Fully Understanding Half-Life in Radiation

[u/_Rexis__]

1. my first question would be, how often does U-235 as an example, shoot out a ray of alpha radiation. Alpha radiation is a helium atom, but how often does that happen? because the half-life of U-235 is 700 million years, it'd take 100 g that many years to become 50 g. But throughout those 700 million years, is the alpha decay a constant drip?
2. If I only have 1 atom of U-235, does that mean its just neutral for 700 million years, until it eventually shoots out 1 helium atom and decays?

[u/Quantumtroll]

An intuitive way to think about radioactive decay is that each individual atom of U-235 has an equal chance of decaying in a given time-span. That chance is equal to 50% in 700 million years. So in 700 million years, you'd expect about half of the U-235 you have today to have decayed.

Now to your questions:

1. The total flux of alpha particles will decrease over time, because the amount of U-235 will decrease over time. After 700 million years, because half of the uranium has decayed, you'd measure half as much alpha radiation coming off your hunk of metal.

2. Yes. That single atom will just sit there until it randomly falls apart without warning.

[u/Hivemind_alpha → u/Quantumtroll]

I’d adjust your answer to (2). That single atom might decay in 15 seconds from you starting to observe it, or 23 hours, or 5 years and 7 months, or 8 centuries, or even still be around and intact in 3,000 million years. All you can say is that there’s a 50% chance it will have decayed before 700 million years from its creation. It could be shorter, it could be longer.

Fortunately predicting single atoms is rarely required (and not possible). What we can talk about with much higher confidence is the group behaviour of a block of uncounted trillions of atoms. Statistically we can be certain that as a group they will follow the decay curve for their known half life.

[u/karlnite → u/Hivemind_alpha]

If you isolated just one atom of U-235 so that no other atoms are around it, will it still decay?

[u/Hivemind_alpha → u/karlnite]

Yes, at an unpredictable time.

[u/Ch3cksOut → u/karlnite]

The decay is driven by intra-nuclear forces, so it does not matter at all what is around it

[u/CosineDanger → u/Ch3cksOut]

There are some exceptions to the rule that the environment does not influence nuclear decay. Electron capture decay requires inner electrons to be present, and how fast it happens can be lightly influenced by chemical bonds.

It generally doesn't though.

It can in the context of nuclei deep inside massive stars or on the crust of a magnetar.

[u/Ch3cksOut → u/CosineDanger]

This is good context in general, but does not apply to the question I responded to, about a lone U atom

[u/BananaResearcher]

1. it's a random process, described statistically such that we can say that any given mass of uranium-235 will be half that mass in a certain amount of time (half-life). This obviously then means that how many alpha particles are being released per unit time depends on how much u-235 you have. The math would be (mass of uranium -> convert to atoms of uranium -> take half and divide by half-life = number of alpha particles per unit time).

I have nothing better to do so 100g of uranium is .42mol Uranium. A mole is 6.02x10²³ atoms.

So: 100g U-235 = 0.42mol U-235 atoms. .42*6.02e23 = 2.53e23 atoms. Half will decay in 700mil years, each releasing an alpha particle (we're ignoring everything else). So 1.265e23 alpha particles. 1.265e23/7e8 = 1.807e14 alpha particle per year = 5.73e6 alpha particles per second, i.e., 5.73 million helium atoms per second. Or something close enough.

1. It's a random process. The single u-235 atom could decay in a second, it could decay in a billion years. It's random. Half-life is usually understood as a bulk property of materials but it's a reflection of the statistical rate of decay, which is random.

[u/sfurbo → u/BananaResearcher]

That's the rate averaged over those 700 million years. The instant rate is 2*ln(2) times, or around 40%, higher.

[u/Ch3cksOut]

If I only have 1 atom of U-235, does that mean its just neutral for 700 million years, until it eventually shoots out 1 helium atom and decays?

This is not how it works. Mathematically, each atom has a probability of decay per unit time as it decay constant:
λ = -(dN)/N/dt
Integrating this gives the familiar exponential rate formula
N = N0*exp(−λt) from which the half life can be expressed as
T½ = ln(2)/λ

For U-235, λ is 9.85 × 10⁻¹⁰ years⁻¹ - your single atom will have this same probability of decay per year, every year throughout its life. The math works out such that, when you start with a large number of atoms, half of them would decay within the half-life. But the smaller this number the larger the statistical fluctuation around the expected number. For your lone atom, there is 50% probability that it'd decay within 700 million years; then, conditional on its survival, it'd have 50% probability for surviving the next 700 million years - and so on.

[u/Frangifer]

If we take any window of time whatsoever , of length t , the probability that the nucleus will still be there @ the end of that window of time is

exp(-λt) ,

where λ = ㏑2/HalfLife . It doesn't matter in the slightest how long that nucleus has already existed - ie @ what particular epoch that window-of-time is set in the total life-history of the nucleus ... or putting it another way: the process of radioactive decay is a totally memoryless one.

 

There's a distribution known as the Weibull distribution that incorporates probability-density of an event being a power-law function of time - the process has a 'memory function' of the form of a power-law woven-into it: if the rate of a thing happening in an ensemble of N things that it hasn't yet happened to is

λ(t/t₀)ᶱN

then the number of things remaining that it hasn't yet happened to is given by

dN/dt = -λ(t/t₀)ᶱN

∴

d⁩㏑P/dt = -λ(t/t₀)ᶱ

or, translating into terms of probability, the probability P of its not having happened to some one particular thing is given by

d⁩㏑P/dt = -λ(t/t₀)ᶱ

∴

P = P₀exp(-λt₀(t/t₀)ᶱ⁺¹/(ϴ+1)) ,

which is the Weibull distribution.

Or we could have a probability distribution for any memory function on the right-hand-side of the differential equation

d⁩㏑P/dt = -λfᐟ(t/t₀)

∴

P = P₀exp(-λt₀f(t/t₀)) .

And the exponential distribution is the case fᐟ(t/t₀) ≡ 1 , or ϴ = 0 in the particular case of the power-law memory function that gives-rise to the Weibull distribution ... ie the absolute value of t is of no significance whatsoever ... ie the distribution is memoryless .

[u/EventHorizonbyGA]

1. my first question would be, how often does U-235 as an example, shoot out a ray of alpha radiation. Alpha radiation is a helium atom, but how often does that happen? because the half-life of U-235 is 700 million years, it'd take 100 g that many years to become 50 g. But throughout those 700 million years, is the alpha decay a constant drip?

This is not correct. You will have 50 g of U-235 and ~50 g of other elements.

1. If I only have 1 atom of U-235, does that mean its just neutral for 700 million years, until it eventually shoots out 1 helium atom and decays?

How often does a single U-235 atom spontaneously decay via alpha emissions? You have about a one in 3.12×10−17 chance of that happening per second. What that means is every second from now there 1.4B years from now you have the exact same chance of it happening.

[u/mfb- → u/EventHorizonbyGA]

What that means is every second from now there 1.4B years from now you have the exact same chance of it happening.

On the condition that the atom is still there, you have the same 3*10^-17 chance for the next second forever. But the chance that the atom is still there decreases. If we take the perspective of today, then a decay within the next year is twice as likely as a decay within a year 700 million years in the future because there is a 1/2 chance it decays before that timespan. There is nothing special about 1.4 billion years here.

[u/EventHorizonbyGA → u/mfb-]

It is believed life on Earth will end in 1.4B years due to the Sun. I some times forget not everyone on reddit was a Physics Professor.

Sorry if my sarcasm was confusing.