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Why is the rate of decay of a substance directly proportional to the amount of nuclei present in the substance?

I don't know much about this topic, my teacher introduced us to this concept in class today, i couldn't wrap my head around this, because it felt really absurd. Some phenomenons which may be predicted based on this law is that the substance may never decay completely. And may keep approaching zero.

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I'd like to start by saying well done. It's really good that you started examining model as soon as you were introduced to it, and thought about the behaviours it predicts. Nice job.

Anyway, as I've alluded to, what you're describing is a model. Rather than the atoms colluding and agreeing that in a given time span a certain percentage of them will decay, what happens is that any given atom has a certain probability of decaying in said time span. (Let's say 5%, just for example.) If there is a statistically large number of atoms (and there will be in any sample you can handle on a lab bench) then almost exactly 5% of the atoms will have decayed. However, as the number of atoms decreases this is less likely to hold true. As an example, if you were to roll 600 dice, you would expect close to 100 dice to show a six. If you were to roll only 6 dice, you wouldn't be surprised if none of them came up six. The same thing happens with the atoms. Measurements of the decay rate will get less and less likely to match the statistical ideal as the population of undecayed atoms drops, until eventually there will be only a single atom, and obviously by that point the only thing that can happen is none of the sample decays or all of the sample decays.

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I will copy from this link, which starts from basics.

Radioactive decay is a statistical process which depends upon the instability of the particular radioisotope, but which for any given nucleus in a sample is completely unpredictable. The decay process and the observed half-life dependence of radioactivity can be predicted by assuming that individual nuclear decays are purely random events. If there are N radioactive nuclei at some time t, then the number ΔN which would decay in any given time interval Δt would be proportional to N: $$ΔN=-λ\cdot N\cdotΔt$$

where $λ$ is the proportionality constant. It is the simplest logical assumption, assuming constant depletion by the $λ$ per unit time, to start describing statistical depletion. This would describe any population depletion,given a constant rate of depletion in time.

It leads to the exponential form as shown analytically here.

$N=N_0\cdot e^{-λt}$

The experimental curves of radioactive decay follow this form and justify the constant rate in time.

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$\let\D=\Delta \let\lam=\lambda$ What @anna v says, her quotation included, is all OK, but for a fundamental point. It is where she writes

It is the simplest logical assumption, assuming constant depletion by the $\lam$ per unit time, to start describing statistical depletion.

To illustrate my point I choose to take a special example: the decay of $^{226}\mathrm{Ra}$, an isotope of radium, $\alpha$-decaying with half-life 1600 years. The first question someone novel to the field should ask is: how so long a time span could be measured? The answer resides in one formula @anna v quotes: $$\D N = -\lam\,N\,\D t.\tag1$$

Assume you are given a sample of substance where you are said there is some radium. If you are an expert chemist you will be able to make a quantitative analysis, in particular to determine the total mass of radium contained in the sample. Then, knowing radium molar mass and Avogadro's constant, you will calculate $N$ (I don't give details, as they are inessential to our purpose).

Now you - as a clever experimental physicist - setup the equipment to count the number $\D N$ of Ra-nuclei that will decay in a certain time $\D t$. Once you hold $N$, $\D t$ and $\D N$ eq. (1) gives $\lam$, and you are done. Well not quite: relation between $\lam$ and half-life is simple, but requires you are familiar with exponentials and logarithms. Are you? It doesn't matter. Content ourselves with a rough idea: half-life is not far from $1/\lam$.

My aim was simply to show how it is possible to determine a half-life of 1600 years without expecting a comparable time. It is possible because $N$ will likely to be an extremely high number. Just to give an example: in a microgram of $^{226}\mathrm{Ra}$ there are about $2.7\cdot10^{15}$ nuclei (almost 3 quadrillions). So also in a small fraction of half-life you will observe a substantial number of decays, and this makes the measurement a feasible affair.


But the above was only a preparation for the real problem. Our friend, a newcomer to radioactivity, has been satisfied as to his first question, and has another ready. He asks "Where does radium come from? I ask this, because 1600 years are a long time on human scale, but not for many natural processes. If we can find some radium around, it must have been created - at most - a few thousands years ago. I heard radium is found in very ancient minerals, maybe billion years old. How did radium enter those rocks in relatively recent times?"

The teacher knows the answer and replies: "Very good question. Actually radium belongs to a 'family' of radioactive elements, whose 'father' is $^{238}\mathrm U$ (uranium). After a chain of several decays I will not detail, radium is born. The point is that the father has a very long half-life: 4.5 billion years. We know that $^{238}\mathrm U$ was generated well before Earth formation, in a primordial supernova explosion. Debris of that explosion formed the cloud whose gravitational collapse gave origin to solar system, about 5 billion years ago. So, even if the subsequent decays are relatively fast, there is always enough uranium to replenish the chain."

An immediate reply follows: "I see, but your explanation raises a serious doubt. If radium nuclei are being continuously created anew, I expect that in my sample there are nuclei of all ages. How can they possibly all follow in their decay the same law? Shouldn't they in some sense 'remember' their birth date?"


Now I'll stop my "fiction" and speak myself. The problem is the following. It is reasonable to assume that all radium nuclei are born in the same state, at different times. They will evolve from that state according to the same law, but each from a different time origin. How can we explain that - as far as we can see - their time evolution (which we see as a decay probability) remains the same starting from present time, if present time means a different previous time span of their life?

I don't want to write equations, but this is a real problem for quantum mechanics. It is not trivial that it has actually been solved in the affirmative, bat with some caveats. It is generally true that the probability of finding an undecayed nucleus decreases exponentially with time. But this is not true for $t$ very small or very large.

Unfortunately mine is a distant memory and I'm not able to give references. Maybe some reader can help?

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