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The decay of a radioactive
material obeys first order
kinetics. This means that the following equations all apply to radioactive
decay:
- Rate = k*X
- ln(X0) - ln(X) =k*t
- k = 0.693/t1/2
Here, X is the amount of radioactive
material, k the 1st order rate constant, and t1/2 the half life for
the material. You can also define an activity
which is a measure of how many readioactive decays occur each second.
Example: 146C is a radioactive isotope of carbon
with a half life of 5720 years. If you start with 1.00 mole of carbon-14 and
wait 1500 years, how much is left?
Solution: This requires the last two equations in the list above. The
second will give us the amount at any time, but we need the rate constant which
we aren't given. The third equation gives a relationship between half life and
k:
- k = 0.693/t1/2
- k = 0.693 / 5720 yr
- k = 1.21*10-4/ yr
Now that we have the rate constant,
simply use the second equation: our initial amount X0 is 1 mole, so
- ln(X0) - ln(X) =k*t
- ln(1.0) - ln(X) = 1.21*10-4 yr*1500 yr
- 0 - ln(X) = 0.181
- ln(X) = -0.181
- X = 0.834 mole
About a sixth of the carbon has decayed. |