K-Ar dating calculation (video) | Khan Academy
Calculation of the decay of potassium in the Earth's crust to make argon shows how to calculate the age of a rock using potassium-argon dating. General assumptions for the Potassium-Argon dating system The resulting 40 Ar* and 40K can be plugged into the age equation as follows: A geologist in the. PROJECT FOR SECTION Potassium-Argon Dating xxiii (Ca) or the gas argon (Ar). decay of K is modelled by the differential equations. dA.
That's the half-life of potassium We know, after that long, that half of the sample will be left. Whatever we started with, we're going to have half left after 1.
Divide both sides by N0. And then to solve for k, we can take the natural log of both sides. The natural log is just saying-- to what power do I have to raise e to get e to the negative k times 1. So the natural log of this-- the power they'd have to raise e to to get to e to the negative k times 1. Or I could write it as negative 1. That's the same thing as 1. We have our negative sign, and we have our k. And then, to solve for k, we can divide both sides by negative 1. And so we get k.
And I'll just flip the sides here. And what we can do is we can multiply the negative times the top. Or you could view it as multiplying the numerator and the denominator by a negative so that a negative shows up at the top. And so we could make this as over 1. Let me write it over here in a different color. The negative natural log-- well, I could just write it this way. If I have a natural log of b-- we know from our logarithm properties, this is the same thing as the natural log of b to the a power.
And so this is the same thing. Anything to the negative power is just its multiplicative inverse. So this is just the natural log of 2. So negative natural log of 1 half is just the natural log of 2 over here. So we were able to figure out our k. It's essentially the natural log of 2 over the half-life of the substance.
So we could actually generalize this if we were talking about some other radioactive substance. And now let's think about a situation-- now that we've figured out a k-- let's think about a situation where we find in some sample-- so let's say the potassium that we find is 1 milligram.
I'm just going to make up these numbers. And usually, these aren't measured directly, and you really care about the relative amounts.What Can Potassium Argon Dating Be Used For?
But let's say you were able to figure out the potassium is 1 milligram. And let's say that the argon-- actually, I'm going to say the potassium found, and let's say the argon found-- let's say it is 0.
So how can we use this information-- in what we just figured out here, which is derived from the half-life-- to figure out how old this sample right over here? How do we figure out how old this sample is right over there? Well, what we need to figure out-- we know that n, the amount we were left with, is this thing right over here. So we know that we're left with 1 milligram.
And that's going to be equal to some initial amount-- when we use both of this information to figure that initial amount out-- times e to the negative kt. And we know what k is. And we'll figure it out later.
So k is this thing right over here. So we need to figure out what our initial amount is. We know what k is, and then we can solve for t. How old is this sample? We saw that in the last video. So if you want to think about the total number of potassiums that have decayed since this was kind of stuck in the lava. And we learned that anything that was there before, any argon that was there before would have been able to get out of the liquid lava before it froze or before it hardened.
So maybe I could say k initial-- the potassium initial-- is going to be equal to the amount of potassium 40 we have today-- 1 milligram-- plus the amount of potassium we needed to get this amount of argon We have this amount of argon 0.
The rest of it turned into calcium And this isn't the exact number, but it'll get the general idea.
And so our initial-- which is really this thing right over here. I could call this N0. This is going to be equal to-- and I won't do any of the math-- so we have 1 milligram we have left is equal to 1 milligram-- which is what we found-- plus 0.
And then, all of that times e to the negative kt. And what you see here is, when we want to solve for t-- assuming we know k, and we do know k now-- that really, the absolute amount doesn't matter. What actually matters is the ratio.
If one of these protons is hit by a beta particle, it can be converted into a neutron. With 18 protons and 22 neutrons, the atom has become Argon Aran inert gas. For every K atoms that decay, 11 become Ar How is the Atomic Clock Set?
When rocks are heated to the melting point, any Ar contained in them is released into the atmosphere.
When the rock recrystallizes it becomes impermeable to gasses again. As the K in the rock decays into Ar, the gas is trapped in the rock. The Decay Profile In this simulation, a unit of molten rock cools and crystallizes.
Calculation of Potassium Decay Into Argon and Crustal Abundance of Argon
The ratio of K to Ar is plotted. Note that time is expressed in millions of years on this graph, as opposed to thousands of years in the C graph.
Click on the "Show Movie" button below to view this animation. How are Samples Processed?
Clicking on the "Show Movie" button below will bring up an animation that illustrates how a K-Ar sample is processed and the calculations involved in arriving at a date.
This is actually a mini-simulator, in that it processes a different sample each time and generates different dates. K-Ar Processing Limitations on K-Ar Dating The Potassium-Argon dating method is an invaluable tool for those archaeologists and paleoanthropologists studying the earliest evidence for human evolution.
As with any dating technique, there are some significant limitations.