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## Class 12 Physics (India)

### Unit 13: Lesson 3

Half life and decay rate- Half-life and carbon dating
- Half-life (qualitative)
- Half life (intermediate)
- Exponential decay formula proof (can skip, involves calculus)
- Activity and Mean life
- Exponential decay problem solving
- More exponential decay examples
- Half-life plot
- Worked example: Fraction of undecayed nuclei
- Exponential decay and semi-log plots

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# Half-life plot

Definition of half-life and graphing the decay of phosphorus-32. Calculating how much phosphorus-32 remains after 57.2 days. Created by Jay.

## Want to join the conversation?

- Well if the radioactive substance keeps getting halved; then does that mean that it never really gets depleted? like from 1 to 1/2, then 1/4,1/8,1/16.....(22 votes)
- Theoretically, that's correct.

In practice, once the numbers get too low, we are no longer able to detect the substance, and we then say that it has disappeared.

Many scientists use an arbitrary cut-off point and say that a radioactive substance has been depleted after 20 half-lives. By that time, only one-millionth of the original sample remains.(8 votes)

- Well, I can imagine how half life of radioactive phosphorus is measured, it's not that long. But how the half life of, for example, uranium isotopes was found? It is measjred in billion years...(5 votes)
- That is not needed because we don't actually measure the half-life, we measure the decay constant. From that it is a simple calculation to get the half-life (or any other fraction you might care to use).

The equation is:

N = N₀ e^(−λt)

Where N is the final amount of the substance, N₀ is the initial amount of the substance, t is time, and λ is the decay constant.

So we just pick some convenient amount of time, measure the other variables and compute λ. Once we know λ, we can compute the half-life or any other convenient fractional life.

λ can also be determined by other methods which involve counting the number of decays per unit time for a given quantity of the material (this is especially useful for radioactive isotopes that exist in trace amounts).

The point being is that half-life is just an easily understood number that we can use for reference. We really measure the λ or the related quantity τ. called "mean lifetime".(12 votes)

- why half life is different for different elements?(4 votes)
- Half life depends on the protons and neutrons, and different isotopes have different number nucleons(3 votes)

- Does an element have to be radioactive for it to decay?(1 vote)
- That's what radioactive means - that it decays.(6 votes)

- So using this chart we see that after the first half life happened in 14.3 days, and half life 2 happened in 28.6 days, which confirms that that the half life 3 will occur in 42.9 days.

Does this means that only 1/8th of the original material is remaining? Is this what actually happens or is this some sort of Achilles and the Tortoise thing?(3 votes) - How do I find the half life of something from the exponential decay equation? (ede = y=sv(df^x) where sv is the start value, and df is the decay value)(2 votes)
- If half a substance decays in one years time why is it incorrect to expect the other half to decay in one more year?(2 votes)
- Decay is a probabilistic occurrence. It is better to think of it as how long does it take for any given atom to have a 50% chance of decaying. If any atom doesn't decay in that half-life, it still has a 50% chance of decaying over the next half-life. The fact that it didn't decay in the first half-life doesn't increase the probability of decay.(2 votes)

- Why decay rate, which is directly proportional to no of atoms present, is constant always constant? As if we see in other random events the highest probability is near to mean value. For example if we take a large numbers of people, they may not decay (die) by above rule, instead most of the people will die near average age.(1 vote)
- I think you might be mixing up the
*number*of atoms that decay with the*percentage*of atoms in your sample that decay. If something has a half-life of one year and you have 100 atoms of it, then in one year 50 of them will have decayed. But if you have 200 atoms, then in one year 100 of them will have decayed. Different numbers of atoms decayed, but either way 50% of the sample has decayed.

Hope this helps!(3 votes)

- Does any substance with a non first order decay reaction exist ?? ( so the half life changes with its quantity)(1 vote)
- For a second order or higher, the decay process itself has to somehow depend on the presence of other molecules and interact with it. This kind of a mechanism is not true for most of spontaneous radioactive decay. But in the presence of any external stimulant, the situation can change, for example in uncontrolled nuclear fission. It is not a true spontaneous decay process, but the nucleus splits in the presence of another projectile.(2 votes)

- since its constantly being cut in half does that mean that it will never reach zero?(1 vote)
- When you reach the limit of a single atom it either decays or not there is no half of an atom of an element.(2 votes)

