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Question

The radioactive decay constant is _____ days-1 if the radioisotope has a half-life of 431 days.

Answer 1

t(1/2) = 0.693 / k

or

k = 0.693 / t(1/2)

k = 0.693 / 431 days

k = 0.001608 1/day to 1 extra sig fig.

I generally tell my students to carry one extra sig fig on intermediate calculations.  

Answer 2

Let k = decay constant.

The Half-life of a radioactive element is the time required for half of the radioactive nuclei present in a sample to decay.

Let yo be the number of radioactive nuclei present initially, then the number y of nuclei present at time t will be given by:

y = yo × e^(kt)

Since we are looking for half-life, we wish to know when

yo × e^(kt) = (½) yo

Cancelling yo’s, we have:

e^(-kt) = ½

-kt = ln(½)

-kt = -ln2

t = ln(2)/k

Thus, Half−life is given by ln(2)/k.

In your case, since you know the Half-life is 431 days, we can solve for &#039;k&#039;, so

431 = ln(2)/k

431k = ln2

k = ln(2)/431 = 0.0016082301

[Answer: see above]