chemistry

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!
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?

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  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 :)

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  2. The same math but in slightly different format.
    k=0.693/t1/2
    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/t1/2*t]

    Now substitute your numbers
    ln(2/0.25) = [0.693/t1/2*42]
    ln 8 = [0.693*42/t1/2]
    2.079 = 29.1/t1/2
    t1/2 = 29.1/2.079 = 13.997 which rounds to 14 days.

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