Phosphorus isotope

Question

 Phosphorus-32 has a half-life of 14.3 days. How many milligrams of phosphorus-32 remain after 71.5 days if you start with 4.00 mg of the isotope

Answer 1

FR = 0.5ⁿ

FR = fraction remaining = ?

n = number of half lives that have elapsed = 71.5 days / 14.3days = 5 half lives

FR = 0.5⁵

FR = 0.03125 which is the fraction remaining

Amount remaining = 0.03125 x 4.00 mg = 0.125 mg remains

Answer 2

4 mg to start

half life = 14.3 days

after 71.5 days what's left?

71.5/14.3 =5

4 to 2 in one half life

2 to 1 in two

1 to 1/2 in three

1/2 to 1/4 in four

1/4 to 1/8 mg in five

1/8 mg or 0.125 mg is left after 71.5 days

4 x (1/2)^5 = 4 x 1/32 = 4/32 = 1/8

or use an exponential function for continuously compounded decay

A = Pe^rt

A= 4e^r(71.5)

2=4e^r(14.3)

1/2 e^14.3r

ln.5 = 14.3r

r = ln.5/14.3 = about -0.0485

A= 4e^-.0485(71.5)

= 1/8 gram