# Exponential Functions: Radioactive Decay

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**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.

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