Exponential Functions: Radioactive Decay

00:00 In this lesson, we’re going to take a look at how we can use the exponential function to model the decay of a radioactive substance.

00:09 Certain substances that have unstable atoms undergo radioactive decay, and the amount of the substance at any given time T can be modeled using an exponential function like this.

00:22 So, in this expression for this function, we’ve got some constant, we’ve got e raised to the power of minus, and then this number of times t divided by this constant capital T. Here, N sub-zero is the amount of the radioactive substance at time t = 0.

00:43 And t = 0 is usually the time where you started measuring the substance. T is the half-life of the radioactive substance, and this is the time in years that it takes for the substance to decay to half of what it started at.

01:00 So if a substance decays from a initial amount of 100 grams to 50 grams in, say, 3.5 years, then the half-life is 3.5 years. The input variable to the function, little t, is measured in years, and so for any given time t, we can compute how much is left, and this is going to be given to us by this numerical value once we put in the value for the little t, do that computation, and then take the exponential of that and multiply it by N sub-zero.

01:35 Now, instead of making up numbers, let’s get some data for a real substance that undergoes radioactive decay. There’s many to choose from, but one that I like in particular—it’s called Californium-252.

01:50 This is the Wikipedia website of Californium. “Californium is a radioactive chemical element with a symbol of Cf and atomic number 98.” Now, for us, we’re interested in getting the half-life of Californium-252.

02:06 There are different versions, or isotopes, of Californium. The one that we want is the 252. It’s got a half-life of 2.645 years. We’ll write a function in Python that gives us, at any given time, the amount left of, say, a certain initial amount of Californium-252 after a certain number of years have passed.

02:30 Let’s define a function. We’ll call it cali_252(). What we’ll do is create a function that gives us the amount of the substance after a certain number of years, and the initial amount is going to be a keyword argument, and the default value will be 1.

02:50 What we want to do here is we want to return—as you saw in the formula, it’s the initial value times the exponential of -0.693 times the input t, all of that divided by the half-life, and the half-life of Californium-252 is 2.645 years. So in other words, whenever we evaluate this function at any given time, we’re going to be returned the amount left once we pass in also a value for the initial amount.

03:28 So if you evaluate this cali_252() function at a time of 2.645, and the initial amount is 100, we should get very close to 50.

03:42 Of course, we’re getting a little bit of roundoff error, but this seems to be working well. It makes sense that if we evaluate at 2.645, an initial amount of 100, which should be at around 50% of that, which is 50, and we’re getting a rounding error here. We’re not getting exactly 50 because probably this 2.645 is off a little bit.

04:02 Now, if we find out how much of this Californium isotope is left, if we evaluate the function, say, at 10 years—again, starting at 100 grams—then there’s 7.28 grams left after 10 years.

04:18 All right, so that is an application there of the exponential function. Coming up next, we’ll take a look at logarithmic functions.