emit less and less radiation. He decided to
measure this decrease in radiation and discovered a simple law of
exponential decay to describe the decrease. The rate of production of radioactive particles depends only on the number of radioactive atoms present. The number of atoms present in a sample decreases according to an exponetial decay and therefore the rate of decay decreases according to the same law.
N
—— = e^{kt} Equation 1
N_{0}
N = no. of atoms left, N_{0} = no. atoms you
started with, e ≅ 2.7183 (a mathematical constant),
k = a rate constant specific to a substance, and t = time
N/N_{0} is the fraction of the material that is left
after some amount of time has passed (for ex., 0.50 or 50%).
The e in the formula above is the base of the
natural logarithm and is a standard constant of mathematics. It
is approximately 2.7183. If you have a scientific calculator, there is
probably a button for this constant and for the logarithm (LN) which is based on it. Use this formula to find the
fraction of a radionuclide that has not decayed (that is, the fraction remaining, N/N_{0}), given the time
and the rate constant. With suitable mathematical manipulations it is possible to use this same formula to calculate the elapsed time using the fraction remaining (ln(N/N_{0}) = –kt). The fraction remaining may also be expressed in terms of relative rates of decay. The rate of decay depends on the number of atoms so as the number of atoms decreases, so does the rate of decay.
The halflife of a radionuclide (a radioactive isotope) is the
amount of time it takes for half of a sample of the material to
undergo radioactive decay. On an atomic level, radioactive decay
is a random event that can happen at any time. It cannot be
hurried up or slowed down. But a large collection of radionuclides will follow a strict statistical law consistent with the random decays that gives an overall constant rate. The way this rate changes over time can best be understood using Equation 1 but it can also be expressed in terms of halflives. Using Equation 1 it is possible to define the period of time needed for a sample to reach the point where the rate has decreased to half of its original value (which is the moment when half of the original sample remains or N/N_{0} = 0.5). The time needed to reach this point is called the halflife. The rate constant that governs the decay of a particular radionuclide can be used to calculate the length of a halflife.
t_{½} = ln(2)/k Equation 2
where t_{½} is the halflife and k is the
rate constant from the previous equation
The primary way to calculate the time elapsed given a certain fraction remaining is to use Equation 1. There is also a way to figure out how old something is by using the number of halflives. At one halflife, N/N_{0} = 0.5 and n (the number of halflives) equals one. At this time one halflife has elapsed. When the fraction remaining is 1/4 then n = 2 and the total elapsed time equals two half lives. The number of halflives can be any number and once you have n you can simply
multiply it by the length of the halflife to find the age of an
object using this formula: t_{total} = nt_{½}. Equation 3 is the fraction remaining in terms of the number of halflives.
ln(N/N_{0})
N/N_{0} = (1/2)^{n} or n = —————————— Equation 3
ln(1/2)
n = the number of halflives
An isotope important to the science of archaeology is Carbon14
(^{14 }_{6}C). It is created in the atmosphere by the action of cosmic rays which enables the neutron bombardment
of nitrogen nuclei. Living things take up this carbon14 and make
it part of themselves while they are alive. The halflife of
carbon14 is 5,715 years. It is possible to measure the amount of
carbon14 in an object and compare it to how much carbon14 there
must have been to begin with. This data can be given in the form
of a fraction of the original amount of carbon14. Using this information it is possible to calculate the amount of time that has elapsed since a living thing died, such as a piece of wood carved into a cup.
page break
Nuclide 
Halflife 
k 
Decay Mode 
^{14 }_{ 6}C 
5,715 yr 
1.213 × 10^{4} 1/yr 
β^{} 
^{40}_{19}K 
1.248 × 10^{9} yr 
5.553 × 10^{10} 1/yr 
β^{} 
^{87}_{37}Rb 
4.97 × 10^{10} yr 
1.394 × 10^{11} 1/yr 
β^{} 
^{238}_{92}U 
4.468 × 10^{9} yr 
1.551 × 10^{10} 1/yr 
α 


Refresher on Logarithms
ln(e^{x}) = x
e^{ln(M)} = M


ln(M·N) = ln(M) + ln(N)
ln(M/N) = ln(M) – ln(N)
ln(M^{x}) = x·ln(M) 

