Question

The following data are for the system $$\mathrm{A}(g) \rightleftharpoons 2 \mathrm{~B}(g)$$ $$\begin{array}{ccccccc}\hline \text { Time (s) } & 0 & 20 & 40 & 60 & 80 & 100 \\ P_{\mathrm{A}} \text { (atm) } & 1.00 & 0.83 & 0.72 & 0.65 & 0.62 & 0.62 \\ P_{\mathrm{B}} \text { (atm) } & 0.00 & 0.34 & 0.56 & 0.70 & 0.76 & 0.76 \\ \hline\end{array}$$ Prepare a graph of \(P_{\Lambda}\) and \(P_{\mathrm{B}}\) versus time and use it to answer the following questions: (a) Estimate \(P_{\mathrm{A}}\) and \(P_{\mathrm{g}}\) after \(30 \mathrm{~s}\). (b) Estimate \(P_{\mathrm{A}}\) after \(150 \mathrm{~s}\). (c) Estimate \(P_{\mathrm{B}}\) when \(P_{\mathrm{A}}=0.700 \mathrm{~atm}\).

Short answer

Question: Estimate the pressures of gases A and B after 30 seconds, the pressure of gas A after 150 seconds, and the pressure of gas B when the pressure of gas A is 0.700 atm, using the provided graph.

Step-by-step solution

Graphing the pressure of A and B over time

First, create a graph with time (seconds) on the x-axis and pressure (atm) on the y-axis. Plot the given data points for both A and B. You can use a separate color or symbol to differentiate between the two gases. A helpful tool to create this graph can be a spreadsheet software or an online graphing calculator.

Estimating P_A and P_B after 30s

To estimate the pressure of A and B after 30 seconds, locate 30s on the x-axis and draw a vertical line upwards, then find where this line intersects the lines representing the pressures of gases A and B. Read the pressure values at this intersection point. Based on the graph, you should be able to find the approximate pressures of A and B at this time.

Estimating P_A after 150s

For this estimation, follow a similar process as in step 2. Locate 150s on the x-axis and draw a vertical line upwards. The line should extend into a region where there is no data, so you might have to extrapolate the trends of the gases' pressures to make an estimation. Find the approximate value where this line intersects the line representing the pressure of gas A. This pressure value should give an estimation of the pressure of A after 150 seconds.

Estimating P_B when P_A = 0.700 atm

To find P_B at this specific pressure for gas A, locate 0.700 atm on the y-axis corresponding to the graph of gas A and draw a horizontal line until it intersects the line representing the pressure of gas B. Read the value at this intersection point to estimate the pressure of gas B at this particular pressure of gas A.

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with the speeds or rates at which chemical reactions occur. Understanding kinetics allows chemists to determine how different conditions, such as concentration and temperature, affect the speed of a reaction. This is crucial for controlling processes in industries and in conducting laboratory experiments.

At its core, a chemical reaction transforms reactants into products, and kinetics analyzes the factors that influence this transformation over time. For instance, a rapid reaction, such as a combustion explosion, happens in a split second, while a slow one, like rusting iron, can take years. By examining changes in concentrations of reactants and products or, as in our textbook case, pressures of gaseous species, reaction rates can be determined.

Reaction Rates

The rate of a chemical reaction indicates how fast reactants are converted into products over time. It's measured by the change in concentration of reactants or products per unit time. In cases involving gases, as seen in the textbook exercise, changes in pressure are often more accessible to measure and can serve as a proxy for concentration changes. This is particularly true under the assumption that the volume of the system is constant and the temperature remains the same, allowing us to use the ideal gas law for conversions if necessary.

In the given exercise, the reaction rate could be approximated by observing the changes in pressures of gas A and B. Initially, the pressure of A decreases while that of B increases, reflective of reactant consumption and product formation. As the reaction progresses towards equilibrium, these changes in pressure will slow down and eventually stop, indicating the reaction rate has decreased to zero.

Pressure vs Time Graph

A pressure vs time graph is an insightful way to visualize the changes in gases' pressures during a chemical reaction. From such a graph, not only can the reaction rates be inferred by examining the slope of the lines, but it also provides information regarding when the system reaches chemical equilibrium — a state where the pressures of gases no longer change significantly.

In our exercise, plotting the pressures of A and B over time allows us to estimate values at intermediate times (like at 30 seconds) and predict future behaviors (like at 150 seconds). The flat region of the graph where the pressures stabilize indicates that chemical equilibrium has been established. Such graphs are invaluable for interpreting kinetic data and are much easier to understand than simply looking at a table of numbers, as they provide a visual representation of the dynamics of the reaction.