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Chapter 13: Problem 13

Analytes \(A\) and \(B\) react with a common reagent \(R\) with first-order kinetics. If \(99.9 \%\) of \(A\) must react before \(0.1 \%\) of \(B\) has reacted, what is the minimum acceptable ratio for their respective rate constants?

Short Answer

Expert verified
The minimum acceptable ratio for the rate constants is approximately 6908.

Step by step solution

01

Understanding the Problem

We are given that analytes \(A\) and \(B\) react with a common reagent \(R\) with first-order kinetics. We need to find the minimum acceptable ratio of their rate constants such that \(99.9\%\) of \(A\) has reacted before \(0.1\%\) of \(B\) has reacted. First-order reactions follow the equation \( [A] = [A]_0 e^{-kt} \).
02

First-Order Kinetics Equations

For a first-order reaction, the remaining concentration is given by \([A] = [A]_0 e^{-k_A t}\) and \([B] = [B]_0 e^{-k_B t}\). The fraction of initial concentration remaining is expressed as \(e^{-k_A t}\) and \(e^{-k_B t}\) respectively.
03

Setting Fractional Remaining Amounts

For \(A\), after \(99.9\%\) reaction, \(0.1\%\) remains: \(e^{-k_A t} = 0.001\). For \(B\), after \(0.1\%\) reaction, \(99.9\%\) remains: \(e^{-k_B t} = 0.999\).
04

Solving for Time in Terms of Rate Constants

Rearrange the equations to express \(t\) in terms of rate constants. For \(A\), \(t = \frac{- ext{ln}(0.001)}{k_A}\), and for \(B\), \(t = \frac{- ext{ln}(0.999)}{k_B}\). For \(99.9\%\) of \(A\) to react before \(0.1\%\) of \(B\) has reacted, \(t_A < t_B\).
05

Derive the Ratio of Rate Constants

Substitute \(t_A\) and \(t_B\) from previous equations: \(\frac{- ext{ln}(0.001)}{k_A} < \frac{- ext{ln}(0.999)}{k_B}\). This implies \(k_A \cdot \text{ln}(0.999) < k_B \cdot \text{ln}(0.001)\).
06

Calculate the Minimum Acceptable Ratio

Calculate the natural logarithms: \(\text{ln}(0.001) = -6.907755\) and \(\text{ln}(0.999) = -0.0010005\). Thus, \(k_A \cdot (-0.0010005) < k_B \cdot (-6.907755)\). The minimum acceptable ratio \(\frac{k_A}{k_B}\) is given by \(\frac{6.907755}{0.0010005}\).
07

Final Ratio Calculation

Calculate the ratio: \(\frac{6.907755}{0.0010005} \approx 6907.755\). The minimum acceptable ratio for the rate constants is approximately 6908.

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Key Concepts

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

Rate Constants
In chemical kinetics, rate constants play a vital role in understanding reaction speeds. A first-order reaction is governed by the equation \[ [A] = [A]_0 e^{-kt} \]where
  • \([A]\) is the concentration of reactant at time \(t\)
  • \([A]_0\) is the initial concentration
  • \(k\) is the rate constant
The rate constant \(k\) determines how quickly a reactant is consumed in a reaction.

In our given problem, analytes \(A\) and \(B\) react under first-order kinetics with a common reagent. Here, the task is to find the minimum ratio of their respective rate constants. This ratio ensures that a much larger fraction of \(A\) reacts compared to \(B\), under given conditions. The rate constant is a distinct factor for each reactant, influencing the speed at which the reactant is consumed.

Being able to calculate and comprehend the implications of rate constants can be incredibly useful, especially when trying to design processes where timing of reactions is crucial.
Analytes
Analytes are the substances whose chemical reactions we are interested in. In the context of this exercise, analytes \(A\) and \(B\) both react with a common reagent \(R\).

The focus here is to understand how much of each analyte reacts under certain conditions.
  • For \(A\), the goal is that \(99.9\%\) needs to be consumed.
  • For \(B\), only \(0.1\%\) should be consumed.
This constraint sets a purpose for calculating the rate constants. By reacting at different extents, these analytes illustrate how specific we can be with reaction control, emphasizing the importance of rate constant ratios.

Analytes are central to reaction kinetics, and understanding their behavior underpins everything from quality control in industrial processes to intricate biochemical pathways in nature.
Reaction Rates
The rate of a reaction tells us how fast a reactant is being converted to a product. In first-order reactions, the reaction rate depends on the concentration of one reactant. Specifically, the rate itself can be expressed as \[ ext{Rate} = -\frac{d[A]}{dt} = k[A] \]Here,
  • \(-\frac{d[A]}{dt}\) is the rate of change of analyte \([A]\)
  • \(k\) is the rate constant
This principle guides how we approach the exercise. We need a specific reaction rate for \(A\) such that it almost completely reacts before \(B\) begins to significantly react.

By manipulating the equations for first-order reactions, we investigate the proportionate responses of different analyte concentrations to their rate constants. This investigation gives rise to an understanding of how reaction conditions could be tailored for desired outcomes.

Grasping the concept of reaction rates in relation to rate constants and analyte properties is crucial for predictive control in chemical processes.

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Most popular questions from this chapter

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