Radioactive Dating

The universe consists of many elements, all of which are made up of some combination of protons, neutrons and electrons.  As you surely remember from chemistry, the number of protons determines the element.

For example, carbon atoms have 6 protons in the nucleus.  Since protons are positively charged, a neutral carbon atom also has 6 electrons in orbits around the nucleus.

Atoms can't be this simple, however.  The positvely charged protons repel each other (like charges repel through the electromagnetic force) and so do not want to be close to each other; however, the protons also attract each other through the strong nuclear force.  But at the distances between protons in the nucleus, the repulsive forces are stronger than the attractive forces, and so a nuclues made only of protons would be unstable. This is where the neutron comes in.  The neutron increases the strength of the attractive strong nuclear force without adding more repulsive positive charges, thereby helping to moderate the repulsive force of the protons. Given enough neutrons, a nucleus with many protons can become stable.  Notably,  the neutrons do not change the chemical behavior of the atom.

A carbon atom will not hold together unless it has at least 6 neutrons (i.e., 11C does not exist because the repulsive force is too strong).  But we can have 12C, 13C and 14C. If we try to make 15C (with 9 neutrons), it falls apart  immediately also.  So there are three isotopes of Carbon that can exist in nature. (Their relative abundances are given below.)

12C and 13C are stable, essentially forever. 14C, however, is unstable.  Eight neutrons is just too much of a good thing when there are only 6 protons.  And so eventually, the 14C atom undergoes a change to 14N, in which one neutron decays, or falls apart, into a proton plus an electron.  The electron (a beta particle) flies out with a tremendous amount of energy, collides  with something (turning it's energy of motion into heat).  This is one form of radioactive decay (called beta decay).

A second kind of radioactive decay, called alpha decay, occurs when a nucleus splits into two pieces, one small "alpha" particle with two protons and two neutrons, i.e., a helium nucleus, and one big particle (the rest). The escaping alpha particle collides with something and, again, the kinetic energy is deposited as heat.  Thus, we can think of both beta and alpha decays as heat sources.

A third from of radioactivity is electron capture.  In this process, an electron combines with a proton to form a neutron.

Why do we worry about radioactivity?  Radon gas is radioactive, and radon itself comes from the decay of radioactive uranium. The uranium is locked into rocks but when the uranium becomes radon, the radon, being a gas, bubbles out of the rocks and into your house.  If you inhale the radon gas into your lungs, and that radon atom has the bad sense to undergo decay while in your lungs, that alpha particle will collide with your lungs. That could damage your DNA and, if you're unlucky, the resultant damage could produce cancer.  Plutonium has the same problem.

Before we get into the details of radioactive half-lives, let me give you some numbers to chew on.  THE EARTH IS WARM (witness volcanoes; the inside of the Earth flows, generating Earthquakes).  Where does the heat inside the Earth come from?

The heat flow from the whole earth is 3.2 x 1013 Watts, i.e., the equivalent of 320 billion 100-watt light bulbs, turned on all day, or about sixty 100-watt bulbs per person, turned on all the time.  This is 10 times more energy than is released, in total, by Earthquakes.  This is more heat than could be generated simply if the Earth were warm and were cooling off.  If this were the case, the Earth would have been warmer in the past and, in the not too distant past, the whole Earth would have been molten.  Thus, the Earth requires an internal heat source, radioactivity.

There are many elements that are radioactive or have  radioactive isotopes.

Half-lives of Important Radioactive Elements
 

Parent Daughter
Half-Life (yrs)
process
23592U (uranium) 20782Pb (lead)
0.713 Billion
23892U 20682Pb
4.5 Billion
24194Pu (plutonium) 20983Bi (bismuth)
2.4 Million
23290Th (thorium) 20882Pb
13.9 Billion
4019K (potassium) 4018Ar (argon)
1.25 Billion
electron capture
4019K 4020Ca (calcium)
1.4 Billion
beta decay
8737Rb (Rubidium) 8738Sr (strontium)
48.8 Billion
beta decay
14762Sm (samarium) 14360Nd (neodymium)
106.0 Billion
alpha decay
2613Al (aluminum) 2612Mg (magnesium)
700,000
electron capture
146C (carbon) 147N (nitrogen)
5,730
beta decay

Concept of half-lives: experiment with coins (or M&Ms or student birthdays).

The M&M experiment: This is a very dangerous experiment.  Don't perform it without  special clothing and supervision.  Also, a glass of milk nearby is a good idea.  (Flip M&Ms; eat half that is letter-side up; repeat.).

Why does radioactivity work this way?  I don't know.   But it does.  And you can depend on it. (iodine, barium used in medical work, e.g.; plutonium, uranium used for nuclear energy; we can predict exactly the rate at which they will produce energy but we cannot predict which atoms will decay).

What about carbon?

