Question

You have just isolated a new radioactive element. If you can determine its half-life, you will win the Nobel Prize in physics. You purify a sample of 2 grams. One of your colleagues steals half of it, and four days later you find that 0.3 gram of the radioactive material is still left. What is the half-life? (Round your answer to three significant digits.)

Answer 1

Staals half
1 gram left
0.3 grams left
half lives are normally written in years
it goes likt this

$A=P( \frac{1}{2})^ \frac{t}{h}$
A=final amount
P=initila amount
t=time elapsed
h=time half life is (same units as time)

let's use days as the units of time


we are given
initial amount=1gram
final amount=0.3gram
time=4 days

so
$0.3=1( \frac{1}{2})^ \frac{4}{h}$
solve for h
$0.3=( \frac{1}{2})^ \frac{4}{h}$
take the ln of both sides
$ln(0.3)=ln( \frac{1}{2})^ \frac{4}{h}$
$ln(0.3)=\frac{4}{h}ln( \frac{1}{2})$
times both sides by h
$hln(0.3)=4ln( \frac{1}{2})$
divide both sides by ln(0.3)
$h= \frac{4ln( \frac{1}{2}) }{ln(0.3)}$
use calculator
h=2.30287
3 sig figs
h=2.303 days