Half Life

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introduction

N = N02−t/τ

radiocarbon dating

Every time a living being dies a stopwatch starts ticking.
Death starts the stopwatch. Science can read it.

Radiocarbon dating is used to determine the age of previously living things based on the abundance of an unstable isotope of carbon. The isotopic distribution of carbon on the earth is roughly 99% carbon 12  (with 6 protons and 6 neutrons) and 1% carbon 13 (with 6 protons and 7 neutrons). These isotopes are stable, which is why they are with us today, but unstable isotopes are also present in minute amounts. About one carbon atom in a trillion (1012) contains a radioactive nucleus with 6 protons and 8 neutrons -- carbon 14. This rare, unstable isotope is produced in the upper atmosphere from ordinary nitrogen 14.

In earth's upper atmosphere, on the edge of what is commonly called outer space, light atomic nuclei from unknown sources outside of our solar system traveling at speeds approaching the speed of light called cosmic rays rain down continuously. These highly energetic nuclear bullets wreak havoc on the atoms in the upper atmosphere: tearing electrons from their orbitals and setting them free, knocking neutrons and protons from the tight confines of the nucleus and setting them free, generating x-rays and gamma rays as they decelerate, and creating exotic particles like muons and pions directly from their excessive kinetic energy. These secondary cosmic rays are also highly energetic and will ionize atoms, transmute nuclei, and generate x-rays themselves. A secondary cosmic ray neutron of sufficient energy striking a common nitrogen 14 nucleus can force it to eject a proton.

This is the process by which all of the carbon 14 on the earth is produced. (Produced naturally to be more precise. More on that later.)

All organic material contains carbon. (By definition, organic chemistry is the chemistry of carbon compounds.) Plants absorb 14C like they absorb other isotopes of carbon -- through the respiration of carbon dioxide -- and then use this carbon to produce sugars, fats, proteins, and vitamins. Bacteria, fungi, and animals eat these plants and each other. In this way, atmospheric carbon is distributed throughout the web of life until every living thing has the same ratio of 14C : 12C as the atmosphere. (Well, nearly the same ratio. Plants and animals tend to favor lighter nuclei just a bit. Serious technicians know how to compensate for this preference when dating samples.)

With a half life of 5730 years, 14C decays by beta emission back into the 14N from which it originated.

Since living creatures are constantly swapping atoms with their environment, the abundance of 14C within them remains fixed. After death, however, no new radioactive carbon comes along to replenish that which has decayed and the abundance of 14C decreases. The ratio of 14C : 12C in a piece of living organic matter will be the same as it is in the atmosphere but larger than in a piece of dead organic material. A timber found in a home built 5730 years ago (one half life) would have half the 14C : 12C ratio that a person living today would. A discarded oyster shell from someone's dinner eaten 11,460 years ago (two half lives) would have one quarter the 14C : 12C ratio that a cotton shirt worn today would. A tusk from a mammoth that died 17,190 years ago (three half lives) would have one eighth the 14C : 12C ratio that a cardboard box manufactured today would. And so on. Death starts the stopwatch. Science can read it.

Radiocarbon Dating of a Hypothetical Organic Sample
age (half-lives) age (years) 14C (atoms) 12C (atoms) 14C : 12C (ppt)*
0 0 128 128 × 1012 1
1 5,730 64 128 × 1012 0.5
2 11,460 32 128 × 1012 0.25
3 17,190 16 128 × 1012 0.125
4 22,920 8 128 × 1012 0.0625
5 28,650 4 128 × 1012 0.03125
6 34,380 2 128 × 1012 0.015625
7 40,110 1 128 × 1012 0.0078125
*  In the United States and a few other countries 1012 is called a trillion. Thus, concentrations of 14C are often stated as parts per trillion (ppt).

Calculations of this sort are based on the assumption that the ratio of 14C : 12C in the earth's atmosphere has remained constant. As a first approximation one can assume this, but more accurate results must take into account fluctuations in the intensity of the cosmic rays entering the earth's upper atmosphere. These deviations were determined from the comparative dating of ancient tree rings (a field called dendrochronology) and the results were then compiled into a calibration curve. Current calibration curves go back as far as 26,000 years ago. The range of radiocarbon dating extends back to about 50,000 years. For items older than this, there isn't enough undecayed 14C left to measure the ratio reliably.

Beginning in the late 1950s, considerable amounts of anthropogenic (human-produced) 14C have been added to the atmosphere; mostly as a result of nuclear weapons testing. This activity reached its peak in the early 1960s when an atmospheric blast occurred somewhere on earth every two to three days. Nuclear bombs generate large numbers of high energy neutrons, which can in turn transmute nitrogen 14 nuclei into carbon 14 nuclei in exactly the same way as naturally occurring secondary cosmic rays. By 1965, atmospheric 14C concentrations were double their pre "atomic age" values. Radiocarbon dating in the future will have to include adjustments for these manmade deviations.


Number of Nuclear Weapons Explosions per Year [magnify]

potassium-argon dating

Potassium-argon dating is used to determine the age of igneous rocks based on the ratio of an unstable isotope of potassium to that of argon. Potassium is a common element found in many minerals. The isotopic distribution of potassium on the earth is approximately 93% 39K and 7% 41K. Since these values are only approximate, the total percent abundance of these two isotopes is not 100%, but 99.9883%. The remaining 0.0117% is 40K -- an unstable isotope with a half life of 1.26 × 109 years (1.26 billion years). Potassium 40 has three decay modes: beta decay, positron emission, and electron capture.

