Problem 27 - General Chemistry: Principles and Modern Applications

Question

Three different sets of data of \([\mathrm{A}]\) versus time are giv the following table for the reaction \(A \longrightarrow\) prod [Hint: There are several ways of arriving at answer each of the following six questions. $$\begin{array}{cccccc} \hline \text { I } & & \text { II } & & \text { III } & \\ \hline \begin{array}{c} \text { Time, } \\ \text { s } \end{array} & \text { [A], M } & \begin{array}{c} \text { Time, } \\ \text { s } \end{array} & \text { [A], M } & \begin{array}{c} \text { Time, } \\ \text { s } \end{array} & \text { [A], M } \\ \hline 0 & 1.00 & 0 & 1.00 & 0 & 1.00 \\ 25 & 0.78 & 25 & 0.75 & 25 & 0.80 \\ 50 & 0.61 & 50 & 0.50 & 50 & 0.67 \\ 75 & 0.47 & 75 & 0.25 & 75 & 0.57 \\ 100 & 0.37 & 100 & 0.00 & 100 & 0.50 \\ 150 & 0.22 & & & 150 & 0.40 \\ 200 & 0.14 & & & 200 & 0.33 \\ 250 & 0.08 & & & 250 & 0.29 \\ \hline \end{array}$$ Which of these sets of data corresponds to a (a) zero-order, (b) first-order, (c) second-order reaction?

Step-by-Step Solution

Answer

Set I corresponds to a first-order reaction, Set II corresponds to a zero-order reaction, and Set III corresponds to a second-order reaction.

Step 1: Examine the Trends for Each Reaction Order

Start by understanding the trends for each reaction order. A zero-order reaction shows a linear decrease in concentration with time. A first-order reaction shows an exponential decay in concentration with time. A second-order reaction typically shows a rapid decrease in concentration at the beginning, which gradually slows down.

Step 2: Identify Zero-Order Reaction

The zero-order reaction should have a linear decrease in the concentration of reactant A over time. When viewing the provided data sets, it can be seen that dataset II matches this description most closely. The concentration decreases from 1.00 M to 0.00 M over 100 seconds, indicating a linear decrease.

Step 3: Identify First-Order Reaction

First-order reactions show exponential decay in the concentration of the reactant over time. In the given data sets, dataset I fits this description. The concentration decreases exponentially from 1.00 M to 0.08 M over 250 seconds.

Step 4: Identify Second-Order Reaction

A second-order reaction shows a rapid decrease in concentration initially, which slows down with time. Dataset III depicts this trend. The concentration rapidly decreases from 1.00 M to 0.50 M in the first 100 seconds, then decreases slowly reaching 0.29 M at 250 seconds.

Key Concepts

Zero-Order Reaction

In a zero-order reaction, the rate at which the reactant is consumed is constant over time. This means the reactant concentration decreases linearly with time. For example, when you have a zero-order reaction, you expect a linear plot with time on the x-axis and concentration on the y-axis. Alike a straight road that does not twist or turn!

One of the characteristics of zero-order reactions is that the rate of reaction doesn't depend on the concentration of the reactant. This is often seen in reactions where the catalyst or surface is saturated. Think of it like a bottleneck effect— regardless of how much reactant you have, the rate remains the same as capacity is maxed out.

• The concentration changes linearly over time.
• The rate is constant and independent of the concentration of reactants.

In terms of mathematical representation, the rate law for a zero-order reaction is given by: \[ [ ext{A}] = [ ext{A}_0] - kt \] where \(k\) is the rate constant. If you were following the data from the exercise, DataSet II beautifully follows this pattern.

First-Order Reaction

First-order reactions are the kind where the rate of reaction is directly proportional to the concentration of one reacting substance. This implies that the concentration of the reactant decreases exponentially over time. As the reaction goes on, the reactant concentration halves successively, demonstrating this exponential decay. Rather like watching a repeatedly halving pie getting smaller and smaller!

If you picture plotting time against the natural logarithm of concentration, you will see a stunning straight line, unlike the curvy profile seen with concentration only. The rule of thumb goes: higher concentration, faster reaction; lower concentration, slower reaction.

• The concentration decreases exponentially over time.
• The rate of reaction depends linearly on the concentration of the reactant.

For first-order reactions, \( ext{Rate} = k[ ext{A}] \), and the logarithmic form is: \[ ext{ln}[ ext{A}] = ext{ln}[ ext{A}_0] - kt \] where \(k\) is the rate constant. Dataset I from the exercise text book illustrates this concept perfectly with an elegant exponential decline.

Second-Order Reaction

Second-order reactions can be a bit more complex as they involve either two molecules of the same reactant or a combination of two different reactants. The rate of reaction is proportional to the square of the concentration of a single reactant or product of concentrations of two reactants. This results in an interesting curve as the concentration of reactant decreases with time—it starts off quickly and gradually slows down, like running in the beginning of a marathon and slowing down towards the end.

This reaction order gives you a nifty hyperbolic decline when plotting concentration against time. If you instead plot the reciprocal of concentration against time, a linear relationship would emerge, revealing its secret.

• The rate is proportional to the square of the reactant's concentration.
• The concentration change with time shows a rapid decrease initially, then slows.

The rate law for second-order reactions is expressed as:\[ rac{1}{[ ext{A}]} = rac{1}{[ ext{A}_0]} + kt \] where \(k\) is the rate constant. In your textbook exercise, Dataset III follows this trajectory, showing a rapid fall followed by a tapering off in concentration.