and Tzara Duchamp, who calculated on the basis of the sample measurement alone to obtain an exponential decay constant of ln (5/3), giving a half-life of 1.36 years. The simplest approach to these questions was adopted by Gerd, Luca U. How many atoms must be in A, B and C initially in order to maximize your chances of getting the precise results you want? Remember, you don’t know the half-life exactly. When you inspect the portions after exactly one more year, you would like A to have exactly 6 undecayed atoms, B to have 7 or 8, and C to have 4 or 5. You need to divide it into three portions - A, B and C - according to the following rules. In the above scenario, for your second year, you obtain 30 atoms of the substance. What is your range for the plausible half-life of unobtainium? Question 4: You want to figure out the half-life of the substance, and like a true scientist you seek a range that has a 95 percent probability of containing the true value. After exactly one year, 2 atoms have decayed. Let us say you manage to obtain 5 atoms of the radioactive isotope of unobtainium, of Avatar fame. So here, instead of giving a unique solution when there isn’t one, I will discuss interesting insights, assumptions and techniques that readers brought to bear on Questions 3 and 4, all of which have some degree of validity considering the ambiguity in the questions. These questions therefore admitted to many possible answers, as I realized after looking at the various ways readers filled in the gaps.
Ralf B is correct in that this will be the expected number of half-lives. For materials other than hydrogen, the number will be slightly smaller. This is equal to about 2^88 atoms, so they will decay to one in 88 half-lives. The pound contains 454 grams, or 454*6.022e23 atoms (assuming hydrogen). Ralf B performed the calculation, and described it as follows (with a correction of the units as pointed out by Rational): From a practical point of view, I said it would be a very large number. Mathematically, the answer is infinite, because there is always a finite probability that some atoms will not have decayed, no matter how much time elapses. This question can be answered from a few different perspectives.
How many half-lives does a pound of radioactive material have? Since the bucket shown is not cylindrical but has sloping sides, this point is valid and presents a new puzzle, which readers are welcome to attempt (you can assume that the sides of the bucket slope upward at an angle of 100 degrees). Gerd drew attention to the fact that the illustration should have shown the bucket filled to a different height to make it half full. One of the many mystical properties of the base of natural logarithms e (2.71828….) that makes it so special is that when you raise it to a variable x, the resulting function’s rate of change is exactly equal to its current value - its rise or fall is, in a sense, perfect. Exponential processes increase or decrease in proportion to their current value. Since the population doubles every day, it had to have been half of the current value the day before, after 47 days. However, what we are dealing with here is an exponential process. The answer that pops into the mind instantly is half of 48 - 24 days - because we are very familiar with linear process.