The half-life for the radioactive decay of Ce-141 is 32.5 days. If a sample has an activity of 4.0 µCi after 130 days have elapsed, what was the initial activity, in microcuries, of the sample?

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Welcome back everyone. A sample of molybdenum 99 has a half life of 2.75 days. After 16.5 days, the activity of the sample was reduced to 378 bells. Calculate the initial activity in bels of the sample. So based on the time frame of 16.5 days and given the half life of our isotope of molybdenum 2.75 days. Let's figure out how many have lives is 16.5 days. We'll set this as a variable N that we can say that N is equal to 16.5 days in which we can then multiply by a conversion factor where we will interpret the half life as in the denominator 2.75 days for one half life in the new raider. This allows us to then cancel out our units of days, leaving us with half life as the final unit. And this will correspond to a total of six half lifes representing 16.5 days from our textbooks. Let's recall that half life corresponds to the time it takes a substance to decay by 50%. Or we can say by half and half life represents a process of exponential decay, which in other words describes the fact that each half-life period reduces the quantity of our molybdenum 99 by half. And we would refer to the following formula where to find the final quantity of our isotope moly 99 that is set equal to the initial quantity of molybdenum 99 multiplied by one half raised to N, which is the number of half lives passed now because we need to find the initial activity. We're going to reformat this formula describing the exponential decay of our isotope. So that instead we isolate for the initial activity of Melito 99. And so we're going to need to divide both sides by one half to the nth power. So that, that term cancels out on the right. And what we would find is that to get the initial quantity and we'll just use I for shorthand, lower case I, the initial quantity of molybdenum 99 is found by taking the final quantity of molybdenum 99 divided by one half to the NTH power. And actually, the nth exponent should be on the outside of the parentheses in the denominator. So plugging in our notes, we've got the amount that milium 99 is reduced to in beryls given in the prompt as 378 bears in the numerator representing the final amount divided by one half which is raised to N where we determined that N is representative of six half lives. And we can actually plug in one half as a decimal being 0.5. So that it's even more clear. And in our calculators, when we simplify, we would find that the initial activity of moli 99 is equal to a value of 24,192. And we have units of bare and we can round this answer to a total of three significant figures that we have 2.42 times 10 to the fourth power bells as our final answer being the initial activity of mole on 99. So I hope this made sense and let us know if you have any questions.

The half-life for the radioactive decay of Ce-141 is 32.5 days. If a sample has an activity of 4.0 µCi after 130 days have elapsed, what was the initial activity, in microcuries, of the sample?

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Key Concepts

Half-Life

Radioactive Decay

Exponential Decay Formula