Since naturally occurring uranium (U-238) has a half life of 4.5 billion years, then it means half of the uranium on earth has decayed into lead by now. But why only half, and why that specific half? What was special about the particles that did decay? Were they different in some way?

In: 76

It doesn’t, it decays with random chance. After 4.5 billion years a single uranium-238 atom has a 50% chance to have decayed.

There are so many atoms in a sample of uranium, that after 4.5 billion years, almost exactly half of them have decayed. It’s just a matter of statistics.

For any given sample, if you could count every atom, it may vary from sample to sample how many have decayed, but there’s so many that it’s just a rounding error.

Every single atom of uranium 238 has at any time a very small chance of decaying. That rate of decay just happens to coincide with half of it decaying every 4.5 billion years. The uranium that hasn’t decayed has essentially just been “lucky” so far, and probability dictates how that distribution is met.

The main thing you’re missing is the impossibly huge number of radioisotopes in a sample. In grams of uranium you’d have about 10 with 21 zeros after it nuclei. At that point, it’s not about individual atoms, you’re safely in the realm of statistics.

If each atom has a certain probability to decay, and that probability is truly random and independent of all the other atoms, then the sample as a whole will exhibit exponential decay. You can calculate the time by which half of the atoms will have decayed, and you have no idea which ones, but probabilities are basically facts when you have a billion trillion of something.

Once you have half of them left, you still have to wait just as long for half of what’s left to decay. You’re starting with half as many, but you’ll get half the decays because there are half as many, so the half life is the same.

Noting is special, all atoms are identical and have not memory

Half-life is the mathematical result if the probability of something happening at each moment in time is constant. It does not matter if a radioactive isotope was formed 1 billion years ago or 10 minutes ago. The probability if it decaying within the next minute is the same. Atoms do not age.

Let’s assume you have 1000 atoms and each has a 5% chance of decaying each minute. That is equal to 95% remaining.

After 1 minute 1000 * 0.95 =950 atoms likely remain. After two minutes it will be 950 *0.95 = 1000 * 0.95 * 0.95= 1000* 0.95^(2) = 902.5 lets round it up to 903 atoms. after n minutes you have 1000* 0.95^(n) atoms left

If you continue that you get the table at the end of the post.

The number of atoms is half to 500 after approximately 13.5 minutes. It is halved again to 250 at 27 minutes which is 13.5 minutes after the first halving. Halved again to 125 after 41.5 minutes, again 13.5 minutes later. If we did not round the number it would continue forever.

The time it takes for half of the atoms is independent of the number of atoms you start with, 1000 is just a constant, and half remains when 0.95^(n)=1/2. If you solve that equation you get n=13.51.

A U-238 atoms have a very low chance of decaying in each moment in time. It has no memory so the chance does not change over time. The result is that it takes a very long time for half of them to decay.

If you put $100 into a bank account and get 5% yearly interest you can calculate the time it takes to double. The double time is the same idea as half-life, the different ei just that one is for grown and the other for reduction each moment in time

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0 1000

1 950

2 903

3 858

4 815

5 774

6 735

7 698

8 663

9 630

10 599

11 569

12 541

13 514

14 488

15 464

16 441

17 419

18 398

19 378

20 359

21 341

22 324

23 308

24 293

25 278

26 264

27 251

28 238

29 226

30 215

31 204

32 194

33 184

34 175

35 166

36 158

37 150

38 143

39 136

40 129

41 123

42 117

43 111

44 105

45 100

46 95

47 90

48 86

49 82

50 78

51 74

52 70

53 67

Imagine you have a big bag of popcorn kernels. Every minute, each kernel has a tiny chance to pop. After a certain time, let’s say an hour, you notice that about half the kernels have popped. That’s similar to the “half-life” of radioactive materials.

Now, for the uranium:

Predictable but Random: Even though the popcorn pops at a predictable rate (half in an hour), you can’t tell which specific kernels will pop first. It’s the same with radioactive atoms; we know half will decay after a certain time, but we can’t say which ones.

Nothing Special About Them: There’s nothing different about the popcorn kernels that popped first. Similarly, the uranium atoms that decayed weren’t special or different.

Always a Chance: Even after an hour, the remaining popcorn kernels can still pop. In the same way, the remaining uranium atoms will keep decaying, but it’ll take another 4.5 billion years for half of those to turn into lead.

So, it’s all about probability. Like rolling dice or flipping a coin, but on a much bigger and longer scale!