Question

Consider the reaction: \(\mathrm{C}_{4} \mathrm{H}_{8}(g) \longrightarrow 2 \mathrm{C}_{2} \mathrm{H}_{4}(g)\) The tabulated data were collected for the concentration of \(\mathrm{C}_{4} \mathrm{H}_{8}\) as a function of time: $$ \begin{array}{cc} \text { Time (s) } & {\left[\mathrm{C}_{4} \mathrm{H}_{8}\right] \text { (M) }} \\ \hline 0 & 1.000 \\ \hline 10 & 0.913 \\ \hline 20 & 0.835 \\ \hline 30 & 0.763 \\ \hline 40 & 0.697 \\ \hline 50 & 0.637 \\ \hline \end{array} $$ a. What is the average rate of the reaction between 0 and 10 s? Between 40 and 50 s? b. What is the rate of formation of \(\mathrm{C}_{2} \mathrm{H}_{4}\) between 20 and \(30 \mathrm{~s}\) ?

Short answer

a. Average rate between 0 and 10 s: -0.0087 M/s; Average rate between 40 and 50 s: -0.0060 M/s. b. The rate of formation of \(\bar{C}_{2}\bar{H}_{4}\) between 20 and 30 s: 0.0144 M/s.

Step-by-step solution

Understand the Average Reaction Rate

The average rate of a reaction over an interval of time can be calculated by taking the change in concentration of a reactant or product and dividing it by the change in time. For reactants, the rate will be negative due to the decrease in concentration over time. The formula to calculate the average rate is: Average Rate = (Final concentration of reactant - Initial concentration of reactant) / (Final time - Initial time).

Calculate the Average Rate between 0 and 10 s

Use the provided concentrations of \( \mathrm{C}_{4} \mathrm{H}_{8} \) at 0 s and 10 s to find the average rate between these times. Plug the initial concentration (1.000 M at 0 s) and final concentration (0.913 M at 10 s) into the formula: Average Rate = (0.913 M - 1.000 M) / (10 s - 0 s) = -0.087 M / 10 s = -0.0087 M/s. The negative sign indicates the concentration of the reactant is decreasing.

Calculate the Average Rate between 40 and 50 s

Similarly, calculate the average rate between 40 and 50 s using the concentrations from the table: Average Rate = (Final concentration of reactant at 50 s - Initial concentration of reactant at 40 s) / (Final time - Initial time) = (0.637 M - 0.697 M) / (50 s - 40 s) = -0.060 M / 10 s = -0.0060 M/s.

Understanding the Relationship Between Reactant and Product Rates

The rate of formation of a product can be determined by understanding the stoichiometry of the reaction. For the reaction \(\bar{C}_{4}\bar{H}_{8}(g) \longrightarrow 2 \bar{C}_{2}\bar{H}_{4}(g)\), the stoichiometric coefficients indicate that one mole of \(\bar{C}_{4}\bar{H}_{8}\) forms two moles of \(\bar{C}_{2}\bar{H}_{4}\). Therefore, the rate of formation of \(\bar{C}_{2}\bar{H}_{4}\) will be twice that of the rate of disappearance of \(\bar{C}_{4}\bar{H}_{8}\).

Calculate the Rate of Formation of \(\bar{C}_{2}\bar{H}_{4}\) Between 20 and 30 s

First, find the average rate of disappearance of \(\bar{C}_{4}\bar{H}_{8}\) between 20 and 30 s as in previous steps: Average Rate = (Final concentration at 30 s - Initial concentration at 20 s) / (Final time - Initial time) = (0.763 M - 0.835 M) / (30 s - 20 s) = -0.072 M / 10 s = -0.0072 M/s. Next, use this rate to find the rate of formation of \(\bar{C}_{2}\bar{H}_{4}\) by multiplying by the stoichiometric factor of 2: Rate of Formation of \(\bar{C}_{2}\bar{H}_{4}\) = 2 * (-0.0072 M/s) = -0.0144 M/s. Note the negative sign indicates the disappearance of the reactant, but since we are interested in the formation and not the disappearance, the final rate of formation will be positive: 0.0144 M/s.

Key concepts

Average Reaction Rate

Understanding the average reaction rate is crucial for studying chemical kinetics. In essence, it represents the speed at which reactants convert into products over a specified time interval. To determine this rate, we examine the decrease in concentration of a reactant or the increase in concentration of a product over time.

For a given reaction, \( \mathrm{C}_{4} \mathrm{H}_{8}(g) \longrightarrow 2 \mathrm{C}_{2} \mathrm{H}_{4}(g) \), we can measure how fast \( \mathrm{C}_{4} \mathrm{H}_{8} \) is consumed. By using the formula \( \text{Average Reaction Rate} = \frac{\text{Final concentration} - \text{Initial concentration}}{\text{Final time} - \text{Initial time}} \), we calculate the rate between two points in time. A negative value indicates the concentration of the reactant is decreasing, which is expected since reactants are used up as a reaction proceeds.

Applying this formula to different time intervals gives insight into how the reaction rate might change as the reaction progresses. For instance, comparing the average rate of reaction between earlier and later intervals can reveal if the reaction slows down or maintains a consistent pace. This insight is invaluable for experimental analysis and for industrial applications where controlling the rate of a reaction is often essential.

Reaction Stoichiometry

Reaction stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It tells us how many moles of each substance react and are produced according to the balanced chemical equation. For our example, the stoichiometry is a 1:2 ratio, where 1 mole of \( \mathrm{C}_{4} \mathrm{H}_{8} \) produces 2 moles of \( \mathrm{C}_{2} \mathrm{H}_{4} \).

This stoichiometric relationship is key when calculating the rate of formation of the products from the rate of disappearance of the reactants. We can simply use the coefficients from the balanced equation to relate these rates. In the case of \( \mathrm{C}_{4} \mathrm{H}_{8} \) turning into \( \mathrm{C}_{2} \mathrm{H}_{4} \), the reaction rate for the formation of \( \mathrm{C}_{2} \mathrm{H}_{4} \) can be found by doubling the reaction rate of the disappearing \( \mathrm{C}_{4} \mathrm{H}_{8} \), which is consistent with the reaction equation.

Concentration-Time Relationship

The concentration-time relationship is fundamental to the study of chemical kinetics, providing insights into the speed of a reaction. By plotting concentration against time, one can visually understand how the reactants are depleted and products are formed over the course of the reaction.

A steep slope on such a plot indicates a rapid change in concentration and, hence, a fast reaction rate. Conversely, a gentle slope suggests a slower reaction. Over time, as reactants are consumed, the slope often decreases, indicating a reduction in the reaction rate. This phenomenon can be attributed to fewer reactant molecules being available to collide and react.

When you're given a set of concentration data over time, as in the exercise, you can calculate the reaction rate over different intervals to see how it changes. This information might help to identify patterns such as the reaction order, or to verify that the reaction follows expected kinetic models. It's important to note, especially when drawing initial rates from such data, that these values provide an average over the time period and might not represent the instantaneous rate at a specific moment.