Question

Experimental data are listed for the hypothetical reaction $$ \mathrm{X} \longrightarrow \mathrm{Y}+Z $$ $$ \begin{array}{lcccccc} \hline \text { Time (s) } & 0 & 10 & 20 & 30 & 40 & 50 \\ {[\mathrm{X}]} & 0.0038 & 0.0028 & 0.0021 & 0.0016 & 0.0012 & 0.00087 \\ \hline \end{array} $$ (a) Plot these data as in Figure 11.3 . (b) Draw a tangent to the curve to find the instantaneous rate at \(40 \mathrm{~s}\). (c) Find the average rate over the 10 to \(50 \mathrm{~s}\) interval. (d) Compare the instantaneous rate at \(40 \mathrm{~s}\) with the average rate over the 40 -s interval.

Short answer

Answer: The comparison depends on the value of the instantaneous rate at 40 seconds, which was found by measuring the slope of the tangent in part (b) of the solution. If the reaction rate is relatively constant, the instantaneous rate will be close to the average rate, which was calculated as 4.825 x 10^{-5} M/s. If the reaction rate changes significantly during the interval, the instantaneous and average rates may be very different.

Step-by-step solution

(a) Plotting the data

To plot the data, create a graph with Time (s) on the x-axis and concentration of X [X] on the y-axis. Plot the given data points and connect them with a smooth curve.

(b) Finding the instantaneous rate at 40 seconds

To find the instantaneous rate at 40 seconds, draw a tangent to the curve at the point corresponding to 40 seconds. Measure the slope of the tangent (rise over run), which will represent the instantaneous rate at that time. The slope of the tangent represents the rate of change of concentration of X with respect to time at that particular point.

(c) Calculating the average rate over the 10 to 50 seconds interval

To find the average rate over the 10 to 50 seconds interval, use the following formula: $$ \text{Average rate} = \frac{\Delta [\text{X}]}{\Delta t} $$ where Δ[X] is the change in concentration of X over the interval, and Δt is the change in time (40 seconds). Here, $$ \Delta [\text{X}] = [X]_{10\,s} - [X]_{50\,s} = 0.0028\,\text{M} - 0.00087\,\text{M} = 0.00193\,\text{M} $$ and $$ \Delta t = 50\,\text{s} - 10\,\text{s} = 40\,\text{s} $$ So we have, $$ \text{Average rate} = \frac{0.00193\,\text{M}}{40\,\text{s}} = 4.825 \times 10^{-5}\, \text{M/s} $$

(d) Comparing the instantaneous rate at 40 seconds with the average rate over the 40-second interval

Now compare the instantaneous rate at 40 seconds (which was found by measuring the slope of the tangent) with the average rate over the interval. If the reaction rate is relatively constant, the instantaneous rate will be close to the average rate. If the reaction rate changes significantly during the interval, the instantaneous and average rates may be very different.