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Chapter 18: Problem 38

The half-life of thorium-234 is 24.10 days. How many days until only one-sixteenth of a 52.0 g sample of thorium-234 remains?

Short Answer

Expert verified
It will take 96.4 days for only one-sixteenth of a 52.0 g sample of thorium-234 to remain.

Step by step solution

01

Understand the Concept of Half-Life

First, understand that it takes one half-life for a substance to decay to half of its original value. Since we're looking for the time it takes for only one-sixteenth of the original 52.0 g thorium-234 sample to remain, we need to figure out how many half-lives it takes to get from the original amount to one-sixteenth of that amount.
02

Calculate Number of Half-Lives

Since each half-life halves the remaining amount of substance, to go from the whole to one-half is one half-life, from one-half to one-fourth is another half-life, from one-fourth to one-eighth is yet another half-life, and from one-eighth to one-sixteenth is still one more half-life. Add these up: one-half life, plus one-half life, plus one-half life, plus one-half life gives 4 half-lives to go from the whole to one-sixteenth of the original amount.
03

Determine the Total Time for the Decay

Now, calculate the total time for the decay process. Since each half-life of the thorium-234 is 24.10 days, and we have calculated that it takes 4 half-lives for the sample to decay to one-sixteenth of its original mass, we can calculate the total decay time. Multiply the half-life of the substance by the number of half-lives it takes for the sample to decay to the desired amount: 24.10 days/half-life * 4 half-lives = 96.4 days.

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