In this activity students will investigate the idea that carbon dating is based on gathering evidence in the present and extrapolating it to the past. Students will use a simple graph to extrapolate data to its starting point and then pool the data to make a graph that simulates half-life. Students will be introduced to solving mathematical problems that involve half-life.
This activity will help prepare your students to meet the performance expectations in the following standards:
- HS-PS3-2: Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative positions of particles (objects).
- Scientific and Engineering Practices:
- Using Mathematics and Computational Thinking
- Analyzing and Interpreting Data
- Engaging in Argument from Evidence
By the end of this activity, students should be able to
- Devise a technique that will allow them to collect data, by means of working backwards, to determine the time (time-zero) that the ice began melting at their lab stations.
- Determine the rate of melting.
- Construct a linear graph to illustrate melting rate.
- Construct an exponential graph, utilizing class data, to illustrate melting rate.
- Understand how scientists use half-lives and carbon dating to try to determine the age of fossils and rocks.
This activity supports students’ understanding of
- Nuclear chemistry
- Half lives
- Radioactive isotopes
- Phase changes
- Melting rates
Teacher Preparation: 30 minutes (or less)
Lesson: 60 minutes
- Ring stand with clay triangle for each lab pair (12 if there are 24 students)
- Funnels that fit into each triangle (using long or short stem funnels are fine; they don’t all have to be the same length)
- Note: All students should have the same type of funnel (glass vs. plastic)
- 50 or 100 mL graduated cylinders placed directly beneath the funnel stem (place stem into graduated cylinder)
- Cubes of ice to be placed into the funnels at the appropriate time before students come into class
- Make sure the cubes are roughly the same size (best if they are obtained from an ice making machine like what might be found in the cafeteria)
- Time keeping mechanism (stopwatch, clock, phone, etc.)
- Always wear safety goggles when handling chemicals in the lab.
- Students should wash their hands thoroughly before leaving the lab.
- When students complete the lab, instruct them how to clean up their materials and dispose of any chemicals.
- This lab could be used as an introduction to half-lives or after students have been made aware of half-lives but don’t necessarily need to know about rates other than a change in amount per time unit. The students will need to know how to solve problems involving half-lives and radiation.
- The lab needs to be set up at least 30 minutes ahead of time so that the ice has already begun to melt when the students come into the classroom. Note the time when the ice is placed in the funnels.
- The teacher can change the times that the culprits were last seen around the room. The times provided are based on a class that would begin at around 11:00 a.m. The times can even be changed to 20 minutes before class starts, 15 minutes before class starts, 10 minutes before class starts, or 5 minutes before class starts.
- Fill the mouths of the funnels with ice cubes (or used crushed ice for faster melting but make sure to use the same type for every group.).
- Students should determine a time interval and should record the volume of water at each interval until the ice is nearly melted. You might want to hold a discussion with the students to come up with an interval that might work best. The student instructions below use 2 minutes, but that number is flexible and can be 1 or 5. They will record this information as points on a “time vs. volume” graph (a reasonably linear plot will emerge after about 45 minutes of melting), and working backwards, determine the exact time the ice was placed in the funnel (time-zero when the volume of liquid water was zero), or the exact time of the crime.
- Students could share data on a common Google or Excel Spreadsheet (if the instructor has access).If students are making graphs by hand then their data should be compiled on the board.
- How fast ice melts will depend on many different factors. It mostly comes down to how fast heat can be transported to the ice to accomplish the melting. The key factor in determining the rate of melting is the rate of heat transfer from the surroundings through the funnel to the ice (this is usually expressed as a temperature difference; the difference between the ice temperature and the surroundings) times a “heat transfer coefficient (which depends on a lot of things, like the thermal conductivity of whatever is surrounding the ice, whether the surroundings are moving to transfer heat by convection, or specifics about the surface).”Until the system has reached thermal equilibrium, the “decay” will be non-linear. This is the reason for placing the ice in the funnel at least ½ hour before class starts. Only late in the melting process (when too little ice remains to keep the funnel at a constant temperature) does the temperature of the funnel rise and the rate of heat transfer to the surroundings drops.
- For the temperature difference, if you have ice "in its own water" they will both be at zero degrees Celsius, so in principle the temperature difference is zero and there will be no melting. That is why you can keep ice water for a long time in an insulated container like a thermos without significant melting. In other situations, there might be a higher temperature someplace (like outside a glass that you've poured ice water into, or in this case, the glass funnel) to provide a driving force for melting.
- With regard to the heat transfer coefficient, liquid water conducts heat much better than air, so ice sitting in liquid would have a higher heat transfer coefficient than ice in air. That is why ice will melt much faster if dropped into warm water than it will in warm air. But another important factor is convection (if the surroundings are moving there will be much more heat transfer). That is why a warm wind will make ice and snow melt faster than if it is sitting in still air, and similarly running warm water over ice will melt it faster than putting it in stagnant warm water.
