Question

The data below were collected for the reaction \(2 \mathrm{HI}(\mathrm{g}) \longrightarrow \mathrm{H}_{2}(\mathrm{~g})+\mathrm{I}_{2}(\mathrm{~g})\) at \(580 \mathrm{~K}\). $$ \begin{array}{lccccc} \text { Time }(\mathrm{s}) & 0 & 1000 . & 2000 . & 3000 . & 4000 . \\ {[\mathrm{HI}]\left(\mathrm{mol} \cdot \mathrm{L}^{-1}\right)} & 1.0 & 0.11 & 0.061 & 0.041 & 0.031 \end{array} $$ (a) Using a graphing calculator or graphing software, such as that on the Web site for this book, plot the data in an appropriate fashion to determine the order of the reaction. (b) From the graph, determine the rate constant for (i) the rate law for the loss of HI and (ii) the unique rate law.

Short answer

The order of the reaction can be determined by plotting concentration of HI versus time, ln[HI] versus time, and 1/[HI] versus time and identifying which plot gives a straight line. Use the slope of the linear plot to calculate the rate constant for the loss of HI. The rate law is derived based on the reaction order, and in this specific case, the unique rate law is identical to the rate law for the loss of HI.

Step-by-step solution

Plotting the Data

Enter the provided concentration and time data into a graphing calculator or graphing software. Plot the concentration of HI against time. For a zero-order reaction, the plot of [HI] versus time should give a straight line. For a first-order reaction, the plot of ln[HI] versus time should yield a straight line. For a second-order reaction, the plot of 1/[HI] versus time should produce a straight line.

Determining Reaction Order

Examine the linearity of each plot: [HI] vs. time, ln[HI] vs. time, and 1/[HI] vs. time. The plot that gives the most linear relationship will indicate the order of the reaction. The slope of this line can be used to determine the rate constant.

Calculating Rate Constant for Loss of HI

If the reaction order has been determined to be first-order from the second step, use the formula \(k = -\text{slope of ln[HI] vs. time plot}\). If the reaction is second-order, use \(k = \text{slope of 1/[HI] vs. time plot}\). Note that the slope will be negative for a zero-order and a first-order reaction because the concentration of the reactant is decreasing over time.

Formulating the Rate Law

Once you have determined the rate constant \(k\), input it into the rate law equation for the order determined. For a first-order reaction, this is \(\text{rate} = k[\text{HI}]\). For a second-order reaction, it is \(\text{rate} = k[\text{HI}]^2\). The rate law for the loss of HI is the same as the overall reaction rate law in this exercise.

Confirming the Unique Rate Law

For the unique rate law in this case, the stoichiometry of the reaction is such that the unique rate law has the same form as that of the loss of HI, \(\text{rate}_{unique} = k[\text{HI}]^{\text{order}}\).

Key concepts

Reaction Kinetics

Understanding reaction kinetics is at the heart of predicting how chemical reactions occur over time. Reaction kinetics, often referred to simply as chemical kinetics, deals with the rates of chemical processes and the factors that affect these rates. It's a cornerstone of physical chemistry and is vital to grasping the dynamics of chemical reactions.

At its core, reaction kinetics involves examining how different variables—such as concentration, temperature, and the presence of a catalyst—alter the speed at which reactants convert to products. One of the primary goals is to determine the reaction order, which tells us how the rate of the reaction depends on the concentration of reactants. For instance, in a first-order reaction, the rate is directly proportional to the concentration of one reactant, implying the reaction rate will double if the reactant's concentration doubles.

Rate Constant Calculation

The rate constant, denoted by k, is a pivotal quantity in reaction kinetics. It provides a quantitative measure of how fast a reaction is proceeding. To calculate the rate constant, you use experimental data and graphical methods, as illustrated by plotting concentration versus time data for different reaction orders—zero, first, and second.

The unit of the rate constant varies with the order of the reaction; for a zero-order reaction, it's in terms of concentration per time, for a first-order reaction, it's in reciprocal time units, and for a second-order reaction, it's in terms of reciprocal concentration per time. This variance in units is not just a mathematical curiosity; it reflects the fundamental ways that different reactions depend on the concentrations of their reactants.

Rate Law Formulation

The rate law formulation of a reaction describes the relationship between the rate of a reaction and the concentration of its reactants. It's an equation that allows chemists to predict the rate of a reaction under varying conditions. The general form of a rate law is \( rate = k [A]^m [B]^n \), where \( [A] \) and \( [B] \) represent the concentrations of the reactants, m and n are the reaction orders with respect to each reactant, and k is the rate constant.

It's crucial to note that the reaction order is an empirical quantity—it must be determined through experiments rather than from the stoichiometry of the reaction's balanced equation. Therefore, understanding how to obtain and apply the rate law is an invaluable skill for anyone studying reaction kinetics.