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?

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

121.38 minutes

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