APPM 2360 Lab 1
Radioactive Decay First Order ODEs

Instructions

This lab is due Friday February 20, 2009 at the beginning of lecture. TA's will hold their regular office hours in ECCR 143(computer lab) during that week. The TAs in the lab are there to help you with Mathematica, MVT and Matlab syntax; they are not there to debug your files.  Please keep this in mind.

• Three (3) students is the maximum group size.You need to hand in only one lab per group.
• Use a cover page
• List the name, recitation number, TA name, and instructor for each student in the group on the cover page
• No loose leaf paper, please use printer paper for all calculations included in appendix
• You are required to submit both, a paper copy, and an electronic copy in MS Word or PDF format to "SafeAssign" in "CuLearn" at culearn.colorado.edu .
• Four decimal point accuracy will suffice for this lab.
• Refer to the lab writing guidelines. They are important and you will be held to them; up to 20% of your final grade can be based on things like writing, introduction, conclusion, etc. Use "Equation Editor" in Microsoft Word for any equations that you include.
• Late projects are not accepted.

  On your title page, clearly mark

• Name(s),  Put the name of the group member you want to get the graded lab in recitation first.
• Student ID number(s)
• Semester (Spring, 2009)
• Professor
• TA
• If you do not include this information, as much as 10% will be deducted from your final score

Format is worth 20% of your grade. Please refer to the following writing guidelines for the expository sections of this report. You are required to know and follow the Writing Guidelines for all labs.

Writing Guidelines for Differential Equations Lab

Take a look at this sample project and examples of "HOW TO" and "HOW NOT TO" present your results.

DO NOT LEAVE ANY OF YOUR FILES SAVED ON THE COMPUTERS IN THE LAB!
IF YOUR FILES ARE FOUND SAVED ON A LAB MACHINE,
IT MAY RESULT IN YOUR GROUP GETTING A 0 (ZERO) FOR THE LAB.

You can either E-mail them to yourself, use webfiles, or save on a USB 'memory stick'

For this lab, you are free to use Mathematica, Matlab, or MVT.  Any or all of them may be helpful. Matlab and Mathematica are installed on the computers in computer labs in ECCR. Guides and tutorials for this software can be found on the web pages for APPM 2360 and APPM 2460

This lab demonstrates the use of first order differential equations to model naturally occurring phenomena, specifically radioactive decay. The purpose of this lab is to use solution techniques and analytical methods to discuss the equations that model radioactive decay chains. The models discussed in this lab will be analyzed both analytically and graphically.

• Start early
• Work in a group, it can help
• Use TA lab hours
• Stay positive, do not take out your frustration on your TA or professors, we're on your side!

Let’s begin with a review of the concepts of radioactive decay and introduce some notation.

Radioactive Decay

Protons and neutrons are the subatomic particles that form the nuclei of atoms. The number of protons in the nucleus determines the identity of the element (for example 26 protons = iron, 8 protons = oxygen). The number of neutrons in the nucleus can vary without changing the identity of the element (for example carbon-12 contains six protons and six neutrons while carbon-14 contains six protons and eight neutrons). Certain atomic nuclei have combinations of protons and neutrons that are unstable and undergo radioactive decay. During the decay process particles are ejected from the nucleus, (this is radiation) and the number of protons and neutrons can be altered. Radioactive decay is a random process, but for very large numbers of atoms we can say that “rate of decay” (number of atoms that decay per second) is proportional to the number of atoms present.

The process of radioactive decay is essentially a rate of change, it is natural to model the process with differential equations. To model the simplest type of radioactive decay, we turn to conservation laws. The rate of change of the particles can be described by the following:

Since no new particles are being created, the RATE IN is zero, and our equation becomes:

Now is a good time to introduce some notation. Call the element being modeled element A. Next, define the number of atoms present at a given time N_A(t). Notice this is a function of time because the number of atoms will change during the process of radioactive decay. Denote the constant of proportionality as k_A. Using the information from the background on radioactive decay, we see that the rate of change is proportional to the number of atoms present. Putting this all together we have:

An important concept in radioactive decay is half-life. This is the amount of time required for half of the atoms in a sample to decay. The notion of a half-life has many applications such as Carbon-14 dating. Carbon-14 is present in many living things and has a half-life of about 5,700 years, which is useful for determining the age of objects. Using the half-life, the constant of proportionality, k_A, can be determined. It follows that the half-life can be computed if k_A is known. For the purposes of this lab, denote the t_1/2 as the half-life. 