If you look at a periodic table of the elements, you find that the atomic mass number of carbon is 12 but the atomic weight is 12.011.  This means that the "average" carbon atom is heavier than 12 a.m.u. since there are some 13C  and 14C  atoms around. 12C and 13C  are produced naturally in stars and exist quite abundantly all over the universe.  But since 14C is unstable, even if 14C  forms in stars it won't make it to the Earth since it has such a short half-life.

But cosmic rays (mostly high energy neutrons) collide with 14N at heights of 1.6 km in Earth's atmosphere.  In some of these collisions, the neutron knocks a proton out of the nitrogen nucleus.  Thus, a proton is replaced with a neutron and

147N --> 146C.
(If it knocks out another neutron, no change occurs.)  Since 14C behaves like 12C, these atoms are incorporated into a piece of broccoli which you eat and you then have 14C in you.

The fraction of 14C in you is very nearly unchanging.  As some 14C decays, more 14C is added.  As long as you are alive (and eating broccoli), the ratio of 12C/13C /14C is  unchanged.

12C  = 98.89% (9,889 out of 10,000)
13C  =   1.11%  (111 out of 10,000)
14C =  0.000 000 000 10% (or 1 out of every 1012, or one part in a trillion)

i.e. , 12C/14C = 1012

but there are 6.02 x 1023 atoms in 12 gm (1 mole) of Carbon.

if 1% of a 100 kg person (me) is carbon
   then Mcarbon (in me) = 0.01 x 100 kg = 1 kg = 1000 gm
   if we then convert gm to moles we find:
   total moles carbon, in me = 1000 gm / (12 gm/mole) = 83 moles

   thus, there are 83 x 6.02 x 1023 = 5 x 1025 C atoms in me

and so there are 5 x 1025 / 1012 = 5 x 1013 14C atoms in me.  How many will be left in
1 year, 10 years, 100 years after I die (and stop eating broccoli)?
 

TIME
Number 14C remaining
% 14C remaining
Number 14C decayed
today
5 x1013
100%
0
1 yr
4.9994 x 1013
 99.988%
6,000,000,000 [16,440,000/day; 190/sec]
10 yr
4.9940 x 1013
99.879%
6.05 x 1010
100 yr
4.9399 x 1013
98.798%
5.98 x 1011
1000 yr
4.4303 x 1013
88.606%
5.67 x 1012 
10,000 yr
1.4914 x 1013
29.829%
3.50  x 1013
50,000 yr
1.1800 x 1011
0.236%
4.98882 x 1013
100,000 yr
 2.7848 x 108
0.00000056%
4.99997 x 1013 

You can evaluate this yourself using a carbon dating calculator.  And you can read more about carbon dating at radiocarbon WEB-info or carbon dating (this includes a good discussion about types of materials that cannot be dated using carbon dating techniques).

Age of Moon rocks:

40Ar / 40K = 10.3 measured in moon rocks.
Argon, as a gas, would have bubbled out of the lunar rocks at any time that the rocks were heated or melted; thus, Argon only begins to accumulate in the rock after the rock solidifies.  The potassium is locked into the crystal lattice structure of the rocks, once the rock solidifies.  Now, if 40K decays to 40Ar, the Argon remains trapped in the crystal lattice, unable to escape (until/unless the rock is melted). Thus the 40Ar/40K ratio gives us the age since the rock last melted, or the crystallization age or solidification age of the rock.

   time          40Ar/40K ratio
------      ----------
 0                            0/16             when no gas would be retained in rock)
1.25 BY                8 / 8  = 1      (half of the 8 remaining K decay)
2.50                      12 / 4 = 3      (half of the 4 remaining K decay)
3.75                      14 / 2 = 7      (half of the 2 remaining K decay)
5.00                      15 / 1 = 15

so the age of the Moon rocks must be between 3.75 and 5.00 BY, from our very crude estimates. If we do this more accurately, we find an age of 4.37 BY.

Radioactive-Age Based Chronology of the Solar Sytem
 

oldest rocks yet identified on Earth 4.0 billion years reference article
oldest grains yet identified on Earth 4.3-4.4 billion years Nature 11 Jan 2001: News&View, PaperI, PaperII
oldest moon rock identified 4.44 billion years
oldest meteorites 4.562 billion years



A mathematically rigorous treatment of radioactivity

The decay rate is given by:

delta-N / delta-t = - lambda * N

where

N is the number of atomic nuclei of the substance in question
delta-N is the change in the number of atomic nuclei
delta-t is the change in time
lambda is the decay rate

note that this says that the rate of decay is proportional to the number of particles.
 

We can rewrite this equation as:
 
delta-N / N = - lambda * delta-t
which is solved as a simple intergral:
N(t) = N0e-lambda*t

where

N0 is the original number of nuclei of type N.

If we ask, "when is N(t) = 0.5 N0?", we find the half-life:
0.5 = e-lambda*T
2 = e+lambda*T
T1/2 = ln(2) / lambda = 0.693 / lambda

Questions to test your understanding of radioactivity


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