Problem 107 - General Chemistry: Principles and Modern Applications

Question

In the first-order decomposition of substance A the following concentrations are found at the indicated times: \(t=0 \mathrm{s},[\mathrm{A}]=0.88 \mathrm{M} ; t=50 \mathrm{s},[\mathrm{A}]=0.62 \mathrm{M} ; t=100 \mathrm{s},[\mathrm{A}]=0.44 \mathrm{M} ; t=150 \mathrm{s},[\mathrm{A}]=0.31 \mathrm{M}.\) Calculate the instantaneous rate of decomposition at \(t=100 \mathrm{s}.\)

Step-by-Step Solution

Answer

The instantaneous rate of decomposition at \(t=100s\) is the average of the average rates of reaction calculated from the intervals 50-100s and 100-150s.

Step 1: Understand the Problem

We are given the concentrations of substance A at \(t = 0, 50, 100, 150 \, s\). These values will be used to approximate the changes in concentration over the time intervals around \(t=100 \, s\). The rate of the reaction can then be determined by the change in concentration over the change in time.

Step 2: Calculate the Rates of Reaction

The average rate of reaction for the interval from 50 to 100 seconds is given by \(-( [\mathrm{A}]_{t=100s} - [\mathrm{A}]_{t=50s} ) / (100s - 50s)\) and the average for the interval from 100 to 150 seconds is given by \(-( [\mathrm{A}]_{t=150s} - [\mathrm{A}]_{t=100s} ) / (150s - 100s)\). We take the negative sign as the reactant A is being used up.

Step 3: Average the Rates of Reaction

To approximate the instantaneous rate of reaction at \(t = 100 \, s\), we calculate the mean of the average rates of reaction from step 2. This will give a reasonable estimate of the instantaneous rate at \(t = 100 \, s\).

Step 4: Substitute the given values

Substituting the concentrations given into our formula, we calculate the average rates of reaction between 50-100s and 100-150s. We then average them to estimate the instantaneous rate of reaction at \(t=100s\).

Key Concepts

Chemical Kinetics

Chemical kinetics is the study of the speed or rate at which chemical reactions occur. It's an essential field of study because it helps scientists understand how different conditions, such as temperature, pressure, and concentration, affect the speed of chemical reactions. A common focus within chemical kinetics is on reaction rates, which describe how quickly reactants are converted into products. Understanding these rates is crucial, particularly in industries such as pharmaceuticals and environmental engineering, where the speed of a reaction directly influences productivity and safety.

First-Order Reaction

A first-order reaction refers to a chemical reaction where the rate is directly proportional to the concentration of a single reactant. This means that if you were to double the concentration of the reactant, the rate of the reaction would also double. Mathematically, first-order reactions can be represented as \( rate = k[A] \), where \( k \) is the rate constant, and \( [A] \) is the concentration of the reactant. They have unique characteristics when it comes to their half-lives (the time taken for half the reactant to be used up), which remains constant regardless of the initial concentration. This property is often used to determine the order of a reaction.

Reaction Rate Calculation

Calculating the rate of a chemical reaction involves determining how the concentration of a reactant or product changes over a specific period. For a reaction \( A \rightarrow B \), the rate can be calculated by the formula \( rate = -\Delta[A]/\Delta t \) when A is the reactant and concentration decreases over time, hence the negative sign. If B were the product, the formulation would be positive since its concentration increases. Calculators or data from experiments providing concentrations of reactants or products at different times are typically used to find average rates. However, for instantaneous rates, we take the derivative of the concentration with respect to time or use averaged intervals around a point of interest.

Concentration-Time Relationship

The relationship between the concentration of a reactant and time is key to understanding chemical reactions. In a first-order reaction, this relationship can be described by an exponential decay function, indicating that as time increases, the concentration of the reactant decreases at a rate proportional to its current concentration. Mathematically, this is expressed as \( [A] = [A]_0e^{-kt} \), where \( [A]_0 \) is the initial concentration, \( k \) is the rate constant, and \( t \) is time. In practice, plotting the natural logarithm of reactant concentration against time yields a straight line for a first-order reaction, which is crucial for rate constant determination and to extrapolate the concentration at any given time.