When 40K undergoes beta decay it transmutes into 40Ca -- the most abundant isotope of calcium. Since calcium is also very common in minerals, it is not possible to distinguish the 40Ca produced from the decay of 40K from the 40Ca present when the rock was formed. However, when 40K undergoes positron emission or electron capture it transmutes into 40Ar. Argon is an inert substance, which means that it basically will not combine chemically with other elements. It is also a gas over an extremely wide range of temperatures, which means that any 40Ar would escape while the rock was molten like carbon dioxide escaping from a glass of soda. After solidification, those 40Ar nuclei that appeared as a result of radioactive decay would be trapped by the crystal structure and accumulate as the mineral aged.

In a hypothetical mineral sample with an initial population of 64 40K atoms, the ratio of 40K : 40Ar would evolve as follows.

Potassium-Argon Dating of a Hypothetical Mineral Sample
age (half-lives) age (109 years) 40K (atoms) 40Ar (atoms) 40K : 40Ar
0 0 64 0 –
1 1.26 32 32 1 : 1
2 2.52 16 48 1 : 3
3 3.78 8 56 1 : 7
4 5.04 4 60 1 : 15
5 6.30 2 62 1 : 31
6 7.56 1 63 1 : 63

Potassium-Argon dating techniques have been used to date minerals covering the entire span of geologic history from 10 thousand to 3 billion years old.

other radioisotopic dating techniques

There are several other dating techniques that rely on the principle of exponential decay and half-life. Each has its own range of validity.

Radioisotopic Dating Techniques
technique range (years past) dateable items
lead 210 1 – 150 lake and ocean sediments, glacial ice
carbon 14 1 – 40,000 previously living things
uranium series 1 – 400,000 bone, teeth, coral, shells, eggs
potassium-argon 10,000 – 3 billion minerals, igneous rocks
uranium-lead 1 million – 4.5 billion minerals, igneous rocks
rubidium-strontium 60 million – 4.5 billion minerals, igneous rocks


Radioisotopic Dating Techniques Compared [magnify]

Summary

Problems

practice

  1. Write something.
    • Answer it.
  2. Write something.
    • Answer it.
  3.  
      [magnify]  
    When the Apollo astronauts landed on the moon they left behind equipment to monitor such things as the moon's internal temperature, its magnetic and gravitational fields, seismic activity caused by moonquakes and meteor impacts, and the moon's extremely thin atmosphere. Known collectively as the ALSEP (an acronym for Apollo Lunar Surface Experiment Packages) these devices were powered by a small and very simple nuclear power plant called a SNAP (an acronym for Space Nuclear Auxiliary Power). A SNAP generator is basically a can of plutonium 238 dioxide surrounded by radiator fins with thermocouples in between. The difference in temperature between the decaying plutonium (600 °C) and the the radiator fins (275 °C) produces a voltage across the thermocouples that can be used to generate electric current. The whole contraption is about the size of an office watebasket and with no moving parts is reliable for very long periods of time.

    The SNAP-27 activated by the Apollo XIV crew on 5 February 1971 used 3.8 kg of plutonium 238 dioxide and generated 73 W of power when first turned on. If 238PuO2 has a half life of 87.74 years and decays via the emission of 5.593 MeV alpha particles, determine …
    1. the initial power radiated by the plutonium fuel
    2. the initial efficiency of the generator
    3. the power of the generator on the anniversary of the Apollo XIV mission from the day it was turned on until its centennial.
    Solutions …
    1. Let's begin by determining the number of atoms in 3.8 kg of plutonium 238 dioxide. Recall that oxygen has an atomic mass of 16 and therefore that 238PuO2 has a molecular mass of 238 + 2(16) = 270.
           
      N =  (3800 g)(6.02 × 1023 atom/mol)  = 8.47 × 1024 plutonium atoms
      (270 g/mol)
           
      Half of this will have decayed after one half life giving an average activity of …
               
      A =  N/2  =  (9.00 × 1024 decay)/2  = 1.53 × 1015 Bq
      t (87.74 yr)(365.25 × 24 × 3600 s/yr)
               
      This is an average value, but we want the initial value. If the activity decreased linearly, the average value would be the average of the initial and final values. But this is radioactivity and radioactivity goes down exponentially not linearly. We will have to resort to calculus for this next step. The average value of a varying quantity is its definite integral divided by the interval.
                    τ            
      A =  1   ⌠
      ⌡      		 A dt =  1   ⌠
      ⌡      		 A02−t/τ dt =  1   A0τ  =  A0
      t τ τ 2 ln2 2 ln2
                    0            
      Rearrange to make the initial activity the subject of the equation and substitute numbers for variables.
       
      A0 = 2 ln 2 A = 2(ln2)(1.63 × 1015 Bq) = 2.12 × 1015 Bq
       
      Multiply this by the energy per decay (in joules) to get the power (in watts).
       
      P0 = A0E = (2.12 × 1015 decay/s)(5.593 × 106 eV/decay)(1.602 × 10−19 J/eV) = 1900 W
       
    2. Efficiency is the ratio of energy output to energy input.
               
      η =  Eout  =  73 J/s  = 3.8%
      Ein 1900 J/s
               
      Not very efficient, is it?
    3. Assume the power output decays in the same manner as the activity, use years as the unit of time, and generate a formula for a spreadsheet or data analysis application.
       
      P = P02−t/τ = 73 W × 2(1971 − t[yr])/(87.74 yr)
       
      The results are listed in the text file snap.txt.
  4. Write something.
    • Answer it.

numerical

  1. Problem

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