ESS 109C Isotope Geochemistry Notes

April 16, 2007

 

The Decay Equation

 

  1. Class notes & homeworks are now available online Ð
                                  http://www2.ess.ucla.edu/~schauble/Isotope_geochemistry/
  2. 2nd homework will be due next Monday (4-23-07).
    I will be traveling Thursday afternoon Ð Sunday, email for help.

  3. Spontaneous radioactive decay
    1. In general, decay reactions are not sensitive to temperature, pressure, or chemistry at geochemically-relevant conditions.
    2. Decay is effectively a random, probabilistic process. All nuclei of a particular isotope have the same probability of decaying in a given period of time.

                                                                  i.      l = P(decay)/sec, called the decay constant

                                                                    ii.      t = 1/l is the mean lifetime (reciprocal of the probability of decay).

                                                                      iii.      Averaging over a large number ÒNÓ of radioactive nuclei, it follows that
dN/dt = ÐlN ˆ dN/N = Ðldt

                                                                     iv.      If each decay produces one stable ÒdaughterÓ nucleus, and N1 is the number ÒdaughterÓ nuclei produced, then
dN1/dt = ÐdN/dt = lN

                                                                   v.      If N0 is the starting number of radioactive parent nuclei, then

INT{dN/N} = ÐlINT{dt}

ln(N) = Ðlt + C

-- and since N=N0 at t=0,

ln(N) = Ðlt + ln(N0)

Ð take exp of each side to simplify,

exp(ln(N)) = N = exp( Ðlt) x exp(ln(N0)) = N0exp( Ðlt)

N = N0exp( Ðlt)

                                                                     vi.      Sketch evolution with time

    1. Relationship to half-life:

                                                                  i.      At t1/2, N = N0/2

N = N0exp( Ðlt1/2) = N0/2

exp( Ðlt1/2) = 1/2

 Ðlt1/2 = ln(1/2) Å Ð0.693

t1/2 Å 0.693/l = 0.693 t

    1. Growth of daughter nuclei (assuming one daughter per decay)

                                                                  i.      dN1/dt = ÐdN/dt

                                                                    ii.      N1 Ð N1(t=0) = Ð(N Ð N0) = N0 Ð  N0exp( Ðlt)

=  N0(1 Ð exp( Ðlt)) = D*

    1. Typically, however, we can only measure the present abundance of parents and daughters, and need to work back to find N0 and N1(t=0)

                                                                  i.      N = N0exp( Ðlt)

N0 = N/exp( Ðlt) = Nexp( lt)

Thus D* = Nexp( lt)(1 Ð exp( Ðlt))

= Nexp( lt) Ð N = N(exp( lt) Ð 1)

D* =  N(exp( lt) Ð 1)

                                                                    ii.      Still donÕt know the initial concentration of daughter!

  1. Ideals for radioactive geochronology
    1. The sample should have formed during the event of interest, without gaining or losing parent or daughter atoms afterwards, except by spontaneous radioactive parent-daughter conversion.
    2. The decay constant is known, and independent of time, temperature, pressure, etc.
    3. The initial concentration of daughter nuclei can be determined.
    4. The present concentrations of parent and daughter nuclei can be determined.
  2. ÒNaiveÓ geochronometry
    1. Early geochronometry got around uncertainty in initial concentration of daughter product by choosing crystals thought likely to have formed with essentially no daughter nuclei present.

                                                                  i.      Guess from homework?

                                                                    ii.      K-Ar dating.