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Chapter 13: Problem 53

OBJECTIVE. Relate half-life and rate constant, and calculate concentration- time behavior from the half-life of a first-order reaction. Calculate the half-life of a first-order reaction if the concentration of the reactant decreases from 0.012 to \(0.0082 \mathrm{M}\) in 66.2 seconds.

Short Answer

Expert verified
The half-life for this first-order reaction is approximately 204.2 seconds.

Step by step solution

01

Identify the first-order integrated rate equation

For a first-order reaction, the integrated rate equation is given by: \[[A] = [A]_0 e^{-kt}\]where \([A]_0\) is the initial concentration, \([A]\) is the concentration at time \(t\), and \(k\) is the rate constant.
02

Use the concentration values to find the rate constant (k)

Rearrange the integrated rate equation to solve for the rate constant \(k\):\[k = -\frac{1}{t} \ln\left(\frac{[A]}{[A]_0}\right)\]Substitute the given values:- Initial concentration \([A]_0 = 0.012 \ M\)- Concentration at time \(t\), \([A] = 0.0082 \ M\)- Time \(t = 66.2 \ seconds\)\[k = -\frac{1}{66.2} \ln\left(\frac{0.0082}{0.012}\right)\]Calculate the value of \(k\).
03

Calculate the half-life using the rate constant

The half-life \(t_{1/2}\) for a first-order reaction is given by:\[t_{1/2} = \frac{0.693}{k}\]Substitute the value of the rate constant \(k\) calculated in the previous step to find the half-life of the reaction.

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Key Concepts

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

First-Order Reaction
In chemistry, a first-order reaction is one where the reaction rate is directly proportional to the concentration of one reactant. This means that as the concentration of this reactant decreases, the rate of the reaction also decreases. A simple example of a first-order reaction is radioactive decay, where exactly half of the material decays over known intervals of time, known as half-life.

First-order reactions have unique characteristics making them easy to analyze.
  • They have a constant half-life that is independent of the reactant concentration.
  • The rate at which the reaction occurs can be described using a straightforward mathematical model.

Understanding first-order reactions helps predict how long a reaction will take under certain conditions.
Integrated Rate Equation
The integrated rate equation for a first-order reaction describes the concentration of a reactant as a function of time. This is given by the formula: \[ [A] = [A]_0 e^{-kt} \] Here,
  • \([A]_0\) is the initial concentration of the reactant.
  • \([A]\) is the concentration at any time \(t\).
  • \(k\) is the rate constant.
This equation helps us to predict how the concentration of a reactant changes over time.

It becomes especially useful when we want to find out how quickly a reaction will proceed or how much of a substance will remain after a given period. By rearranging this formula, we can solve for any of the variables if the others are known.
Rate Constant
The rate constant, \(k\), is a crucial element in the study of reaction kinetics. It quantifies the speed of a reaction by linking it to the concentration of the reactants. For first-order reactions, you can calculate \(k\) using the integrated rate equation: \[ k = -\frac{1}{t} \ln\left(\frac{[A]}{[A]_0}\right) \]
  • By substituting values for the initial and current concentrations \([A]_0\) and \([A]\), as well as the time \(t\) that the reaction has been occurring, we can determine \(k\).
  • A larger \(k\) value indicates a faster reaction.
The rate constant is only truly constant at a given temperature, as it can vary with changes in environmental conditions.

Therefore, understanding \(k\) aids in anticipating how swiftly reactants convert to products.
Concentration-Time Behavior
The concentration-time behavior of a reactant in a first-order reaction describes how its concentration decreases over time. This is elegantly captured by the integrated rate equation, which enables us to model the exponential decay of the reactant. As time progresses, the concentration of the reactant exhibits a decrease characterized by an exponential decay pattern.

The half-life, symbolized as \(t_{1/2}\), provides significant insight into concentration-time behavior:
  • For a first-order reaction, it remains constant and is calculated as \(\frac{0.693}{k}\).
  • It indicates the time required for half of the reactant to be consumed, providing a convenient measure of how quickly a reaction proceeds.

Analyzing concentration-time behavior is important for predicting reaction completion times and optimizing conditions to alter reaction rates when desired.

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Most popular questions from this chapter

A catalyst reduces the activation energy of a reaction from 215 to \(206 \mathrm{~kJ} .\) By what factor would you expect the reaction-rate constant to increase at \(25^{\circ} \mathrm{C}\) ? Assume that the pre-exponential term (A) is the same for both reactions. (Hint: Use the formula \(\ln k=\ln \mathrm{A}-E_{a} / R T\).)

List the factors that affect the rate of reaction.

OBJECTIVE. Use stoichiometry to relate the rate of reaction to changes in the concentrations of reactants and products. The chromium(III) species reacts with hydrogen peroxide in aqueous solution to form the chromate ion. $$ 2 \mathrm{CrO}_{2}^{-}+3 \mathrm{H}_{2} \mathrm{O}_{2}+2 \mathrm{OH}^{-} \rightarrow 2 \mathrm{CrO}_{4}^{2-}+4 \mathrm{H}_{2} \mathrm{O} $$ Under particular experimental conditions, the chromate ion, \(\mathrm{CrO}_{4}^{2-},\) is produced at an instantaneous rate of 0.0050 \(M /\) s. Calculate the instantaneous rates at which the other species change concentration and the instantaneous rate of reaction under these same conditions.

When formic acid is heated, it decomposes to hydrogen and carbon dioxide in a first-order decay. $$ \mathrm{HCOOH}(\mathrm{g}) \rightarrow \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{H}_{2}(\mathrm{~g}) $$ The rate of reaction is monitored by measuring the total pressure in the reaction container. $$ \begin{array}{rc} \text { Time (s) } & P \text { (torr) } \\ \hline 0 & 220 \\ 50 & 324 \\ 100 & 379 \\ 150 & 408 \\ 200 & 423 \\ 250 & 431 \\ 300 & 435 \end{array} $$ Calculate the rate constant and half-life, in seconds, for the reaction. At the start of the reaction (time \(=0)\), only formic acid is present. (Hint: Find the partial pressure of formic acid [use Dalton's law of partial pressure and the reaction stoichiometry to find \(P_{\mathrm{HCOOH}}\) at each time]).

OBJECTIVE. Use stoichiometry to relate the rate of reaction to changes in the concentrations of reactants and products. Consider the combustion of ethane: $$ 2 \mathrm{C}_{2} \mathrm{H}_{6}(\mathrm{~g})+7 \mathrm{O}_{2}(\mathrm{~g}) \rightarrow 4 \mathrm{CO}_{2}(\mathrm{~g})+6 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) $$ If the ethane is burning at the rate of \(0.20 \mathrm{M} / \mathrm{s}\), at what rates are \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) being produced?
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