First Order ODEs
Lab Goals and 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.
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.Free advice:
Let’s begin with a review of the concepts of radioactive decay and introduce some notation.
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
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 NA(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 kA. 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, kA, can be determined. It follows that the
half-life can be computed if kA is known.
For the purposes of this lab, denote the t1/2 as the half-life.
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
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 NB(t) and NC(t).
For most of the calculations in this section, you should express your answers symbolically, that is, in terms of kA
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
For this section we will use the same methods to solve for and graph NA(t), NB(t), and NC(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 NA(0)=0, NB(0)=0, and NC(0)=0, but let P, and the constants of proportionality, kA, kB, be parameters. For graphs in section 2, use the numerical values for the half-lives from section 1 (that is, keep kA, kB the same), the initial conditions from this section, and let P = 15,000 atoms / s. For all other calculations refer to these quantities symbolically!