Quantitative concepts: exponential growth and decay, probablility
How Does Radioactive Decay Work?created by Jennifer M. Wenner, Geology Department, University of Wisconsin-Oshkosh
Jump down to: Isotopes | Half-life | Isotope systems | Carbon-14 | Examples & Exercises
Generally, there are four main concepts that students struggle with when thinking about radioactive decay:
- the spontaneity (or randomness) of radioactive decay,
- the reason isotopes are important,
- the concept of half-life, and
- knowing which system is appropriate
Radioactivity: A steady but unpredictable (spontaneous) process
Radioactivity and radioactive decay are spontaneous processes. Students often struggle with this concept; therefore, it should be stressed that it is impossible to know exactly when each of the radioactive elements in a rock will decay. Statistical probablity is the only thing we can know exactly. Often students get bogged down in the fact that they don't "understand" how and why radioactive elements decay and miss the whole point of this exercise. If they can begin to comprehend that it is random and spontaneous, they end up feeling less nervous about the whole thing.
Radioactive decay involves the spontaneous transformation of one element into another. The only way that this can happen is by changing the number of protons in the nucleus (an element is defined by its number of protons). There are a number of ways that this can happen and when it does, the atom is forever changed. There is no going back -- the process is irreversible.
This is very much like popping popcorn. When we pour our popcorn kernels into a popcorn popper, the is no way to know which will pop first. And once that first kernel pops, it will never be a kernel again...it is forever changed! (And coincidentally, much yummier!)Teaching example using popcorn to teach radioactive decay
Isotopes: same element, different atomic mass
- "A variety of a chemical element (strictly, of one particular element) which is distinguished from the other varieties of the element by a different mass number but shares the same atomic number and chemical properties (and so occupies the same position in the periodic table)."
The atoms that are involved in radioactive decay are called isotopes. In reality, every atom is an isotope of one element or another. However, we generally refer to isotopes of a particular element (e.g., Rubidium-87 (87Rb) or Lead-206 (206Pb)). The number associated with an isotope is its atomic mass (i.e., protons plus neutrons). The element itself is defined by the atomic number (i.e., the number of protons).
Only certain isotopes are radioactive and not all radioactive isotopes are appropriate for geological applications -- we have to choose wisely. Those that decay are called radioactive (or parent) isotopes; those that are generated by decay are called radiogenic (or daughter) isotopes. The unit that we use to measure time is called half-life and it has to do with the time it takes for half of the radioactive isotopes to decay (see below).
Half-life: a useful way of telling geologic time
Half-life is a very important and relatively difficult concept for students. Mathematically, the half-life can be represented by an exponential function, a concept with which entry-level students may not have much experience and therefore may have little intuition about it.
I find that entry-level students in my courses get stuck on the term "half-life". Even if they have been given the definition, they interpret the term to mean one-half the life of the system. Instead, it is really the lifetime of half of the isotopes present in the system at any given time.
Problem solving in the geosciences was forever changed with the discovery of radioactivity. Radioactive elements can be used to understand numerical age of geological materials on time scales as long as (and even longer than) the age of the Earth. In order to determine the age of a geologic material, we must understand the concept of half-life.
Half-life is a term that describes time. The definition is:
- The time required for one-half of the radioactive (parent) isotopes in a sample to decay to radiogenic (daughter) isotopes. --modified from Webster's Third International Dictionary, Unabridged
In other words, it is the lifetime of half the radioactive isotopes in a system. The units of half-life are always time (seconds, minutes, years, etc.). If we know the half-life of an isotope (and we can measure it with special equipment), we can use the number of radiogenic isotopes that have been generated in a rock since its formation to determine the age of formation. Radiometric dating is the method of obtaining a rock's age by measuring the relative abundance of radioactive and radiogenic isotopes.
Using demonstrations of half-life such as a coin toss for large classes or M&M demonstration for smaller classes can help students to better understand what is happening. Plotting the results of these demonstrations results in a curve of an exponential decay function. Showing this plot and asking them questions about the shape and changes in number of isotopes through time may help students to develop some intuition about half-life. Although most introductory students may not be prepared for the equation for exponential decay, discussion of half-life and radioactive decay prepares entry-level students for the introduction of more mathematical discussion of exponential growth and decay in upper level classes.
So many systems, how do we choose?Most students don't really know how isotopes are used to determine age. In particular, they have a hard time understanding that different systems are appropriate for different types of radiometric dating and why. There are several important points that can be emphasized to help avoid confusion when talking about the various systems:
With all these systems, how do we choose? Geologists use a number of criteria to decide which of the systems to use:
- Is the half-life of the system appropriate for the rock that you are trying to date? Most geologists have an idea of the age of a rock (if age is less than 6 half-lives, it'll work).
- Does the rock have minerals that can be used for the isotope system you want to use? You need to have minerals in your rock that contain the element(s) you want to use.
- What event do you want to date? Some systems are very good for dating igneous events, others are very good for dating metamorphic events. (Remember, it is impossible to date sedimentary minerals because they eroded from some igneous or metamorphic rock.)
What about Carbon-14?Most students have heard of Carbon-14; yet, it doesn't appear in the table of isotopes used to date rocks and minerals. Why not? Carbon-14 is not appropriate for rocks because it must involve organic carbon. Rocks are made of minerals that are by definition inorganic. The discussion of 14C below is a great way to illustrate important points of how to choose a system.
- With 14C, we can only calculate the age of something that was once living (contains organic carbon). Since (most) rocks were never alive, we can't use this to date a rock.
- The half life of 14C is geologically short -- 5730 years -- and is therefore not useful for materials older than about 35,000 years. That's well over 4 billion years of geologic history that we can't touch.