First Order Half Life

Hey, guys, In this new video, we're gonna take a look at radioactive decay rates. Now we're gonna say when it comes to radioactive decay, it's gonna follow a first order mechanistic step. So we're gonna say here it's a first order process whose rate is proportional to the number of radioactive nuclei. N four K is a first order rate constant called the decay constant. Now, this kind of follows what we learned earlier when it comes to kinetics. So if you've seen my earlier videos on chemical kinetics, you should be very familiar on how exactly we're gonna solve these types of questions. Because when it came to chemical kinetics, we did the integrated rate loss. And we're gonna say when it comes to radioactive decay, we're gonna basically follow those integrated rate loss just a little bit differently. Now, when it comes to Kinetics were gonna say that rate law equals R rate constant k times the concentration off our reactant to some power that power we call a reaction order. Now, since we're dealing with radioactive systems now, it's basically this equation just changed a little bit. So here we still have rate, but instead, instead of saying rate law were saying decay rate here. This is our rate constant K But now it's our decay constant. Okay, here this was the concentration off our reacted. But now it's the concentration off our radioactive nuclei. So similar, but yet different. Now we're gonna say again, radioactive decay follows a first order process. Like I said earlier, and here we're just gonna reuse the first border integrated rate law from the chapter on chemical kinetics. Remember, Kinetics just looks at rape for speed of reactions we're looking at How quickly do my reactions breakdown to give me product? But since we're talking about radioactive systems now were saying How quickly will my radioactive compound breakdown? So that's the just the fundamental difference in regular kinetics we're looking at how faster reacting basically becomes product here. Looking at how fast is my radioactive compound? Just two k away. So here, when we dealt with chemical kinetics, we said that the integrated great law for the first order process waas Alan off my final concentration of my reactant equals negative great constant times time plus Ellen off the initial concentration of my reacted Okay, so again, this was your final concentration of your reactant. This right here was your rate constant here. This was your time. And here this was the initial concentration of your reactant. And since you're gonna need some room guys to finish writing all the notes that we need, I'm gonna remove myself from the image. So remember this was in chemical kinetics, the first order integrated rate law. But now we're dealing with radioactive systems, so the equation is going to change a little bit. So here, we're gonna have Ln and t and here NT stands for what? We're gonna stay here. NT stands for the final concentration off our radioactive nuclei. So just think of it as your radioactive reacted. Okay? So radioactive nuclei is equivalent to, in a sense, to your radioactive reacted. So that equals negative, Katie Plus Ellen off, you know, and you can see that they're very similar. All we're really doing is substituting in and and replacing the variable A. That's the only real difference here. Okay, Here, instead of this being our rate constant, it's gonna be now, are decay constant? And here this is still time. And here this would just represent your final concentrate on your initial concentration off your radioactive nuclei, but they're very similar equations. So if you know how to use the one from the Chemical Kinetics chapter that you'll have no problem adopting it to these radioactive systems because it's basically the same type of process.

so here. Since we're gonna continue to need more space, guys, I'm gonna stay out of the image a little bit longer here. We're gonna say a sample of Radan to 22 has an initial Alfa particle activity. So just think of that as your initial concentration off 8.5 times 10 to the four. That should be a superscript DPS, which means disintegration is per second. So basically, it's breaking down by that much per second. After 7.3 days, it's activity A is 3.7 times 10 to the four. Disintegration is per second. What is the half life off Radan to 22. So again, we're dealing with first order kinetics here. But now it's just first order radioactive kinetics. So here it's gonna be Ellen anti equals negative, K T plus Allen and up. And here they're asking us for half life. Now, remember, for first order, half life was half life equals Alan two divided by K. And here we're gonna say that Ellen to is the same thing as 0.693 When you punch it into your calculator, they're asking me to find half life. So what I need to find is K. So remember, we're dealing with first order processes because radioactive decay occurs by a first order process. So it has the same exact half life, whether you're dealing with radioactive systems or not. So we got to do now is just plug in what we know. I tell you that initially, this is what you have. So this is your initial concentration. Then we're gonna say here that this is your activity after given amount of time 7.3 days, to be exact. So that represents your final concentration equals we don't know what K is. That's what we're looking for. So negative K time is 7.3 days now, remember, because your time is in days. That means you're k will be in days in verse. They have to agree in terms of time. So if K is in minutes in verse, for example, time has to be in minutes. They have to agree. All we gotta do now is separate. Just the K by itself. So what we're gonna do next is we're gonna subtract Alan 85 times, 10 to the four from both sides here. So this gets taken out over here is gonna give me negative 0.831733 That equals negative K time 7.3 days in verse. Next, we wanna just isolate K by itself. So we're gonna divide out negative 7.3 days from both sides here. Okay, so the negative gets canceled out and the 7.3 days gets canceled out now, because your days is on the bottom. When we find K, it'll be in days in verse, the negative and the negative Cancel out. So I have 0.114 days in verse, those days in verse, I just plug in over here 0.114 days in verse And because days is on the bottom here, when I find half life, it's just gonna be in days because I'm bringing it up back up top. So he it'll give you 6.8 days now. We might be talking about radio activity now, but fundamentally, it's still the same thing When it came to chemical Kinetics. We're dealing with a first order process. So we're using an incense. The first order integrated rate law with the Half life for a first order reaction, if you could just see it, is that it's the same exact things that we've done earlier before in the earlier chapters. Remember that and just apply it so that you can get through problems like this here, instead of talking about polarity off are reacting. We're talking about activities were talking about this integrations per second. It's the same thing. Just apply what you learned earlier to what we're doing now.

