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Mobile App Chapter 9: Problem 44
Show that time required to complete \(99.9 \%\) completion of a first order reaction is \(1.5\) times to \(90 \%\) completion.
Short Answer
Expert verified
For a first-order reaction, \(t_{99.9} = 1.5 \cdot t_{90}\), proving that the time to achieve 99.9% completion is 1.5 times that required for 90% completion.
Step by step solution
01
Understand the relationship between completion percentage and reaction rate
For a first-order reaction, the time required to reach a certain percentage of completion can be expressed using the formula \( -\ln(1 - \text{fraction completed}) = k \cdot t \), where \(k\) is the rate constant and \(t\) the time required for completion.
02
Set up the equation for 90% completion
First, express 90% as a fraction, which is 0.9, and insert it into the equation. The expression becomes \( -\ln(1 - 0.9) = k \cdot t_{90} \), where \(t_{90}\) is the time required to reach 90% completion.
03
Simplify the equation for 90% completion
Simplify the equation by calculating the natural logarithm: \( -\ln(0.1) = k \cdot t_{90} \).
04
Set up the equation for 99.9% completion
Now express 99.9% as a fraction which is 0.999, and insert it into the original equation resulting in \( -\ln(1 - 0.999) = k \cdot t_{99.9} \), where \(t_{99.9}\) is the time required to reach 99.9% completion.
05
Simplify the equation for 99.9% completion
Similar to step 3, simplify by calculating the natural logarithm: \( -\ln(0.001) = k \cdot t_{99.9} \).
06
Compare the two time expressions
To show that \(t_{99.9}\) is 1.5 times \(t_{90}\), divide the expression for 99.9% completion by the one for 90% completion: \( \frac{-\ln(0.001)}{-\ln(0.1)} = \frac{k \cdot t_{99.9}}{k \cdot t_{90}} \) and simplify.
07
Calculate the ratio
Calculate the ratio of the logarithms, which turns out to be \( \frac{\ln(0.1)}{\ln(0.001)} = \frac{2}{3} \). Therefore, \(t_{99.9} = 1.5 \cdot t_{90}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chemical Kinetics
Chemical kinetics is the area of chemistry that studies the rates at which chemical reactions occur and the factors that affect these rates. It's essential for understanding how quickly reactions progress and what conditions might slow them down or speed them up.
In the context of a first-order reaction, the rate at which the reactants are converted into products is directly proportional to the concentration of the reactant. This means the rate of the reaction decreases as the reactant is consumed. Interestingly, the time it takes to reach a certain completion percentage doesn't depend on the initial concentration but rather on the rate constant and the specific logarithmic ratio representing the completion fraction.
Therefore, when studying the time required for a first-order reaction to reach 99.9% completion relative to 90% completion, kinetic analysis through mathematical equations involving the natural logarithm becomes a powerful tool.
In the context of a first-order reaction, the rate at which the reactants are converted into products is directly proportional to the concentration of the reactant. This means the rate of the reaction decreases as the reactant is consumed. Interestingly, the time it takes to reach a certain completion percentage doesn't depend on the initial concentration but rather on the rate constant and the specific logarithmic ratio representing the completion fraction.
Therefore, when studying the time required for a first-order reaction to reach 99.9% completion relative to 90% completion, kinetic analysis through mathematical equations involving the natural logarithm becomes a powerful tool.
Reaction Rate Constant
The reaction rate constant, denoted as 'k', is a pivotal parameter in chemical kinetics. It offers a quantitative measure of how fast a reaction proceeds. For first-order reactions, this constant is unique in that it remains constant, irrespective of the concentration of the reactant.
By knowing the rate constant, you can predict the time required for a reaction to reach a certain completion percentage. For instance, the rate constant can be used to relate the timing of 90% and 99.9% reaction progressions, as seen in the exercise solution.
The rate constant is influenced by various factors, including temperature, presence of catalysts, and the physical state of the reactants. Understanding 'k' provides not just an insight into the reaction speed, but it is also a crucial step in calculating the time to completion for any percentage of a first-order reaction.
By knowing the rate constant, you can predict the time required for a reaction to reach a certain completion percentage. For instance, the rate constant can be used to relate the timing of 90% and 99.9% reaction progressions, as seen in the exercise solution.
The rate constant is influenced by various factors, including temperature, presence of catalysts, and the physical state of the reactants. Understanding 'k' provides not just an insight into the reaction speed, but it is also a crucial step in calculating the time to completion for any percentage of a first-order reaction.
Natural Logarithm in Kinetics
The natural logarithm, typically represented as 'ln', is a mathematical operation that is extensively used in chemical kinetics, especially for first-order reactions. Its role in the field becomes clear when analyzing the relationship between the concentration of the reactant and time.
In kinetic equations, the natural logarithm helps us determine the time required for a reaction to reach a certain percentage of completion. As demonstrated in the provided exercise, the manipulation of the 'ln' function leads us to understand the proportionality between the time taken for different completion stages.
The negative natural logarithm of one minus the completion fraction (expressed as a decimal) gives us a value that, when multiplied by the rate constant, equates to the time for that level of completion. This is why when comparing times to reach 90% and 99.9% completion, the natural logarithm becomes an essential tool for simplifying and solving the rate equations.
In kinetic equations, the natural logarithm helps us determine the time required for a reaction to reach a certain percentage of completion. As demonstrated in the provided exercise, the manipulation of the 'ln' function leads us to understand the proportionality between the time taken for different completion stages.
The negative natural logarithm of one minus the completion fraction (expressed as a decimal) gives us a value that, when multiplied by the rate constant, equates to the time for that level of completion. This is why when comparing times to reach 90% and 99.9% completion, the natural logarithm becomes an essential tool for simplifying and solving the rate equations.
