To the right is a brief review of how the differential equation for the rate-change expression for the amount of daughter is derived (click on the picture for a larger version). The parameters have the usaul names,

In this exercise you can either use numerical methods to solve this equation or you can use the analytically derived mathematical solution. We suggest that you do both.

Here we have introduced the disintegration rate

How this equation is derrived is explained in basic math courses and has probably also been explained by your NRC teacher. It is not very complicated, but involves a few standard "tricks" and takes some time. We will not explain it here, but you can find it explained in e.g. Walter Lovelands's textbook "Modern Nuclear Chemistry" in Chapter 3.3 (page 67).

If you use this equation and write a program that plots the disintegration rate A2 of the daughter (you should also plot the rate A2 of the mother to get the complete picture), you will be able to learn a lot about radioactive equilibria by playing with the mother and daughter halv-lives. This is explained below. In addition, you can train your skills in using numerical methods to solve differantial equations. For this example, an analytical solution is possible, but for more complex problems the numerical methods might be the only solution. It will be a good exercise to use the numerical approach, since you can check the results against the solution obtained with the analytical formula.

The error in each step (the discrepancy between our approximated point and the true value) is going to increase the further away from the initial point we get. It is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the overall error is proportional to the step size.

A good description of Euler's method can be found on Wikiepedia, please refer to this or to a course book in numberical methods for details.

Use equation (4) and (7) to calculate the amount of mother and in-growth of daughter you will get for a selected time span. Make sure that the step-size you use is small enough to make the stepwise error negligible.

From the analytical solution to the mathematical problem you can see how the half-lives of the two nuclei determine the genetic relationship. In particular you should notice and investigate how the fraction with the

As a concrete and also very important example, you should in particular investigate the 99Mo/99mTc-radionuclide generator. 99mTc is the most commonly used medical radionuclide in the world and the hospitals use generators with 99Mo attached to a column to produce the 99mTc.

We suggest that you write your program step by step and thoroghly check that each step works before you continue. You might want to follow the following path:

- The timestep for the numerical solution needs to be small enough for the method to be precise.

- Short-lived mother, long-lived daughter (no equilibrium)
- Short-lived mother, short-lived daughter (no equilibrium)
- Relatively long-lived mother, short-lived daughter (transient equilibrium, T1/2(mother) >> T1/2(daughter)
- Very long-lived mother, short-lived daughter (seqular equilibrium, T1/2(mother > 104x T1/2(daughter))
- Examples of operation of a radionuclide generator, e.g. the 99Mo -> 99mTc generator. How much activity will you get after one, three and 10 daugher half-lives? What happens if you milk the generator two times in a row?