Hypofluorous acid, HOF, is extremely unstable at room temperature. The following data apply to the decomposition of HOF to \(\mathrm{HF}\) and \(\mathrm{O}_{2}\) gases at a certain temperature. $$\begin{array}{cl}\hline \text { Time (min) } & \text { [HOF] } \\\\\hline 1.00 & 0.607 \\\2.00 & 0.223 \\\3.00 & 0.0821 \\\4.00 & 0.0302 \\\5.00 & 0.0111 \\\\\hline \end{array}$$ (a) By plotting the data, show that the reaction is first-order. (b) From the graph, determine \(k\). (c) Using \(k\), find the time it takes to decrease the concentration to \(0.100 \mathrm{M}\) (d) Calculate the rate of the reaction when [HOF] \(=0.0500 M\).

Understanding the order of a chemical reaction is crucial for predicting how the concentration of reactants will change over time. To determine the reaction order, one must analyze how the reaction rate depends on the concentration of reactants.

For the decomposition of hypofluorous acid (HOF), we are given concentration data at various time points. By plotting these data points, we can deduce the reaction order by looking at the nature of the graph that results from the relationship between concentration and time. For a first-order reaction, when we plot the natural logarithm of the concentration of HOF (ln[HOF]) against time, we should expect a straight line. This is because the rate of reaction is directly proportional to the concentration of a single reactant for first-order reactions.

If the data do not line up on a straight line when plotted as ln[HOF] versus time, one may need to consider possible errors in the experiment or in the data recording process. Otherwise, it could suggest that the reaction is not first-order, prompting a re-evaluation of the reaction kinetics.

Once we've established the order of the reaction, calculating the rate constant (\( k \)) becomes a straightforward task. For a first-order reaction, the rate constant is the slope of the line obtained from the plot of ln[HOF] versus time. Since the slope of this line is equal to the negative rate constant, it is essential to take the absolute value of the numeric slope to gain the correct value of the rate constant.

The rate constant is a vital parameter in chemical kinetics as it provides insight into the speed of the reaction under the given conditions. Once we have determined the value of the rate constant, it can be used to predict the rate of the reaction at different concentrations of the reactant, as well as to calculate the half-life of the reaction. If the slope obtained from the graph is not linear or consistent, it could signal an error in the experimental setup or data measurement.

Chemical kinetics is the study of the speed or rate at which chemical reactions occur and the factors affecting these rates. Kinetics allows chemists to understand and predict the behavior of reactions, helping to control processes such as chemical synthesis and degradation.

In the context of our example, kinetics examines how the unstable hypofluorous acid decomposes over time. By tracking the concentration changes in HOF at regular intervals, we glean information about the reaction's kinetics, such as the rate constant and reaction order. Chemical kinetics also involves the study of reaction mechanisms, which are the step-by-step pathways that reactions follow from reactants to products. When a student encounters discrepancies in reaction data, understanding kinetics can guide them through troubleshooting, whether the issue lies in the reaction mechanism, changes in environmental conditions, or measurement precision.

The integrated rate law expresses the relationship between the concentration of reactants and time for a given order reaction. It allows us to calculate either concentration or time when the other variables, including the rate constant (\( k \)), are known. For first-order reactions, this law takes the form of a natural logarithmic equation:

\( \text{ln} \frac{[A]_0}{[A]_t} = kt \)where \( [A]_0 \) is the initial concentration, \( [A]_t \) is the concentration at time \( t \) and \( k \) is the rate constant.

Using the rate law, we can solve various problems, such as predicting how long it will take for a reactant to reach a certain concentration, as illustrated in the exercise. For more practical considerations, if the calculated time or concentration does not match the expectations or returns a non-physical result (e.g., negative time), this could point to an error in the input data or misunderstandings regarding the law's application.