chemistry
posted by Ryan on .
How do you find the halflife 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!
the problem is:
After 42 days, a 2.0g sample of phosphous32 contains only 0.25g of isotope. What is the halflife of phosphorus32?

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 :) 
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.