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Radioactive isotopes are used as contrast dyes to study soft tissues such as the gastrointestinal tract. The radiocontrast dyes are considered safe because the isotopes have short half-lives and the isotopes coat the GI tract and are not absorbed by normal tissue. If a patient ingests 7.50 mg of barium sulfate, how long until the barium-141 is 99% out of the patient's system if the half-life barium-141 is 18.27 minutes?

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The radioactive decay of the isotopes is an exponential decay.

In consequence, the half-life law is:

Remaining amount of isotope = Initial Quantity * [1/2]^n, where n is the number of half-lives that have elapsed.


=> Remaining amount of isotope / initial quantity = [1/2]^n.


In this case we want that the remaining quantity be 1%.


So, Remaining amount = 1% * initial quantity


=> Remaining amount / Initial quantity = 0.01


=> 0.01 = [1/2]^n


  => n log(1/2) = log(0.01)


=> n = log (0.01) / log (0.5) = 6.6439 half-lives


Now, you just must multiply the number of half-lives times the time of a half-life


=> T = 6.6439


=> n = 6.6439 * 18.27 minutes = 121.38 minutes


Answer: 121.38 minutes

For 99% of the barium-141 to be gone, 1% (0.075mg) of the starting material remains

For 7.50mg,  

after the 1st half-life 3.75mg remain  

after the 2nd half-life 1.875mg remain

after the 3rd half-life 0.9375mg remain

after the 4th half-life 0.46875mg remain

after the 5th half-life 0.234375mg remain

after the 6th half-life 0.1171875mg remain

after the 7th half-life 0.05859375mg remain and so on.  

So it takes 7 half-lives to reach 99% of the barium-141 gone.

7*(18.27 minutes)= 127.89 min = 2.13 hr

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