How do you find the half-life of an element? Please explain and show me how to do this. I know the answer is 14.3 days, but I don't know how to work it. thank you!

Question

the problem is:
After 42 days, a 2.0-g sample of phosphous-32 contains only 0.25g of isotope. What is the half-life of phosphorus-32?

Answer 1

mass at t = mass at beginning times e^-t/T
where T is some decay time
m = mo e^-t/T
now when will m = 1/2 mo ?
1/2 = e^- thalf/T where I am calling thalf the half life

-ln (1/2) = thalf/T
but -ln(1/2) = .6931
so
T = thalf/.6931
NOW for this problem
.25 = 2 e^-42/T
ln (.125) = -42/T
T = 20.2 days
so
thalf = .6931(20.2 = 14 days

rough check
after 14 days I have (1/2)2 = 1 gram
after 28 days I have (1/2)1 = .5 gram
after 42 days I have (1/2)(.5)=.25 grams
well, amazing, it worked :)

Answer 2

The same math but in slightly different format.
k=0.693/t_1/2
ln(No/N) = kt where No = beginning grams, N = ending grams, k is the constant from equation 1 and t is the time.
substitute from equation 1 into k in equation 2 to obtain
ln(No/N) = [0.693/t_1/2*t]

Now substitute your numbers
ln(2/0.25) = [0.693/t_1/2*42]
ln 8 = [0.693*42/t_1/2]
2.079 = 29.1/t_1/2
t_1/2 = 29.1/2.079 = 13.997 which rounds to 14 days.