# Radioactive Half-life Experiment – Part 3 – Calculations and Results

Frostbite Theater presents… Cold Cuts! No baloney! Just science! Hi! I’m Joanna! And I’m Steve! We’re finally ready to calculate the half-life of metastable barium-137 using the data that you collected earlier. If you haven’t collected any data, you may want to watch the first two videos in this series. We’re going to start with an equation. Don’t panic. As far as equations go in physics, this one isn’t too bad. This equation tells us that the activity of a radioactive substance at any time ‘t’ is equal to some initial activity multiplied by ‘e’ raised to the power of negative lambda ‘t,’ where lambda is the decay constant of the particular radioactive isotope and ‘t’ is the time. Our first goal is to find lambda, the decay constant. In order to do that, we need to know how the activity of the sample changed over time. Now, the scaler keeps track of events like a car’s odometer keeps track of distance. The numbers just keep going up and knowing the total number of events after five minutes isn’t really that useful. What we need to know is how many events happened within each of your timing windows.

So, if you took data every fifteen seconds, you need to figure out how many events happened within the first fifteen second window, how many happened within the second fifteen second window, and so on all the way out to five minutes. Now, luckily, the math for this is pretty easy, although it’s a little tedious. Again, imagine that it’s a car’s odometer and you want to know how far you’ve driven in a day. You take an odometer reading at the beginning of the day and again at the end of the day and you subtract the two numbers to find out how far you’ve gone. It’s the same thing with this. Subtract two subsequent readings to find the number of events within the time period.

If you like, you can do this old school using paper and a calculator or, if you’re a little bit more sophisticated, you can easily knock-out these calculations using a spreadsheet. Now you know the total number of events that occurred during each of your timing windows. Unfortunately, our detectors can’t tell the difference between an event caused by a decay of the barium source and an event caused by some other source of ionizing radiation in the room. Earlier, Joanna and I recorded the background levels in this room. With the barium source gone, the GM tube records, on average, 46.1 counts per minute while the photomultiplier tube records, on average, 3,454.4 counts per minute. These extra events need to be subtracted out of your data. Now, you need to be a little careful doing this. The background value you subtract needs to be scaled to match your timing window. If you took readings every 15 seconds, you’ll need to divide the appropriate background rate by four. If you took readings every 10 seconds, you’ll need to divide the appropriate background rate by six.

And, again, you can work this out using paper and a calculator, or just add a new column to your spreadsheet. Now, it’s time to graph! Plot the background corrected number of events in each of your timing windows versus time. If you use regular graph paper, you should end up with a nice, exponential decay curve. While it may look really nice, it isn’t really that useful. It’s better to make your graph on semi-log paper because when you graph an exponential function on semi-log paper, you’ll end up with a nice, straight line. Use the log scale for the number of events and the linear scale for time.

Once you’ve plotted your data, draw a single, straight line that best fits your data. We can use this graph to find the decay constant. All you need to know are two points on the line of best fit and the amount of time between them. The points you pick don’t even need to be actual data points. Any two points on the line that are convenient to use will work. Once you have those points, you need to solve this equation for lambda. We aren’t going to work this out for you step-by-step. That’s something you can do on your own. But, this is what you should get… Just fill in the appropriate values… Do the division within the parentheses and take the natural log of that number. And, finally, divide by the time. At last, we’ve found the decay constant! If you’re using a spreadsheet, make a scatter plot of background corrected events versus time.

The spreadsheet should have a function that will calculate the line of best fit for you. Make certain you tell it to fit your data to an exponential function. Once you have that, ask the spreadsheet to display the equation of the line of best fit. The variables may not have the right names, but you shouldn’t have a problem figuring out what lambda is. Knowing the decay constant is nice, but the decay constant isn’t the same thing as the half-life. Fortunately, it’s easy to calculate the half-life once you know the decay constant. Remember this equation? Again, we aren’t going to work it out for you, but it’s a fairly simple matter to use this equation to show that the half-life of a substance is equal to the natural log of 2 divided by its decay constant. You know the decay constant and you know that the natural log of 2 is about .693. Do the division and you’ll end up with the half-life, in seconds! The half-life of metastable barium-137 is listed as being 2.552 minutes, which is about 153 seconds.

So, how did you do? Did you get something close to that value? I did! Really? Um hum! All right! Yea! Thanks for watching! I hope you’ll join us again soon for another experiment! That was a lot of work! Yes, it was. And we didn’t even get to use any liquid nitrogen. I know… Can we use some next time! Sure! Maybe we’ll use balloons! Yeah! 