gives the relationship between the rate and the concentration of the reactant present in the chemical reaction.

Rate order: The order of each reactant in a reaction is represented by the exponential term of the respective reactant present in the rate law and the overall order of the reaction is the sum of all the exponents of all reactants present in the chemical reaction.  The order of the reaction is directly proportional to the concentration of the reactants.

Half-life: It is defined as the time required reducing the concentration of the reactant to one of its initial value.

### Explanation of Solution

Examining the given rate law of the reaction clearly says that the given reaction is a first order reaction hence the expressions for first order reaction should be used.

In order to find the rate constant at given temperature the following first order rate reaction should be used.

k=1tln[A]0[A]

With known concentration of the reactant at given time the rate constant is determined as follows,

k=1tln[A]0[A]=113ln[A]00

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