 If we start with a sample of radioactive atoms, in a time interval delta T, some of them will undergo radioactive decay. Let's say that the number of atoms that do undergo decay is delta N, so the remaining atoms are N-delta N. Here we can write the rate of this decay. That will be the number of atoms that decayed, that is delta N, per unit time. So the rate of decay becomes delta N by delta T. In this video, we will talk about what this rate of decay depends upon, and also how we know if the rate is fast or slow. To begin with, we can say that the number of radioactive nuclei, they are decreasing with time. We started with N and now we have N-delta N. So delta N here, the final, final Nf-ni, final number of remaining nuclei minus initial number of nuclei. That will be a negative number, since we have less active, radioactive nuclei after time delta T. So the rate of decay is a negative quantity, and therefore we had a minus sign before it. And turns out that this rate of decay, it is directly proportional to the number of radioactive nuclei. Let's try and get some intuition of this. Let's say here you have, here you have a group of, let's say a group of 10,000, 10,000 carbon-14 radioactive atoms of carbon-14. And in a different group, we have, let's say we have 100, 100 radioactive carbon-14 atoms. So the same radioactive isotope of carbon, but we have different N. We have 10,000 here and we have 100 over here. And after some time, let's say after one second, after one second, it's possible that 100 carbon atoms over here, the decay, they undergo a radioactive decay. You wouldn't really see that with carbon-14. These are just random, completely random numbers, but let's assume them just for the sake of our intuition. So in one second, 100 carbon atoms undergo decay from the first group. And here, here we have 100 of the number of carbon atoms in this sample as this one. We have only 100 carbon-14 atoms over here and over here we have 10,000. So here we have 100 of the carbon-14 atoms compared to this one. So for every 100 atoms we saw decaying here, we will really expect to see one carbon atom, one carbon atom undergoing a decay from this group. One carbon atom's decay per second, just because we have a smaller amount. So if you think about delta N by delta T, delta N by delta T is much less for this sample. Only one, here delta N is just one, only one carbon atom undergoing decay in one second, but over here 100 undergoing decay in one second. And we can also see this if we have some data on half-life of any random sample. Let's say we have some random sample X and if we start with 100 radioactive nuclei of that random sample, after one half-life, only 50 will remain. And after one more half-life, 25 will remain. And after one more half-life, around 12 will remain. So we see that in the same interval of one half-life, in the same delta T of one half-life. Initially when we had 100 radioactive nuclei, 50 of them decayed. And then when we had 50 nuclei remaining, only 25 decayed. And then when we had 25, only 13 decayed. So delta N is decreasing, delta N by delta T is decreasing as the number of remaining radioactive nuclei that is N as that number decreases. So the rate of decay delta N by delta T, it is proportional to the number of radioactive nuclei that is N. And when we remove this proportionality, we get a constant which is called the decay constant. This right here is called the decay, decay constant. And this is denoted by lambda. This constant is specific for a particular nuclei. It will have one value for carbon-14, it will have a different value for some other radioactive atom. Let's say some radioactive isotope of carbon, maybe carbon-15. It will have a different value for some radioactive isotope of nitrogen. And it has different values for all the different radioactive nucleates. But what does that different value tell us? So we can take a similar looking example to understand that. Let's say again we have two groups. Now we have, instead of carbon-14, let's say we have 10,000, we have 10,000, some other radioactive isotope. We can take anything we can take. Let's take nitrogen-16. And another group we have 100 of the same nitrogen-16. And let's say that the value of lambda is more for nitrogen-16 compared to carbon-14. So how would that look like? If the value of decay constant is more for n-16, it's possible that in one second, in one second, out of 10,000, we have 500 decays, 500 decays per second. And here, because the number is 100th of this, we will have 5 decays per second. But if we compare this nitrogen-16 to carbon-14, whose decay constant would be less than that of nitrogen, we say that lesser number of decays happen in one second. So what decay constant really tells us is how likely a radioactive atom will undergo decay. What are the chances that a radioactive atom will undergo decay? If the chances of an atom undergoing decay are large, then the decay will happen quicker. So a large value of lambda tells us that there will be a rapid decay. The decay will happen quicker because higher value of lambda tells us the chance or the probability that an atom will undergo decay is large. In this case, in this case, we see that the probability of one nitrogen-16 atom undergoing decay is much more than the probability that one carbon-14 will undergo decay. And a small value of lambda tells us that decay will be slower. This will be a slower decay. So this right here is a very important equation, which is that the rate of decay that equals lambda into n. And the rate of decay, it's called, this is sometimes called as, this is sometimes called as activity, activity of the sample. We can spend some time talking about its units. So one common unit for activity is, and I'm writing all that over here, one common unit is called curie, C-U-R-I-E, and it is denoted by capital C, small i. And this is equal to 3.7 into 10 to the power 10 dKs per second, dKs per second. But there is one SI unit of activity, which is called becural. This is called becural, and it is denoted by capital B, small q. And the relation is that one curie, this is equal to 3.7 into 10 to the power 10 becurals. So one becural is basically equal to 1 dK per second. These are the units for activity. We can also try and represent activity in the form of a graph. We know that a sample of radioactive atoms, they undergo dK exponentially. So this n varies as e to the power minus lambda t. That is how it, that is how it decays exponentially. So if activity is equal to lambda into something that varies exponentially, activity also varies exponentially. So the same graph that we drew for an exponential dK for any sample of radioactive atoms, the same graph holds true for activity as well. We can say that there is some initial activity, which reduces to half a0 by 2 in one half-life. And this reduces to a0 by 4 in one more half-life. This number can also be verified experimentally. And the instrument that is used to count the number of radioactive dKs per second or to measure the activity of a radioactive sample, that instrument is called a GM counter or a Geiger-Muller counter. So Geiger-Muller counter. This instrument measures the activity of any radioactive sample. So that's why it becomes very important to know the activity of any sample because it can be experimentally verified. You can actually count the number of dKs happening per second with the help of this instrument. And it's very interesting how this instrument works. We are not going to the detail of that. But as soon as it detects one radioactive dK, there is a clicking sound or there's a crackling sound which this instrument emits and one can actually take a note of that. So we saw that the rate of dK, it depends upon the number of radioactive nuclei present and also this dK constant lambda. This dK constant tells us whether a dK would happen rapidly or slowly. Turns out there is one more quantity which can give us some insight into whether a dK will happen rapidly or slowly. And that is called a mean life. So let me, let me make some space. Let's say we have this group of radioactive atoms and because radioactive dK, it's a random process really. You can never predict when a radioactive dK will occur. All of these radioactive atoms, even though they must be of the same radioactive isotope, they will undergo dK at different time instants. That could look somewhat like this. Maybe for this much amount of time, nothing happens to the topmost atom and then it immediately decays and it differs for different atoms. So a mean life is nothing but calculating all of these times. Maybe this is T1, T2, T3, this could be T4, this could be T5. So it's just adding up all of this, dividing by the number of the radioactive atoms present. And there is some integration and calculus involved in this. We will not be going into that. But turns out after doing all of that calculation, you get mean life, mean life, which is denoted by tau, this to be equal to 1 by lambda. And let's see if this equation makes sense. Lambda is a dK constant, which tells us how likely it is that a radioactive atom will undergo dK. So if the value of lambda is high, if an atom has a very high chance of undergoing dK, then it's true that the mean life of that radioactive nuclei will be very less. And if this lambda is less, if the rate of dK is very slow, then the radioactive nuclei will remain radioactive for a longer period of time. And so the mean life also would be more.