A random-number generator can be used to simulate the probability of a given atom decaying over a given time. For example, the formula "=RAND()" in an Excel spreadsheet returns a random number between 0 and \(1 ;\) thus, for one radioactive atom and a time of one half-life, a number less than 0.5 means the atom decays and a number greater than 0.5 means it doesn't. (a) Place the "=RAND()" formula in cells A1 to A10 of an Excel spreadsheet. In cell B1, place "=IF(A1 \(<0.5,0,1\) )." This formula returns 0 if \(\mathrm{A} 1\) is \(<0.5\) (the atom decays) and 1 if \(\mathrm{A} 1\) is \(>0.5\) (the atom does not decay). Place analogous formulas in cells B2 to B10 (using the "Fill Down" procedure in Excel). To determine the number of atoms remaining after one half-life, sum cells \(\mathrm{B} 1\) to \(\mathrm{B} 10\) by placing \("=\mathrm{SUM}(\mathrm{B} 1: \mathrm{B} 10) "\) in cell \(\mathrm{B} 12 .\) To create a new set of random numbers, click on an empty cell (e.g., B13) and hit "Delete." Perform 10 simulations, each time recording the total number of atoms remaining. Do half of the atoms remain after each half-life? If not, why not? (b) Increase the number of atoms to 100 by placing suitable formulas in cells Al to A100, B1 to B100, and B102. Perform 10 simulations, and record the number of atoms remaining each time. Are these results more realistic for radioactive decay? Explain.

Step by step solution

01

Insert Random Number Generator in Excel

Open an Excel spreadsheet. In cells A1 to A10, type the formula =RAND() in each cell. This will generate random numbers between 0 and 1.

02

Apply IF Function to Determine Decay

In cell B1, type the formula =IF(A1<0.5, 0, 1). This formula checks if the value in A1 is less than 0.5. If true, it returns 0 (atom decays); if false, it returns 1 (atom does not decay).

03

Fill Down the Formulas

Copy the formula in cell B1 and paste it into cells B2 to B10. To do this quickly, use the 'Fill Down' feature by dragging the small square at the bottom-right corner of the cell B1 down to B10.

04

Calculate the Sum of Remaining Atoms

In cell B12, type the formula =SUM(B1:B10). This will sum up the values in cells B1 to B10, giving the total number of atoms that did not decay.

05

Perform Multiple Simulations

Click on an empty cell, such as B13, and hit 'Delete' to generate new random numbers in cells A1 to A10. Perform this step 10 times and each time, record the number of atoms remaining in cell B12.

06

Analyze Results for 10 Atoms

Observe if about half the atoms remain after each half-life simulation. Discuss if the results consistently show half remaining and why or why not.

07

Increase the Number of Atoms to 100

Repeat Steps 1 through 4 but this time for cells A1 to A100 and B1 to B100. Modify the SUM formula to =SUM(B1:B100) in cell B102.

08

Perform Multiple Simulations for 100 Atoms

Click on an empty cell, such as B103, and hit 'Delete' to generate new random numbers in cells A1 to A100. Perform this step 10 times and record the number of atoms remaining each time in cell B102.

09

Analyze Results for 100 Atoms

Compare the results of 10 simulations for 100 atoms. Discuss if these results are more realistic and consistently show that roughly half the atoms decay each time and why.

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Key Concepts

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

half-life

The concept of a half-life is essential in radioactive decay. It represents the time taken for half the atoms in a radioactive sample to decay. After one half-life, you can expect approximately half of the original atoms to remain, while the other half will have decayed into new elements or isotopes. This probabilistic nature means that the decay process is somewhat predictable over large samples, but individual atoms decay randomly. Understanding half-life helps with simulating and predicting the behavior of radioactive materials over time.

random number generator

A random number generator is a tool that creates a sequence of numbers without any predictable pattern. In this simulation, Excel's =RAND() function is utilized. It outputs a random number between 0 and 1 for each cell it's placed in. Each number represents the random chance of an atom decaying or not. If the random number is less than 0.5, the atom decays; if it's greater than 0.5, it doesn't. By using these randomly generated numbers, we can model the randomness of radioactive decay.

probability

Probability is the measure of how likely an event is to occur. In the context of our radioactive decay simulation, there is a 50% probability (or a probability of 0.5) that any given atom will decay during one half-life period. This probabilistic approach aligns with the nature of radioactive decay, where each atom has an independent chance of decaying. Using probability, we can predict the behavior of a large group of atoms even if we can't predict the behavior of a single atom with certainty.

decay process

The decay process of radioactive atoms involves the transformation of an unstable atom into a more stable form, typically releasing radiation. During a simulation, this decay is modeled using random numbers. For example, an atom will 'decay' if the random number generated is less than 0.5. The process captures the randomness of actual atomic decay, allowing us to simulate the decay behavior of a group of atoms over a specific period.

Excel spreadsheet

Excel is a powerful tool for running simulations. In this exercise, we use Excel to create and analyze a radioactive decay simulation. By inputting the =RAND() and =IF() functions into specific cells, we can generate random numbers and determine if an atom decays. Excel's features, such as the 'Fill Down' option, allow efficient duplication of formulas across multiple cells, facilitating the simulation of a larger number of atoms.

IF function

The =IF() function in Excel is used to make logical comparisons. It returns one value if a condition is TRUE and another value if the condition is FALSE. In this simulation, =IF(A1<0.5, 0, 1) checks whether the value in cell A1 is less than 0.5 (decay) or not (no decay), outputting 0 or 1 accordingly. This logical test effectively models whether an atom decays based on the randomly generated number.

SUM function

The =SUM() function in Excel helps tally values across a range of cells. In our simulation, it is used to count how many atoms have not decayed by summing up the outputs of the =IF() function across multiple cells. For example, =SUM(B1:B10) adds together the values in cells B1 to B10, giving us the total number of atoms that did not decay after one half-life. This summary is crucial for analyzing the simulation results.

simulation analysis

Simulation analysis involves interpreting the results of multiple runs of the simulation. By performing 10 simulations and recording the number of remaining atoms each time, we can draw conclusions about the decay process. For instance, with only 10 atoms, the results might vary significantly from the expected 50% decay due to randomness. However, with 100 atoms, the results should be closer to the expected value, illustrating the law of large numbers. This analysis helps understand the reliability and accuracy of our simulation model.