Decay Chains

We now turn our attention to a decay chain, where element A decays to element B which decays to element C. Note C does not decay in this discussion.

As the situation becomes more complicated it can be helpful to think in terms of “production rates” (PR) and “loss rates” (LR). The PR and LR for a given element are the rates at which that element is produced or lost due to the various decays. This concept leads to a template for setting up differential equations:

For the example from the last section, the PR is zero. Now we can use this model to develop differential equations to describe N_B(t) and N_C(t).

Questions and Issue to Address

Section 1:

For most of the calculations in this section, you should express your answers symbolically, that is, in terms of k_A etc. For your graphs, however, you will need to use numerical values. In this section, use the following numerical values for your graphs: half-life of A = (ln 20) seconds, half-life of B = (ln 32) seconds, A₀=50,000 atoms. For each plot you will also need to choose an appropriate range for t. For all other calculations refer to these quantities symbolically!

1. Solve equation (1) for N_A(t), given the initial condition N_A(0)=A₀ and using the half-life of A to compute k_A (include units for k_A).
2. Produce a plot of N_A(t) using the ODE Integrator in MVT or using Matlab or Mathematica. Explain the graph and discuss whether or not it agrees with the result from question (1).
3. If the half-life for element A is 25 s instead of (ln 20) s, determine new value of the constant of proportionality, k_A . Remember to include units! Will this change cause slower or faster decay?
4. Using the PR-LR template, set up a differential equation for N_B(t).
5. Temporarily assume that particle B is not a part of the decay chain. If given N_B(0) = B₀, use the half life of  B to compute k_B (include units and remember the answer should be given symbolically).       
Secondly, give the numerical k_B.
6. Using the integrating factor method, and your solution for N_A(t) from question (1), determine the analytical solution for N_B(t) (Answer in terms of A₀,B₀,k_B,k_A). Now simplify your equation by using N_B(0)=B₀=0.
7. Set up a differential equation for N_C(t) using the PR-LR template. Solve for N_C(t) using your result in question (6) with  with B₀=0. Use N_C(0)=C₀=0.
8. Add N_A(t),N_B(t), and N_C(t) with B₀=0 and C₀=0.  What do you get? (Hint: this is a check to make sure you did everything correctly) .
9. Use MVT, Mathematica, or Matlab to plot N_B(t) and N_C(t) with the above numerical values. Describe the behavior of each graph as t approaches infinity. Compute the local maximum of N_B(t) symbolically directly from N_B(t) with B₀=0. Why does the graph of N_B(t) have a local maximum? Explain what is happening physically in the graph of N_C(t). Is this behavior expected?

Section 2:

For this section we will use the same methods to solve for and graph N_A(t), N_B(t), and N_C(t), but we will slightly alter the situation. Now assume that element A is produced by some natural process at a constant rate P > 0. For this section use N_A(0)=0, N_B(0)=0, and N_C(0)=0, but let P, and the constants of proportionality, k_A, k_B, be parameters. For graphs in section 2, use the numerical values for the half-lives from section 1 (that is, keep k_A, k_B the same), the initial conditions from this section, and let P = 15,000 atoms / s. For all other calculations refer to these quantities symbolically!

10. Set up and solve a differential equation for N_A(t) using the given initial condition.
11. Use a function plotter to graph your result from question (10). Discuss the behavior as t approaches infinity and explain what is happening physically.
12. Set up and solve a differential equation for N_B(t) using the given initial conditions (Answer in terms of P,k_B,k_A).
13. Use a function plotter to graph your result from question (12). Explain how the asymptotic behavior compares to N_A(t). Is this expected?
14. Set up and solve a differential equation for N_C(t). Plot N_C(t) and describe what is occurring, especially the behavior as t approaches infinity. Does this make sense? Explain.
15. The solution N_C(t) has a specific asymptotic behavior as t approaches infinity. For large t, deduce the dominant term and justify your choice.
16. This discussion of radioactive decay is somewhat simplistic. We assumed, for very large numbers of atoms, that the rate of decay is proportional to the number of atoms present. Briefly comment on additional assumptions that may affect radioactive decay and chains of decay as examined in this lab. This may require (a little) additional reading on the subject…(The response to this question will be graded merely based on the reasonableness of your response. i.e. there is no right or wrong to this question )