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Chapter 1: Problem 55

In a laboratory you measure the decay rate of a sample of radioactive carbon. You write down the following measurements: $$\begin{array}{lrrrrrrr} \hline \text { Time (min) } & 0 & 15 & 30 & 45 & 60 & 75 & 90 \\\ \text { Decays/s } & 405 & 237 & 140 & 90 & 55 & 32 & 19 \\\ \hline \end{array}$$ (a) Plot the decays per second versus time. (b) Plot the natural logarithm of the decays per second versus the time. Why might the presentation of the data in this form be useful?

Short Answer

Expert verified
Answer: Presenting the data in the form of ln(decays/s) versus time can be useful because it transforms the exponential decay curve into a linear curve, allowing for easier understanding of the relationship between decay rate and elapsed time. This linear representation also aids in deriving parameters such as the decay constant or half-life.

Step by step solution

01

Convert time to seconds

Since the decay rate is given in decays per second, we should convert the time from minutes to seconds for better consistency. To do this, multiply each time value by 60.
02

Create a scatter plot for the decay rate against time

Now, you can create a scatter plot using the converted time values (in seconds) and the corresponding decay rates. Label the x-axis as "Time (s)" and the y-axis as "Decays/s".
03

Calculate the natural logarithm of the decay rate

To plot the natural logarithm of the decay rate, use the natural logarithm function (ln) to find the ln(decays/s) for each time point in the dataset.
04

Create a scatter plot for the natural logarithm of the decay rate against time

Now, create a new scatter plot using the same time values (in seconds) but with the ln(decays/s) values you calculated in step 3. Label the x-axis as "Time (s)" and the y-axis as "ln(Decays/s)".
05

Discuss the usefulness of plotting ln(decays/s) vs. time

The presentation of this data in the form of ln(decays/s) versus time may be useful because radioactive decay generally follows an exponential decay process. By taking the natural logarithm of the decay rate, we can transform the exponential decay curve into a linear curve, which can help us better understand the relationship between decay rate and elapsed time. Additionally, in a linear curve, we can more easily derive parameters such as the decay constant or half-life.

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