A moon rock collected by the Apollo mission is estimated to be...

Question

A moon rock collected by the Apollo mission is estimated to be 3.80 billion years old by uranium/lead dating. Assuming that the rock did not contain any lead when it was formed, what is the current mass of 206 Pb in the rock, given that it currently contains 1.180 g of 238 u? The half-life of ²³⁸U is 4.47×10⁹ years.

Accepted Answer

Alright. So in this practice problem, we're talking about this very old rock in Canada um using uranium lead. Radio metric dating, it has been found to be 4.28 billion years old. So this is the basically how old this rock is. If we have seen that the rock only contained uranium in the beginning, what would be the mass of lead in the sample of the rock if it currently contains this many grams of uranium? So we have the half life of uranium, which is that many years. Okay, now, So what what do we need here? What we need here is actually mass of lead. But at the beginning it was only uranium. Right? So uranium after a while obviously decays and it decays into lead. Okay, So the mass that it decays into lead is basically going to be the same mass. So think of conservation of mass. Right, So the original mass that we start with. So let's say original mass massive this rock When it only contained uranium. Right? So this is gonna be our original mass. But it consists of the 1. 00 g of uranium plus let's say X grams of lead. Right? So the grams of lead is basically how much is the mass of uranium being decayed into lead. But altogether if you put them together, that's going to give us the original mass. Right, conservation of mass here. Alright, So what we need to find is actually the very original mass that only contained uranium. And we can do that by using the integrated rate law equation. Now we're talking about radioactive processes here and we know that this is going to be a first order reaction. Right? So for the first order reaction we have this equation, we have Ln of A or N. Whatever. So it's going to be our radioactive isotope right after a certain amount of time is going to equal to negative Katie. So, okay, here is decay constant or re Constantine's time plus Ln of The initial concentration of that isotopes. So here basically the initial is going to be the original mass when it only contained uranium and then the final is going to be how much uranium contains. Now, which is 1.500 g. So we can use this equation to find the original mass. And once we have original mass we can solve for this X grams of lead. Okay, now, the only problem is uh, you know, we have time, we have initial and final concentrations, we don't have K which is the rate constant or decay constant. So anytime a problem gives you half life, you know, you're gonna go ahead and use that half life most likely. So for the first order reaction, half life Is the simplest equation is just a line of two divided by K. Okay, so we do know half life. It is that many years and one of two is basically just a constant. Right? So let's go ahead and solve for K. So K. Here is going to equal to a line of two divided by the half life. Okay. Which is a lot of Come on. One of two Divided by half life is four 0.47 Times 10 to the 9th. And that's years. Alright, so once we find that K here is going to equal to 1.5, 5 times 10 to the negative 10. And because here we had years on the bottom of the denominator, the units are going to be in years in verse four K. Okay, so now that we have K we can go ahead and plug it into right here. Okay, And now we're going to be solving for the initial concentration of of this rock, which is going to give us the original mass of when it was just uranium. Alright, so let's go ahead and do that. So we're going to have A land of so final right now, after so many years after 4.28 billion years, what's the concentration of uranium right now? It's 1.500 g. Right? So that is the number we put plug in. That's going to equal to negative case. Okay, here is this number right here that we plug in 1. years, inverse time time is just how old this truck is and that is how many years? It is so billion years. There's going to be to the times 10 to the ninth. Right? So it's going to be four 4. Times 10 to the 9th. That's billion Of course years plus a line of what it was originally. And that's what we're solving for here. Right? Because we don't really know that. So we're gonna leave that blank. That's what we're striving for. So now let's go ahead and multiply these two numbers together. Notice that the years here will cancel out because here we have years on the numerator here. We have ears on the denominator. So they cancel when you multiply them together. Now we're gonna go ahead and isolate Ln of a initial by by itself first. Okay, so these two numbers multiplied together and then notice that they're negative. Right? So after that we're going to go ahead and add whatever you get here to both sides. So it's going to be a lot enough. 1.500 plus whatever you got here for those two numbers when you multiply them together and that should be 1. 6.886. Okay. Now we just want a we just want the concentration the mass of a. Which is the original mass. Right? That's the original mass right there. So how do we get rid of Ellen? What? We're just gonna go ahead and take E to the power of this number. So in order to get rid of Ln I'm going to take e. You should find a button on your calculator and just take it to the power of that number that we found here. 886. Okay, so this will give you 2.912. And of course here we were working with g. So that will give us of g. So that is going to be the original mass. Right? That's the original mass. So going back to this equation right here that we talked about at the beginning, we said original mass is going to consist of uh the amount of uranium that's left right now. Plus how much of the lead was made from the decay of uranium. Right, We should still have the same original mass due to conservation of mass here. So we're going to say to find x grams of lead, We're going to take the original mass which is 2. grams and notice how it's it's a bigger number than what we have of uranium now, Which makes sense. Right? We started with a bigger number of uranium and now we have less so minus 1.500 g of uranium. And that will give us 1.412. So let me fix that. 1.4 12 grams of lead Have led to of six isotope. Okay, so folks, that is our finally answered their 1.412. That is how much of lead was basically made um After 4.20 billion years from uranium. Okay, so we're all done here. If you have any questions please let us know and we'll see you the next problem.

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