- First-order reactions
- First-order reaction (with calculus)
- Plotting data for a first-order reaction
- Half-life of a first-order reaction
- Half-life and carbon dating
- Worked example: Using the first-order integrated rate law and half-life equations
- Second-order reactions
- Second-order reaction (with calculus)
- Half-life of a second-order reaction
- Zero-order reactions
- Zero-order reaction (with calculus)
- Kinetics of radioactive decay
- 2015 AP Chemistry free response 5
The nuclei of radioactive elements decay according to first-order kinetics. As a result, the half-life equation and integrated rate law for radioactive decay processes can be derived from the rate laws for first-order reactions. The resulting equations can be used to find the rate constant k for a decay process and determine the amount of radioactive isotope remaining after a certain time period. Created by Jay.
- [Instructor] Strontium-90 is a radioactive isotope that undergoes beta decay. Because radioactive decay is a first-order process, radioactive isotopes have constant half-lives. Half-life is symbolized by t1/2, and it's the time required for 1/2 of a sample of a particular radioactive isotope to decay. For example, the half-life of Strontium-90 is equal to 28.8 years. Let's say we start with 10 grams of our Strontium-90 radioactive isotope. And on the y-axis, we're gonna put the mass of our Strontium isotope in grams. And on the x-axis, we're going to have time. So when time is equal to zero, we have 10 grams of our isotope. Since the half-life of Strontium-90 is 28.8 years, if we wait 28.8 years, we'll go from 10 grams to five grams. So the next point on our graph will be at five grams, and this time should be 28.8 years. If we wait another 28.8 years, we're gonna go from five grams to half of that, which would be 2.5 grams. So the next point would be here at 2.5. If we wait another 28.8 years, we'd go from 2.5 grams down to 1.25, so approximately here on our graph. So this graph shows exponential decay. Let's say we were asked to find out how much of our radioactive isotope Strontium is left after 115.2 years. So what we would do is take 115.2 years and divide that by the half-life of 28.8 years. And by doing that, we realize that 115.2 is really just four half-lives. So one approach with this problem would be starting with our 10 grams. And we think about one half-life taking us to five grams, another half-life taking us to 2.5, another half-life taking us to 1.25, and then finally, one more half-life that takes us to 0.625 grams. So just doing the problem this way, we can see that's one, two, three, four half-lives. So our final answer is 0.625 grams remain after 115.2 years. Another approach to do the same problem would be to start with our 10 grams, and we multiply that by 1/2 to get the amount remaining after one half-life. And we could do that three more times to get the amount that remains after four half-lives. We could have also have written this as 10 times 1/2 to the fourth power since we needed to wait four half-lives. All these approaches will get you the answer of 0.625 grams of our radioactive isotope remaining after 115.2 years. For a chemical reaction with reactant A that's first-order, the rate law says that the rate of reaction is equal to the rate constant k times the concentration of A to the first power. Since radioactive decay is a first-order process, we can write that the rate of decay is equal to the rate constant k times N to the first power, where N is the number of radioactive nuclei in a sample. Since radioactive decay is a first-order process, we can also use this equation for the rate constant, which comes from first-order kinetics, which says that the rate constant k is equal to 0.693 divided by the half-life. For example, if we wanted to find the rate constant for the radioactive decay of Strontium-90, the rate constant would be equal to 0.693 divided by the half-life of Strontium-90, which we saw was 28.8 years. So when we do that math, we find that k is equal to 0.0241 one over years. Another equation from first-order kinetics is the integrated rate law for a first-order reaction. And the integrated rate law says that the natural log of the concentration of reactant A at some time t is equal to negative kt plus the natural log of the initial concentration of reactant A. Since we're using N, which is the number of radioactive nuclei in our sample, instead of the concentration of A, we can write the integrated rate law for our first-order radioactive decay process, which says the natural log of the number of radioactive nuclei at some time t is equal to negative kt plus the natural log of the initial number of radioactive nuclei. So let's say we start with 1.000 grams of our Strontium-90 radioactive isotope, and our goal is to find out how much remains after two years. So we're gonna wait two years and find out how much of our radioactive isotope remains. So we're going to use our equation for the integrated rate law. Let's go ahead and plug in what we know. We already know what k is, we found that in our earlier problem, so we can write this is equal to negative times k, which is 0.0241. Next, we also know the time period we're interested in. So we know what t is, t is two years, so let's go ahead and write 2.00 in here for the time. Next, we're gonna add the natural log of the initial number of radioactive nuclei in our sample. And while we don't have the initial number, we do have the mass. And since the mass is proportional to the number of radioactive nuclei, it's okay to go ahead and plug that into our equation. So we're gonna plug in the natural log of one. And all of this is equal to the natural log of the initial number of radioactive nuclei at some time t. So we have the natural log of that. The natural log of one is zero, so now we have the natural log of N is equal to, and when we do this math, we're gonna get negative 0.0482. Next, we have to get rid of the natural log. And we can do that by taking e to both sides. So if we take e to both sides, the natural log cancels out, and we get that N is equal to 0.953 grams. So that's how much of our radioactive isotope remains after two years.