- An example of a linear graph for a pair of students is illustrated below. Using the slope of the line (y = 0.4551x + 24.36) students should be able to solve for x when y = 0. This shows that x would be -53 or 53 minutes before class started.
- The culprit in this case would be the Human Torch (again this is showing data for a class that started around 11:00 a.m.)
- An example of a melting curve generated (power best line of fit) from an entire class set of data (differences in volumes over time) is illustrated below. Notice how this curve represents the classic exponential decay that occurs with radioactive isotopes. This curve is produced from class data and it represents the differences in volumes between each time interval. Example: At the beginning of the experiment there is 25.0 mL of water in the graduated cylinder. At 5 minutes there is 30.4 mL. That is a difference of 5.4 mL (30.4 – 25.0). The first data point would be (5, 5.4). At 10 minutes the volume is 33.4 mL. The difference is 3.0 mL (33.4 – 30.4) which makes the second data point (10, 3.0). At 15 minutes if the volume is 35.6 mL then the difference is 2.2 mL thus making the third data point (15, 2.2). This is illustrated below but with all the group’s data.
- After the data has been collected, and the graphs have been developed, the class can be led to the assumptions being made, for example, to illustrate the concept of radioactive decay. Uranium 238 will decay into lead 206. In this activity the melting of the ice has a rate that will be similar to the rate of nuclear decay of elements.
Background on radioactive decay:
- The AACT Half-life Investigation Simulation will help students understand half-life, and could be used either before or after this activity.
- Radioactive atoms decay into stable atoms by a simple mathematical process. Half of the available atoms will change in a given period of time, known as the half-life. For instance, if 1000 atoms in the year 2000 had a half-life of ten years, then in 2010 there would be 500 left. In 2020, there would be 250 left, and in 2030 there would be 125 left.
- By counting how many carbon-14 atoms in any object with carbon in it, we can work out how old the object is, or how long ago it died. So we only have to know two things, the half-life of carbon-14 and how many carbon-14 atoms the object had before it died. The half-life of carbon-14 is 5,730 years. However knowing how many carbon-14 atoms something had before it died can only be guessed. The assumption is that the proportion of carbon-14 in any living organism is constant. It can be deduced then that today's readings would be the same as those many years ago. When a particular fossil was alive, it had the same amount of carbon-14 as the same living organism today.
- The fact that carbon-14 has a half-life of 5,730 years helps archaeologists date artefacts. Dates derived from carbon samples can be carried back to about 50,000 years. Potassium or uranium isotopes which have much longer half-lives are used to date very ancient geological events that have to be measured in millions or billions of years.
- For example, suppose a piece of charcoal from a fire pit is found in a cave.How long ago did the wood exist as a living tree?If the charcoal sample in the cave holds 1/8 the amount of carbon-14 found in a piece of living wood today, the carbon-14 in the sample has survived three half-lives (½ x ½ x ½ = 1/8). Thus, the sample is 3 x 5730, or 17190 years old.
- 1 half-life for C-14 is 5,730 years; 2 half-lives is double that, or 11,460 years; 3 half-lives is 3 x 5,730 years, or 17,190 years; 4 half-lives is 4 x 5,730, or 22,920 years; 5 half-lives is 5 x 5,730 or 28,650 (as can be seen on the graph).
For the Student
Beloved holiday icon, television personality, and song inspiration, Frosty the Snowman lies melting in the funnels at your lab station. He was brought into class from outside, subjected to a particular frequency of electromagnetic radiation in the classroom (i.e., the lights) and now lies in a puddle at your lab station. Don’t cry though...he will be back again someday. There were no eyewitnesses, but there are several suspects. All the suspects have holes in their alibis. You need to determine the exact time at which Frosty was put into the funnels to melt away, leaving no trace.
Heat Miser was seen around the classroom at approximately 10:00 a.m.(He had a thing about sharing the limelight with his brother Snow Miser and hates everything to do with winter).
Johnny Storm (the Human Torch) was seen around the classroom at approximately 10:15 a.m. (It was known in social circles that he was upset that Frosty was dating his old flame Crystal).
Sun Girl (Deborah Morgna from the Teen Titans who has the ability to use the power of the sun to manipulate flames) was seen entering the classroom at 10:30 a.m. (She despises anything cold).
Wasp (Jan Van Dyne and Ant Man’s good friend) was seen exiting the classroom at 10:45 a.m. She is known to have heat-related energy-blast powers and suffers from trauma due to a winter snowball fight gone badly, with her friends Anna and Elsa).
Frosty should be melting at a particular rate, one which can be labeled a rate of decay. Decay rates can be used to explain the disappearance of froth on root beer; enzyme-catalyzed reactions in the body; how pressure decreases with increasing height above sea level (changing 12% per 1000 m); and the conversion of radioactive nuclei to non-radioactive ones.
This activity will allow you to develop graphs to illustrate this type of decay; apply it to the concept of nuclear decay; and use the information to capture the criminal red handed!
- Pay close attention to the lab set up. Looking at your lab set up, determine the units for the rate at which Frosty melted.
