radioactive decay

Question

Suppose we have two radioactive nuclei in a closed box (we can't see inside without opening it). Half life of these nuclei is 12 hrs.

1. What will we find if we open the box after 12 hrs?
2. What will we find if we open it after 24 hrs?
3. Suppose we first look after 12 hrs and find only one nuclei has decayed; then we close the box and look again after another 12 hrs. What will we find now?

I think, if we look after 12 hrs we will find only one nucleus decayed but not sure about other two cases.

Please answer with explanation. Hope it will increase my understanding.

Answer 1

Radioactive decay can be explained by a simple differential eqution. Experiments have shown that number of decays per unit time is propotional the current number of nucleis always. So $$\frac{dN}{dt}=-\lambda N$$ where $\lambda>0$ is decay constant, and larger $\lambda$ leads to faster decay. The solution is $$N(t)=N(0)e^{-\lambda t}$$ Clearly, $N(t)\neq 0$ whenever the decay time increases.

Half life definition: $$N(T_{1/2})=\frac{1}{2}N(0)=N(0)e^{-\lambda T_{1/2}}$$ So, we have $$T_{1/2}=\frac{ln2}{\lambda}$$ which means after time $T_{1/2}$, current number of nucleus will be half of the original number. After two Half life, $t=2T_{1/2}$, current number is $$N(t)=N(0)e^{-2\lambda T_{1/2}}=\frac{N(0)}{4}$$ So, we can obtain a simple relation $$N(nT_{1/2})=\frac{N(0)}{2^n}$$ Obviously, it is never to reach zero.

Answer 2

Suppose you have two pennies in the box and you shake the box once every 12 hours, then open it. Any penny that lands on tails "decays" (since decay means that it becomes a more stable nucleus, and ultimately reaches a state where it cannot decay any further, we will represent this by gluing the penny to the bottom of the box).
After 12 hours (and one shake) what will you find? It's a matter of probability. You might find two heads, you might find two tails, you might find one of each. But if you repeated this with 1000 boxes (or 1000 pennies in a single box), then on average you would find 50/50 heads and tails.
Half life is not a guarantee, it's a probability. But with larger sample sizes, actual outcomes match probabilities with greater reliability.

Answer 3

Radioactive decay is a random process so anything could have happened. The idea of half life makes sense only if you have a very large number of nuclei. Consider tossing a coin.The probability of head and tail are 1/2 each.Assume you are going to perform two toss. Say the first one gives you a head. Now that head has occurred once doesn't mean tail should necessarily occur in the second toss just because he probability of each is 1/2. However if you toss a coin millions of times then within reasonable error you will get equal number of heads and tails.