Learning Objectives

Learning Objectives

By the end of this section, you will be able to do the following:

  • Explain radioactive half-life and its role in radiometric dating
  • Calculate radioactive half-life and solve problems associated with radiometric dating
Section Key Terms
activity becquerel carbon-14 dating
decay constant half-life radioactive dating

Half-Life and the Rate of Radioactive Decay

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 Figure 22.27, 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, t12t12. 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.

Tips For Success

A more precise definition of half-life is that each nucleus has a 50 percent chance of surviving for a time equal to one half-life. If an individual nucleus survives through that time, it still has a 50 percent chance of surviving through another half-life. Even if it happens to survive hundreds of half-lives, it still has a 50 percent chance of surviving through one more. Therefore, the decay of a nucleus is like random coin flipping. The chance of heads is 50 percent, no matter what has happened before.

The probability concept aligns with the traditional definition of half-life. Provided the number of nuclei is reasonably large, half of the original nuclei should decay during one half-life period.

The figure shows a graph and table with the readings. The exponential decay graph shows number of nuclides on the y axis and time (in half-lives) on the x-axis. The graph starts from 1000 at t=0 and decreases to half after every half-life. The range on the x-axis is from 0 to 10 half-lives.
Figure 22.27 Radioactive decay reduces the number of radioactive nuclei over time. In one half-life (t12t12), the number decreases to half of its original value. Half of what remains decays in the next half-life, and half of that in the next, and so on. This is exponential decay, as seen in the graph of the number of nuclei present as a function of time.

The following equation gives the quantitative relationship between the original number of nuclei present at time zero (NO)(NO) and the number (N)(N) at a later time t


where e = 2.71828... is the base of the natural logarithm, and λλ is the decay constant for the nuclide. The shorter the half-life, the larger is the value of λλ, and the faster the exponential eλteλt decreases with time. The decay constant can be found with the equation


Activity, the Rate of Decay

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 ΔNΔN is the number of decays that occur in time ΔtΔt.

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,

1 Bq = 1 decay/second.1 Bq = 1 decay/second.

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

22.491 Ci = 3.70×1010 Bq,1 Ci = 3.70×1010 Bq,

or 3.70×10103.70×1010 decays per second.

Radiometric Dating

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 14N14N 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 1.3×10121.3×1012 to find the number of 14C14C nuclei within the object. Over time, carbon-14 will naturally decay back to 14N14N 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 14C14C is not replenished. By comparing the abundance of 14C14C 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 14C14C 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 Figure 22.28). 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 14C14C found in living tissues, allowing the shroud to be dated (see Figure 22.28).

A photo of Shroud of Turin and its negative imprint.
Figure 22.28 Part of the Shroud of Turin, which shows a remarkable negative imprint likeness of Jesus complete with evidence of crucifixion wounds. The shroud first surfaced in the 14th century and was only recently carbon-14 dated. It has not been determined how the image was placed on the material. (credit: Butko, Wikimedia Commons)

Worked Example

Carbon-11 Decay

Carbon-11 has a half-life of 20.334 min. (a) What is the decay constant for carbon-11?

If 1 kg of carbon-11 sample exists at the beginning of an hour, (b) how much material will remain at the end of the hour and (c) what will be the decay activity at that time?


Since NONO refers to the amount of carbon-11 at the start, then after one half-life, the amount of carbon-11 remaining will be NO/2NO/2. The decay constant is equivalent to the probability that a nucleus will decay each second. As a result, the half-life will need to be converted to seconds.




Since half of the carbon-11 remains after one half-life, N/NO=0.5N/NO=0.5.


Take the natural logarithm of each side to isolate the decay constant.


Convert the 20.334 min to seconds.

22.530.693=(λ)(20.334 min)(60s1 min)0.693=(λ)(1,220.04s)0.6931,220.04s=λλ=5.68×104s10.693=(λ)(20.334 min)(60s1 min)0.693=(λ)(1,220.04s)0.6931,220.04s=λλ=5.68×104s1

(b) The amount of material after one hour can be found by using the equation


with t converted into seconds and NO written as 1,000 g

22.55N=(1,000 g)e(0.000568)(60.60)N=129.4gN=(1,000 g)e(0.000568)(60.60)N=129.4g

(c) The decay activity after one hour can be found by using the equation


for the mass value after one hour.


(a) The decay constant shows that 0.0568 percent of the nuclei in a carbon-11 sample will decay each second. Another way of considering the decay constant is that a given carbon-11 nuclei has a 0.0568 percent probability of decaying each second. The decay of carbon-11 allows it to be used in positron emission topography (PET) scans; however, its 20.334 min half-life does pose challenges for its administration.

(b) One hour is nearly three full half-lives of the carbon-11 nucleus. As a result, one would expect the amount of sample remaining to be approximately one eighth of the original amount. The 129.4 g remaining is just a bit larger than one-eighth, which is sensible given a half-life of just over 20 min.

(c) Label analysis shows that the unit of Becquerel is sensible, as there are 0.0735 g of carbon-11 decaying each second. That is smaller amount than at the beginning of the hour, when R=(0.000568decays)(1,000g)=0.568R=(0.000568decays)(1,000g)=0.568 g of carbon-11 were decaying each second.

Worked Example

How Old is the Shroud of Turin?

Calculate the age of the Shroud of Turin given that the amount of 14C14C found in it is 92 percent of that in living tissue.


Because 92 percent of the 14C14C remains, N/NO=0.92N/NO=0.92. Therefore, the equation N=NOeλtN=NOeλt can be used to find λtλt. We also know that the half-life of 14C14C is 5,730 years, and so once λtλt is known, we can find λλ and then find t as requested. Here, we assume that the decrease in 14C14C is solely due to nuclear decay.


Solving the equation N=NOeλtN=NOeλt for N/NON/NO gives




Taking the natural logarithm of both sides of the equation yields


so that


Rearranging to isolate t gives


Now, the equation λ=0.693t1/2λ=0.693t1/2 can be used to find λλ for 14C14C. Solving for λλ and substituting the known half-life gives


We enter that value into the previous equation to find t.

22.64 t=0.08341.21×10−4=690years.t=0.08341.21×10−4=690years.

This dates the material in the shroud to 1988–690 = 1300. Our calculation is only accurate to two digits, so that the year is rounded to 1300. The values obtained at the three independent laboratories gave a weighted average date of 1320 ± 60. That uncertainty is typical of carbon-14 dating and is due to the small amount of 14 C in living tissues, the amount of material available, and experimental uncertainties (reduced by having three independent measurements). That said, is it notable that the carbon-14 date is consistent with the first record of the shroud’s existence and certainly inconsistent with the period in which Jesus lived.

There are other noncarbon forms of radioactive dating. Rocks, for example, can sometimes be dated based on the decay of 238U238U. The decay series for 238U238U ends with 206Pb206Pb, so the ratio of those nuclides in a rock can be used an indication of how long it has been since the rock solidified. Knowledge of the 238U238U half-life has shown, for example, that the oldest rocks on Earth solidified about 3.5 × 1093.5 × 109 years ago.

Virtual Physics

Radioactive Dating Game

Learn about different types of radiometric dating, such as carbon dating. Understand how decay and half-life work to enable radiometric dating to work. Play a game that tests your ability to match the percentage of the dating element that remains to the age of the object.