Chapter 23 Worksheet 3 (ws23.3)
Kinetics of radioactive decay and dating objects
Radioactive decay follows first order kinetics. The first order rate law is:
rate  activity  A 
 dN
 kN
dt
where N is the number of radioactive nuclei present at time t and k is the first order rate constant (time-1)
The rate of radioactive decay is also called activity (A)
Units of activity
1 cpm = 1 count per minute (this is what you measure with a Geiger counter)
1 dpm = 1 disintegration per minute = cpm / efficiency of counter
1 Ci (curie) = 2.2 x 1012 dpm (activity of 1g of radium, named in honor of Marie and Pierre Curie)
1 Bq (becqerel) = 1 dps (SI unit for activity but rarely used in practice. Named for Henri Becquerel.)
The integrated first order rate law
Integration of the first order rate law and evaluating at time zero gives:
ln N = -kt + ln No (a plot of ln N vs. t yields a straight line with slope of k)
Rearranging gives several useful expressions:
N/No = e-kt
[the fraction of radioactive nuclei (or activity) remaining at time t]
N = Noe-kt
[the number of radioactive nuclei (or activity) remaining at time t]
 ln( N / N o )
(time at which a given fraction of radioactive nuclei remain)
k
N is the number of radioactive nuclei remaining at time t
N0 is the number of radioactive nuclei at time zero
k is the first order rate constant (unit is reciprocal time, s-1, min-1, hr-1, yr-1, etc)
t
(note that N/No = A/Ao = n/no = m/mo where n is moles and m is mass)
Half-life (time at which ½ of the radioactive nuclei remain)
When t = t½ (after one half-life) then N/ No = 0.5 so:
t1/ 2 
 ln( 0.5) 0.693

k
k
or
k
0.693
t1/ 2
1
Calculations Based on half-life
If we know t½ (and thus k), N, and No, then we can calculate the time that has elapsed since time zero
(BUT WHAT IS TIME ZERO??).
 N 
t
1  N 
   1/ 2 ln 

t   ln 
k  No 
0.693  N o 
(REMEMBER: N/No = A/Ao = n/no = m/mo)
WHENEVER YOU DO A HALF-LIFE PROBLEM ALWAYS ASK YOURSELF:
“HOW MANY HALF-LIVES HAVE PASSED?”
Carbon-14 dating of organic material (t½ = 5715 yr)
Carbon 14 is constantly produced in the upper atmosphere by neutron capture:
14
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N  01n146C 11p
Carbon-14 dating is based on two assumptions:
1. The ratio of carbon-14 to carbon-12 in the atmosphere has remained relatively constant for at least
50,000 years (~ 10 half-lives).
2. Since living organisms exchange carbon with the environment, they maintain the same ratio
throughout their lives.
Once an organism dies, then there is no more carbon exchange and the amount of carbon-14 begins to drop
due to radioactive decay (the half-life of carbon-14 is 5715 yr). The amount of carbon-14 remaining in an
organic sample, is a measure of the time elapsed since the organism died. (Thus, time zero is the moment
of death!) This dating method is accurate up to about 50,000 years (~ 10 half-lives). After this length of
time, the radioactivity is too low to be measured accurately.
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1. Carbon-14 decays by beta emission. Write the balanced nuclear equation.
2. If the activity of carbon-14 is ¼ that in a living organism. How long ago did the plant die? (How many
half-lives?)
3. What are the upper and lower limits on the age of an object that can be determined by this technique?
(Assume that the activity of 14C can be measure with an accuracy of ±1%.)
4. A wooden object from an archeological site is subjected to radiocarbon dating. The activity of the sample
due to carbon-14 is measured to be 11.6 dps (disintegrations per second). The activity of a carbon sample of
equal mass from fresh wood is 15.2 dps. What is the age of the sample?
A = ________________
Ao = _______________
t½ = ________________
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Lead-206 dating of rocks
Uranium-238 decays to lead-206 by a series of steps (see fig. 23.3). The half-life for 238U is 4.5 x 109 years.
Thus, the ratio of lead-206 to uranium-238 in a uranium-containing mineral is a measure of the time since
the mineral was formed. A very sensitive and precise method (mass spectrometry) exist for measuring the
mass of these two isotopes in a rock sample.
1. This method assumes that all of the lead-206 in a rock came from decay of uranium-238. Is this
assumption always valid? How could you tell if some of the lead-206 came from a different source? (No
other radioactive decay process produces lead-206.)
2. What are the youngest rocks that could be accurately dated by this technique?
3. A rock contains 0.257 mg of lead-206 for every milligram of uranium-238. How old is the rock? To
solve this problem, we need to know the amount of uranium-238 that was present when the rock was formed
(No). This is the amount of uranium-238 present now plus the amount that decayed.
How many moles of uranium-238 decayed to give 0.257 mg of lead-206?
How many moles of uranium-238 were present when the rock was formed?
Calculate the age of the rock.
N = _____________
No = ____________
t½ = ______________
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