## Video transcript

- [Voiceover] Phosphorus-32 is radioactive and undergoes beta decay. So we talked about beta
decay in the last video. Here's our beta particle, and the phosphorus is going to turn into sulfur. Let's say we started with four
milligrams of phosphorus-32. And we wait 14.3 days, and we see how much of
our phosphorus is left. You're going to find two
milligrams of your phosphorus left. The rest has turned into sulfur. And this is the idea of half-life. Let's look at the definition
for half-life here. It's the time it takes for 1/2 of your radioactive nuclei to decay. So, if we start with four milligrams, and we lose 1/2 of that, right, then we're left with two milligrams. And it took 14.3 days for this to happen. So 14.3 days is the
half-life of phosphorus-32. And this is the symbol for half-life. So, 14.3 days is the
half-life for phosphorus-32. The half-life depends on
what you're talking about. So if you're talking about
something like uranium-238, the half-life is different,
it's approximately 4.47 times 10 to the ninth, in years. That's obviously much
longer than phosphorus-32. We're going to stick with
phosphorus-32 in this video, and we're going to actually start with four milligrams every time in this video just to help us understand
what half-life is. Next, let's graph the rate
of decay of phosphorus-32. Let's look at our graph here. On the Y-axis, let's do the amount of phosphorus-32, and we're working in milligrams here, so this will be in milligrams. On the X-axis, let's do time,
and since the half-life is in days, it just makes it
easier to do this in days. Alright, we're going to start with four milligrams of our sample. Let's go ahead and mark
this off so this would be one milligram, two
milligrams, three, and four. So we're going to start
with four milligrams. So when time is equal to
zero, we have four milligrams. Let's go ahead and mark this off. So one, two, three, and four. We wait 14.3 days, so this is 14.3 days, and half of our sample should be left. So what's half of four,
it's of course, two. And so, we can go ahead and
graph our next data point. There should be two milligrams left after 14.3 days so that's our point. Alright, we wait another 14.3 days, so we wait another half-life, so after two half-lives, that should be 28.6 days. So we know that after 28.6
days, it's another half-life, so what's 1/2 of two, it's one, of course. So that's our next point. So after 28.6 days, we should have one milligram of our sample. Let's wait another half-life. 28.6 plus 14.3, should be 42.9. So that's our next point. And what's half of one? It's 0.5, of course, so, in here, that's about 0.5, and so that gives us an idea about where
our next data point is. And we could keep going, but this is enough to give you an idea
of what the graph looks like. Right, so if I think about this graph, this is exponential decay. That's what we're talking about when we're talking about
radioactive decay here. We'll talk a little bit more about exponential decay in the next video. But this just helps you
understand what's happening. So as you increase the
number of half-lives, you can see the amount of radioactive material is decreasing. Alright, let's do a very
simple problem here. If you start with four
milligrams of phosphorus-32, how much is left after 57.2 days? So if you're waiting 57.2 days, well, the half-life of
phosphorus-32 is 14.3 days. So, how many half-lives is that? 57.2 days divided by 14.3 days would give us how many
half-lives, and that's four. So there are four half-lives,
so four half-lives here. We're starting with four milligrams, so one very simple way of doing this is to think about what happens
after each half-life. So four milligrams, if
we wait one half-life, goes to two milligrams. Wait another half-life,
goes to one milligram. Wait another half life,
goes to 0.5 milligrams. And, if we wait one more half-life, then that would go to 0.25 milligrams. So that would be our answer,
because that's four half-lives. Here's one half-life,
two, three, and four, which is how many we
needed to account for. That's one way to do the math. Another way, would be
starting with four milligrams, we need to multiply that by 1/2, and that would give us two, and then multiply by 1/2 again, and 1/2 again, and 1/2 again. So that's four half-lives, right? So this represents our four half-lives. And that's the same thing as going four, times 1/2 to the fourth power, which mathematically, is four times one over 16, so that's 4/16, so that's
the same thing as 1/4, and so that's 0.25 milligrams. So it doesn't really matter how you do the math, there are
lots of ways to do it. You should get the same answer. You could also get this on the graph if you had a decent graph. After four half-lives, you would be you would be over here somewhere. And so you could just find where that is. So let me use red, so you could find where that is on your graph, and then go over to here, so that would be approximately right here, and then read that off your graph. And that looks like about
0.25 milligrams as well. We'll talk more about
graphing in the next video.