After 8 days, a sample of bismuth-210 decayed to 33% of its original mass.
(a) Determine the half-life of this element.
(b) Determine the mass remaining after 12 days.
a.) Recall the formula for radioactive decay,
m(t)=m0e−rt which r=ln2h
Where,
m(t)= the mass remaining at time t
m0= initial mass
r= rate of decay
t= time
h= half life
If the mass remaining is 33% of the original mass, then
m(t)=0.33m0 ,so0.33m0=m0e−(ln2h)(8)Divide both sides by m00.33=e−(ln2h)(8)Take ln of each sideln(0.33)=−(ln(2)h)(8)Recall lne=1h=−8ln2ln(0.33)Multiply h both sidesh=5.0022 days or 5 days
The half life of bismuth −210 is approximately 5 days.
b.) If t=12 days, then
m(12)=m0e−r(12);where r=ln2h=ln25m(12)=m0e−(ln2h)(12)m(12)=0.1895m0
It shows that after 12 days, the mass remaining will be approximately 19% of the original mass.
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