- Looking at your lab set up, determine how you will make a graph using only your data.
- Determine which axis you will use for volume and which axis for time, recall that slope is rise (y-axis) over run (x-axis). Look at which units you decided to use for the rate of melting.
- What volume will you start with at the origin of your graph? Why did you choose that number?
- Think about making a graph using the data from the whole class. Why would this be a better option than just using your own data?
- At what point did Frosty the Snowman start melting in class and which culprit was responsible for poor Frosty’s demise?
- How will Frosty’s death help illustrate what occurs during radioactive decay?
- Graphing program
- Always wear safety goggles when handling chemicals in the lab.
- Wash your hands thoroughly before leaving the lab.
- Follow the teacher’s instructions for cleanup of materials and disposal of chemicals.
- Immediately record the volume of Frosty's melted remains (water) in your graduated cylinder and note the time on the clock.
- Make a data table and, at regular intervals (every 2.00 minutes), record the time on the clock and the volume of water in the graduated cylinder.
- Stop after about 30 minutes, unless Frosty has completely melted earlier.
Use the following table to collect your data:
|Time (minutes)||Actual Time of Reading on Clock (a.m./p.m.)||Volume of Water in Graduated Cylinder (mL)||Difference in Volumes between Readings (mL)|
- Determine the difference in volume between your first reading and your second reading and place that in the data table.
- Determine the difference in volume between your second reading and your third reading and place that in the data table.
- Determine the difference in volume between your third reading and your fourth reading.
- Continue doing this this all the way until you have the table filled up to the differences between each consecutive reading.
- Make a graph with time on the x-axis and volume on the y-axis. Plot your data on the graph and draw a best line of fit.
- Determine the slope of your line.
- Using the function of a linear relationship, y = mx + b, work backwards to find the exact time that Frosty first started melting (this will require you to go back beyond time zero on your graph).
- Determine the exact time that Frosty began melting. Who was the culprit?
- Make a second graph involving the data from the whole class. Plot the time, in minutes, on the x-axis and the DIFFERENCE IN VOLUMES between each time reading on the y-axis. For example, if the first reading you took was 20.00 mL at time zero and two minutes later it was 22.00 mL, the difference would be 2.00 mL. Add a trend line (a line indicating the general course or tendency of the information collected). The trend line should be a curve (computer programs label this “a power” trend line).
- What does your second graph tell you about the rate at which Frosty melted and how is this like radioactive decay?
- Radioactive atoms decay into stable atoms by a simple
mathematical process. Half of the available atoms will change in a given period
of time, known as the half-life. For instance, if 1000 atoms in the year 2000
had a half-life of ten years, then in 2010 there would be 500 left. In 2020,
there would be 250 left, and in 2030 there would be 125 left.
By counting how many carbon-14 atoms in any object with carbon in it, we can work out how old the object is, or how long ago it died. So we only have to know two things, the half-life of carbon-14 and how many carbon-14 atoms the object had before it died. The half-life of carbon-14 is 5,730 years. However knowing how many carbon-14 atoms something had before it died can only be guessed at. The assumption is that the proportion of carbon-14 in any living organism is constant. It can be deduced then that today's measurement of carbon-14 in living things would be the same as those many years ago. When the organism that formed a particular fossil was alive, it had the same amount of carbon-14 as the same living organism today.
The fact that carbon-14 has a half-life of 5,730 years helps archaeologists date artefacts. Dates derived from carbon samples can be carried back to about 50,000 years. Potassium or uranium isotopes which have much longer half-lives are used to date very ancient geological events that have to be measured in millions or billions of years.
For example, suppose a piece of charcoal from a fire pit is found in a cave. How long ago did the wood exist as a living tree? If the charcoal sample in the cave holds 1/8 the amount of carbon-14 found in a piece of living wood today, the carbon-14 in the sample has survived three half-lives (½ x ½ x ½ = 1/8). Thus, the sample is 3 x 5,730, or 17,190 years old. The diagrams below show the amount of carbon-14 remaining in the sample after each half-life. If one half life is 5,730 years, how many years is:
- 2 half-lives?
- 3 half-lives?
- 4 half-lives?
- 5 half-lives?
|© Wikimedia Commons/Kurt Rosenkrantz|
problems involving half-lives:
- A rock was analyzed using potassium-40. The half-life of potassium-40 is 1.25 billion years. If the rock had only 25% of the potassium-40 that would be found in a similar rock formed from a volcano today, calculate how long ago the rock was formed.
- Ash from an early fire pit was found to have 12.5% as much carbon-14 as would be found in a similar sample of ash today. How old is the ash in fire pit?
- How old is an Egyptian scroll made of papyrus that contains 75% of the amount of carbon-14 that would be found in a piece of paper today.
- What percentage of carbon-14 would you expect a piece of 34,000 year-old fossilized bone from a mastodon to have when compared to a similar piece of bone from a modern elephant?
Looking at the graph that you constructed from the class data on the melting of Frosty, what generalizations can you make about the decay of a radioactive isotope in a sample over time?