Hey, guys, In this new video, we're gonna continue with our discussion of radioactive decay rates. So because guys, we're gonna need a lot of room to work out these problems. I'm gonna remove myself from the image so we have way more room to work with. So here it says gallium citrate containing the radioactive isotope. Gallium 67 is used medically as a tumor seeking agent. It has a half life off 78.2 hours. How long will it take for a sample of gallium citrate to decay to 20% of its original activity. So we know here that radioactive decay occurs by a first order process, so we know that our equation are integrated. Equation is Alan NT equals negative. Katie, plus Alan and l here I'm asking, how long will it take? So here I'm really asking you to find time. I also talked about half life. Now, why do I give you half life? Because if I give you half life, that's gonna help us find Kay. And if we know what K is, we can plug it into this equation to help us find the missing variable t. So let's just do that first. So half life is 78.2 hours here, So that equals 0.693 over K. Multiply both sides by K. So we're gonna say here, K times 78.2 hours equals 0.6 93. Divide both sides by 78.2 hours. Since ours is on the bottom, care will be in hours in verse. So that gives me 8.862 times 10 to the negative. Three hours in verse now, because our case and hours in verse, that means when we find time, time will be in hours. Now, be careful in your exam. Make sure it is. Your professor wants your time and ours Or do they want it in days or years? If so, you'd have to convert further from time and hours to maybe days, seconds, minutes, whatever. So we just found out what K is so we can plug that in for right now. So negative. 8.862 times 10 to the negative, three hours in verse. We need to find time and to find time, we need to know what the initial and the final concentrations are. Now I hear, I hear, I hear I say to decay to 20% of its initial concentration. Since we're dealing with percentages, we can assume that were started out with 100%. Now we have to just figure out what the final amount is. Now what's in a word award has ah lot of power because the word can mean completely different things. Depending on how you use it, I say to decay, too 20% because I use the word to That means that your final concentration would be 20. But if I had say decay by 20% then it wouldn't be 24. Final by would mean it's 100% minus 20% so it be 80% as your final. So again, what's in a word award has tremendous power. One word changes everything. So be careful when you read it. Does it say to a number? If it does, it's that number. If it says by a number, that means you need toe work from 100% subtracted by that new number to give you your final concentration. Okay, so we have everything we need, except for time. So let's just solve for T now. So subtract Allen 100 from both sides here. So these cancel out. So it's gonna give me negative. 1.60944 equals negative. 8.862 times 10 to the negative. Three hours in verse. Times teeth. You wanna isolate just time by itself. So divide out. Negative. 8.862 times 10 to the negative, three hours in verse. So this cancels out with this because ours and verses on the bottom. When it comes up to the top, it'll just be in hours. So the answer here would be 181.6 hours. So that would be our answer again. Following the things that we've learned earlier about chemical kinetics were just following the same basic method here. Instead of dealing with a regular reacting. Now we're now dealing with a radioactive compound. But the same methods apply

Alright. So here says, what percentage of carbon 14 which has a half life of years, remains in a sample estimated to be 16,230 years old. All right, so we have to figure out what the percentage of the compound remaining they're giving us half life here. Remember, all these radioactive processes are first order, so half life equals Ln two over k, we know what the half life is. And because of that, we can calculate what my K value will be. Multiply both sides here by K. Then we're gonna divide both sides here by 5715 years. So that's gonna give me K equals 1.21 times 10 to the negative, four years in verse. But remember, we're not looking for K here. We're looking for the percentage of my carbon 14 that's remaining. So now we're gonna use the integrated rate law for this first order process here, which is Ln NT, which is your disintegration per second equals negative, Katie, Plus Ln and, uh oh, yes. We're going to say here that they're giving us the we figured out what K is so negative. 1.21 times 10 to the negative four years in verse time is 16,230 plus l N n o. What we're gonna do now is we're going to subtract Allen and oh from both sides here at this point, what we need to realize in terms off math that we've learned is that Ln of a value minus Ellen of another value, this can be rewritten where this really means. Ln of anti divided by N o, we're gonna multiply these two terms together. When we do that, that's gonna give me negative 1. Now, to figure out the percentage of my material remaining, I have to figure out what the ratio is of my final amount divided by my initial amount. So this ratio is what I'm trying to isolate. Toe isolated. I need to get rid of this natural log here. Remember, to get rid of natural log, we're gonna take the inverse of the natural log to do that. That's gonna become e to the value that we have. So we do e to the negative 1. That's gonna equal 0.1397 to 9. That they're represents the fractional amount of my reactant remaining. But remember, we're not looking for the fractional amount of carbon 14. We're looking for its percentage. So how do I convert a fractional amount to a percentage? I'm just gonna multiply it by 100 so we're gonna have 13.97% remaining. So after 16,230 years, the amount of carbon remaining in the sample would be 13.97%. So just remember they didn't ask us for the final amount initial amount. They're asking for the percentage remaining anytime they're asking us for the percentage remaining, you have to isolate the ratio of your final disintegration per second, divided by your initial disintegration per second and then multiply that by 100 to get the final percentage. Doing that, you always get your answer