Radio-isotopes of different elements have different half-lives. Magnesium-27 has a half-life of 9.45 minutes. What is the decay constant for Magnesium-27? Round to five decimal places.
Since the decay rate is given in terms of minutes, then time t will be in minutes. However, I note that there is no beginning or ending amount given. How am I supposed to figure out what the decay constant is? I can do this by working from the definition of "half-life": in the given amount of time (in this case, 9.45 minutes), half of the initial amount will be gone. That is, from t = 0 to t = 9.45, I will have gone from 100% ("1") of however much I started with to 50% ("0.5") of that amount. Since the half-life does not depend on how much I started with, I can either pick an arbitrary beginning amount (such as 100 grams) and then calculate the decay constant after 9.45 minutes, at which point only 50 grams will remain (the other 50 grams have mutated into some other isotope or element). Or else I can just deal with the 50% that is left. Either way, I will end up dealing with this equation:
0.5 = e9.45k