Radioactive Decay

Differential Equation Applications

Radioactive Decay

A piece of human bone is found at an archeological site. If 10% of the original amount of radioactive carbon-14 was present, estimate the age of the bone. The half-life of C-14 is 5600 years. (Half-life is the the time it takes for half of it to decay.)

Write down what is known.

     We don’t know what the original amount is, so we’ll call it N_o.  

        t = 0      N = N_o

        And from the definition of half-life:

        t = 5600      N = N_o/2

        Also:      dN/dt = kN

    And what is it that we are looking for?

        t = ?      when N = 0.1N_o

Now, separate variables and integrate.

   dN/N = k dt

    ln N = kt + C

    N = C e^kt

Solve for the constants:

Use     t = 0      N = N_o

    N_o = Ce^k(0)

    C = N_o

    Now use     t = 5600     N = N_o/2

    N_o/2 = N_oe^k(5600)

    k = -0.0001238

    So the equation is:         N = N_oe^-.0001238t

We can now solve for time:

Use  t = ?      when N = 0.1N_o

    0.1N_o = N_oe^-.0001238t

    t = 18,600 years

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