A study of the rate of the reaction represented as \(2 A \longrightarrow B\) gave the following data: $$\begin{array}{|c|c|c|c|c|c|c|c|} \hline \text { Time (s) } & 0.0 & 5.0 & 10.0 & 15.0 & 20.0 & 25.0 & 35.0 \\ \hline \text { \([A](M)\) } & 1.00 & 0.775 & 0.625 & 0.465 & 0.360 & 0.285 & 0.230 \\ \hline \end{array}$$ (a) Determine the average rate of disappearance of \(A\) between \(0.0 \mathrm{s}\) and \(10.0 \mathrm{s}\), and between \(10.0 \mathrm{s}\) and \(20.0 \mathrm{s}\). (b) Estimate the instantaneous rate of disappearance of \(A\) at 15.0 s from a graph of time versus \([A]\). What are the units of this rate? (c) Use the rates found in parts (a) and (b) to determine the average rate of formation of \(B\) between 0.00 s and 10.0 s, and the instantaneous rate of formation of \(B\) at 15.0 s.

Step by step solution

01

Determine the Average Rate of Disappearance of A

Calculate the average rate of disappearance of A by using the formula: Average rate = (Change in concentration of A) / (Change in time). For the first interval, between 0.0 s and 10.0 s, we have: Average rate = ([A] at 10.0 s - [A] at 0.0 s) / (10.0 s - 0.0 s).

02

Calculate Average Rate between 0.0 s and 10.0 s

Substitute the given concentrations into the formula: Average rate = (0.625 M - 1.00 M) / (10.0 s - 0.0 s) = (-0.375 M) / 10.0 s = -0.0375 M/s. Note that the rate is negative, indicating the concentration of A is decreasing.

03

Calculate Average Rate between 10.0 s and 20.0 s

Similarly, calculate the rate for the interval between 10.0 s and 20.0 s: Average rate = ([A] at 20.0 s - [A] at 10.0 s) / (20.0 s - 10.0 s) = (0.360 M - 0.625 M) / 10.0 s = -0.0265 M/s.

04

Estimate Instantaneous Rate at 15.0 s

To estimate the instantaneous rate at 15.0 s, plot the given data points of concentration vs time, draw the tangent line at 15.0 s, and calculate its slope. Since a graph cannot be drawn here, we provide a general approach. The slope of the tangent line at that point corresponds to the instantaneous rate of disappearance.

05

Determine the Units of Rate

The units of rate are derived from the change in concentration over change in time, which is mol/L/s or M/s. This is the unit for the rate of disappearance of A.

06

Calculate the Average Rate of Formation of B between 0.00 s and 10.0 s

Since the reaction shows that 2 moles of A produce 1 mole of B, the rate of formation of B is half the rate of disappearance of A. Calculate using the previous result for 0.0 s to 10.0 s: Average rate of formation of B = (1/2) * (-0.0375 M/s) = -0.01875 M/s.

07

Calculate the Instantaneous Rate of Formation of B at 15.0 s

Use the same relationship as in step 6 to find the instantaneous rate of formation of B at 15.0 s. The instantaneous rate of formation of B is half the rate of disappearance of A at the same time point. Assuming the estimated instantaneous rate of disappearance of A at 15.0 s was found to be -0.0300 M/s: Instantaneous rate of formation of B = (1/2) * (-0.0300 M/s) = -0.0150 M/s.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Rate of Reaction

When studying chemical kinetics, it's essential to understand the average rate of a reaction. This rate measures the speed at which reactants are consumed or products are formed over a given time period. To calculate it, you need the change in concentration of a reactant or product and the time interval for that change.

For instance, consider a reaction where substance A is transforming into substance B. Let's say we observe the concentration of A decreases from 1.00 M to 0.625 M over 10 seconds. The average rate of disappearance of A is computed by dividing the change in concentration (1.00 M - 0.625 M) by the time span (10.0 s), resulting in an average rate of -0.0375 M/s. The negative sign indicates the concentration of A is decreasing with time.

Instantaneous Rate of Reaction

Unlike the average rate, the instantaneous rate of reaction refers to the rate at a particular moment in time. It's akin to checking the speedometer of your car to see exactly how fast you're going at a specific instance.

To estimate the instantaneous rate at 15.0 s, for our example reaction, we'd plot the concentration of A against time and draw a tangent to the curve at 15.0 s. The slope of this tangent line represents the sought-after rate. It is a snapshot of how quickly A is disappearing right at that moment. In practice, because the precise numerical value depends on the steepness of the tangent, the accuracy of your estimation will depend on the quality of the graph and your ability to accurately draw tangents.

Reaction Rate Calculation

Calculating reaction rates is central in chemical kinetics as rates can inform us about the mechanism of the reaction and can be the basis for determining other important factors such as yielding efficient production processes. If we know the stoichiometry of the reaction, like in our case, 2 molecules of A produce 1 molecule of B, it allows us to also calculate the rate of formation of B from the rate of disappearance of A.

Let's say we have an extreme situation where we can't draw tangents accurately. In that case, using secant lines over very short time intervals can provide an approximation for the instantaneous rate. The finer the interval, the closer the approximation will be to the true instantaneous rate.

Rate of Disappearance

The rate of disappearance is a vital concept when looking at a specific reactant in a chemical equation. It tells us how quickly that reactant is being consumed as the reaction progresses. For every chemical reaction, the rate of disappearance of a reactant is tied directly to the rate of appearance of the product(s) according to the stoichiometry of the reaction.

In our exercise, the disappearance of reactant A is straightforward to follow through the given concentration values over time. Remember that the rate is typically expressed with a negative sign because the concentration of the reactant is decreasing. For every 2 moles of A that disappear, 1 mole of B appears, and this stoichiometric relationship can help us find the rate of appearance of the product, offering insights into both sides of the reaction.