# The Atom

## Half Life and Radiometric Dating


### Section Key Terms


### Half-Life and the Rate of Radioactive Decay

Unstable nuclei decay. However, some nuclides decay faster than others. For example, radium and polonium, discovered by Marie and Pierre Curie, decay faster than uranium. That means they have shorter lifetimes, producing a greater rate of decay. Here we will explore half-life and activity, the quantitative terms for lifetime and rate of decay.
Why do we use the term like half-life rather than lifetime? The answer can be found by examining , which shows how the number of radioactive nuclei in a sample decreases with time. The time in which half of the original number of nuclei decay is defined as the half-life,  
  . After one half-life passes, half of the remaining nuclei will decay in the next half-life. Then, half of that amount in turn decays in the following half-life. Therefore, the number of radioactive nuclei decreases from N to N / 2 in one half-life, to N / 4 in the next, to N / 8 in the next, and so on. Nuclear decay is an example of a purely statistical process.
The following equation gives the quantitative relationship between the original number of nuclei present at time zero  
   and the number  
   at a later time t
where e = 2.71828... is the base of the natural logarithm, and  
   is the decay constant for the nuclide. The shorter the half-life, the larger is the value of  
 , and the faster the exponential  
   decreases with time. The decay constant can be found with the equation

### Activity, the Rate of Decay

What do we mean when we say a source is highly radioactive? Generally, it means the number of decays per unit time is very high. We define activity R to be the rate of decay expressed in decays per unit time. In equation form, this is
where  
   is the number of decays that occur in time  
  .
Activity can also be determined through the equation
which shows that as the amount of radiative material (N) decreases, the rate of decay decreases as well.
The SI unit for activity is one decay per second and it is given the name becquerel (Bq) in honor of the discoverer of radioactivity. That is,
 
 

Activity R is often expressed in other units, such as decays per minute or decays per year. One of the most common units for activity is the curie (Ci), defined to be the activity of 1 g of 226Ra, in honor of Marie Curie’s work with radium. The definition of the curie is
or  
   decays per second.

### Radiometric Dating

Radioactive dating or radiometric dating is a clever use of naturally occurring radioactivity. Its most familiar application is carbon-14 dating. Carbon-14 is an isotope of carbon that is produced when solar neutrinos strike  
   particles within the atmosphere. Radioactive carbon has the same chemistry as stable carbon, and so it mixes into the biosphere, where it is consumed and becomes part of every living organism. Carbon-14 has an abundance of 1.3 parts per trillion of normal carbon, so if you know the number of carbon nuclei in an object (perhaps determined by mass and Avogadro’s number), you can multiply that number by  
   to find the number of  
   nuclei within the object. Over time, carbon-14 will naturally decay back to  
   with a half-life of 5,730 years (note that this is an example of beta decay). When an organism dies, carbon exchange with the environment ceases, and  
   is not replenished. By comparing the abundance of  
   in an artifact, such as mummy wrappings, with the normal abundance in living tissue, it is possible to determine the artifact’s age (or time since death). Carbon-14 dating can be used for biological tissues as old as 50 or 60 thousand years, but is most accurate for younger samples, since the abundance of  
   nuclei in them is greater.
One of the most famous cases of carbon-14 dating involves the Shroud of Turin, a long piece of fabric purported to be the burial shroud of Jesus (see ). This relic was first displayed in Turin in 1354 and was denounced as a fraud at that time by a French bishop. Its remarkable negative imprint of an apparently crucified body resembles the then-accepted image of Jesus. As a result, the relic has been remained controversial throughout the centuries. Carbon-14 dating was not performed on the shroud until 1988, when the process had been refined to the point where only a small amount of material needed to be destroyed. Samples were tested at three independent laboratories, each being given four pieces of cloth, with only one unidentified piece from the shroud, to avoid prejudice. All three laboratories found samples of the shroud contain 92 percent of the  
   found in living tissues, allowing the shroud to be dated (see ).

### Section Summary

1. Radioactive half-life is the time it takes a sample of nuclei to decay to half of its original amount.
2. The rate of radioactive decay is defined as the sample’s activity, represented by the equation  
  .
3. Knowing the half-life of a radioactive isotope allows for the process of radioactive dating to determine the age of a material.
4. If the half-life of a material is known, the age of the material can be found using the equation  
  .
5. The age of organic material can be determined using the decay of the carbon-14 isotope, while the age of rocks can be determined using the decay of uranium-238.

### Key Equations


### Critical Thinking


### Test Prep Short Answer