(a^{b}· a^{c} = a^{b + c})
(a^{b} ÷ a^{c} = a^{b – c})
((a^{b})^{c} = a^{b·c})


Exercises
Answer the following questions of perform the calculations required, showing work.
 Write a balanced nuclear equation for
the decay of each of the isotopes in the table given above.
 When using the decay rates of naturally occurring radioactive isotopes to measure the age of objects there are some natual limits. These limits depend on the isotope being used.
 What fraction of any isotope remains after nine halflives?
 If lab techniques are unable to measure the amount of an isotope that remains after it reaches this level of decay, then approximately what is the oldest age that can be established using carbon14?
 What is the oldest object whose age can be established by measurements of the isotope potassium40? Assume that it can be detected down to the amount that would remain after 9 halflives.
 Although uranium238 has a much longer halflife than carbon14 it is not used to find the ages of fossils that come from the Mesozoic era (the age of dinosaurs). It is only used to find the ages of rocks. Explain.
 Compare 10 μg samples of carbon14,
potassium40, rubidium87, and uranium238. After one million years, all
of them have decayed to some extent. Put them in order from most
decayed to least decayed without doing any calculations. Explain your reasoning.
 The risks of exposure to a radioactive material depends on two things: how many atoms of the material there are and how rapidly it decays. Based on the data in the table at the bottom of the first page of this packet, place the isotopes in order from least risk to greatest risk for exposure to a sample with the same number of atoms. Explain your reasoning.
 How much ^{14 }_{ 6}C
would remain after 5 halflives? How old would a sample be that
had gone through five halflives?
page break
 Use algebra to show the steps necessary to find the relationship between the halflife and the rate constant starting from the basic decay equation (N/N_{0} = e^{–kt}). Consider that when one halflife has elapsed exactly one half of the original amount of a radioactive isotope remains in its original form and that fractions such as 1/2 can be written using exponents: 1/2 = 2^{–1}.
 The rate constant for a certain
radioactive nuclide is 1.0 × 10^{3} 1/hr. What is
the halflife of this nuclide in hours?
 In 1988, three teams of scientists found
that the Shroud of Turin, which was reputed to be the burial
cloth of Jesus, contained 91% of the amount of carbon14 contained in
freshly made cloth of the same material. How old is the Shroud
according to the data? What year do you suppose it was actually made?
 A wooden artifact from a site in Europe has a ^{14}C activity of 39.0 counts per minute. By comparison, the expected rate of decay for an object at the present time is 58.2 counts per minute. Since the rate of decay decreases in exactly the same way as the number of atoms, a ratio of decay rates can be used in the same way as a ratio of the number of atoms. What is the age of the wooden artifact?
 A rock containing potassium minerals is
analyzed. The sample shows that only 75% of the original
^{40}_{19}K is still
present. How old is the sample?
 Radioactive copper64 decays with a halflife of 12.8 days.
 What is the value of k in 1/days?
 A chemist obtains a fresh sample of
^{64}Cu and measures its radioactivity. She then
determines that to do an experiment, the radioactivity cannot
fall below 25% of the intial measured value. How long does she
have before the ^{64}Cu is too depleted to work
with? (There are two ways to do this calculation; show work for both).
 How long will it be until the amount of ^{64}Cu remaining is only 9%?
 The first atomic explosion was detonated
in the desert north of Alamogordo, New Mexico, on July 16, 1945.
What fraction of the strontium90 (t_{½} = 28.8
years) originally produced by that explosion still remains as of
December 31, 2018?
 Iodine131 is used in the diagnosis and treatment of thyroid
disease and has a halflife of 8.02 days. If a patient with
thyroid disease consumes a sample of Na^{131}I containing
10 μg of ^{131}I, how long will it take for the amount
of ^{131}I to decrease to 0.1 μg?
 Radioactive dating is accomplished by establishing the ratio by numbers of atoms between the initial nucleus and the final nucleus to which it decays. A rock contains 0.688 mg of
^{206}Pb for every 1.000 mg of ^{238}U. Assuming
that no lead was originally present, that all the
^{206}Pb formed over the years has remained in the rock,
and that the number of nuclides in the intermediate stages of
decay between ^{238}U and ^{206}Pb is negligible,
calculate the age of the rock. (The exact mass of ^{206}Pb is 205.974449 g/mol and the exact mass of ^{238}U is 238.050784 g/mol).