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Exponential Functions: Radioactive Decay
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.
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.
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
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.
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.
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