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# Half Life

Half-life is a measure of the time it takes a radioactive sample to decrease its mass or quantity by half and, among other things, its danger. However, the half-life isn’t just about the danger of radioactive substances – we can also use it for many other applications, such as carbon-14 dating techniques.There are certain elements in nature whose atoms have an excess of particles or…

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# Half Life

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Half-life is a measure of the time it takes a radioactive sample to decrease its mass or quantity by half and, among other things, its danger. However, the half-life isn’t just about the danger of radioactive substances – we can also use it for many other applications, such as carbon-14 dating techniques.

## What is nuclear decay?

There are certain elements in nature whose atoms have an excess of particles or energy, making them unstable. This instability causes nuclei to emit particles to achieve a stable state with a different number or configuration of particles in the nucleus.

The emission of particles by nuclei is known as nuclear decay (or radioactive decay). It is a quantum effect whose characterisation for samples with a large number of atoms is very well known.

The consequence of decay being a quantum effect is that it occurs with a certain probability. This means that we can only speak about the probability of a certain decay happening over a certain period.

For instance, if we predict that the probability of a particular nucleus decaying into another one is 90% after one day, it may happen in one second or one week. However, if we have a lot of identical nuclei, 90% of them will have decayed after one day.

This is the general equation that models this effect:

$N(t) = N_0 \cdot e^{-\lambda t}$

N(t) is the number of unstable nuclei at time t, N0 is the initial number of unstable atoms in our sample, and λ is the decay constant, which is characteristic of each decay process.

See our article on Radioactive Decay for a graph and more examples.

## What is half-life?

Half-life is the time it takes a sample of a certain unstable isotope to half its number of unstable nuclei.

At first, this concept seems odd since we would expect that the time it takes for a sample to lose half of its components is constant. We are used to a constant rate of phenomena, like losing a fixed amount of unstable nuclei in a certain period. However, the equation implies that this is not the case for nuclear decay.

### The half-life symbol and half-life equation

Suppose we look at a sample at a specific time t1 > 0 and then at a later time t2 > t1. If we want to find the ratio of the number of unstable atoms in the sample, we only need to divide their expressions:

$\frac{N(t_2)}{N(t_1)} = \frac{N_0 \cdot e^{-\lambda t_2}}{N_0 \cdot e^{-\lambda t_1}} = e^{-\lambda (t_2-t_1)}$.

This relation gives us two important (related) facts:

1. The ratio between the numbers of unstable nuclei at two different times is independent of the initial number of unstable nuclei. Since the decay constant for a specific element is given, we know that for a specific time interval t1 - t2, the number of unstable nuclei will decrease in the same percentage (ratio).
2. Given that the percentage decrease of unstable nuclei is the same for a fixed interval, the decrease is much faster at earlier times because the total number of unstable nuclei is bigger.

An example showing radioactive decay as a function of time where the y-axis gives the number of particles as a percentage of the initial value

When we divide the number of unstable atoms at different times for a fixed interval, we obtain the same quantity.

• For instance, if we consider time intervals of 1 second, we can divide the amount at 1 second by the amount at 0 seconds and obtain 1/2. If we do the same with the amounts at 2 seconds and 1 second, we obtain the same rate, and so on.

These quantities reflect that the percentual decrease is constant for fixed time intervals. For one second, the percentual decrease is 50%, while for 2 seconds, it has a value of 75%, and so on.

The percentual decrease also has a relevant effect regarding the total number of unstable atoms in the sample, which shows us that the rate of decrease of the total number of unstable nuclei is faster at earlier times.

• For instance, if we consider time intervals of 1 second, the number of unstable atoms decreases by 5 during the first second, while the decrease is only 2.5 for the next second. If we consider two seconds, the decrease will be 7.5 for the first second and 1.875 for the next two seconds.

This is why radioactive samples become less and less dangerous as time passes. Although their perpetual decay rate is constant (which is helpful for applications like date samples), the absolute number of decays decreases with time. Since fewer atoms are decaying with time, fewer particles will be emitted from the nuclei in these decaying processes.

If we now focus on a ratio of one-half, we can find the expression for the half-life. The symbol for half-life is usually $$\tau_{1/2}$$.

$e^{-\lambda \tau_{1/2}} = \frac{1}{2} \rightarrow \tau_{1/2} = \frac{\ln(2)}{\lambda}$

This expression confirms that the time it takes for a radioactive sample to lose half of its unstable nuclei depends only on the isotope (decay constant) and not on the number of unstable nuclei. Thus, it is constant.

Below is a table with some values for the half-lives of certain isotopes.

 Element Half-Life Radium-226 1600 years Uranium-236 23,420 million years Polonium-217 1.47 seconds Lead-214 26.8 minutes

Here you can see that some isotopes have a very short half-life. This means they decay very fast and almost don’t exist in nature. However, like uranium-236, others have a very long half-life, making them dangerous (like the radioactive waste from nuclear power plants).

## What are some applications of half-life?

Half-life is a valuable indicator of the age of a sample or the needed containment time of a particular material. Let’s look at this in more detail.

### Carbon-14 dating techniques

Carbon plays an essential role in the functioning of organic beings. Although carbon-12 and carbon-13 are stable isotopes, the most abundant is carbon-12, which we typically find in every organic structure. We also find an unstable isotope (carbon-14) on Earth, which is formed in the atmosphere due to radiation from outer space.

If you refer to our explanation on Radioactive Decay, you can find more information and examples about carbon-14 dating. Just know that we can accurately estimate the deaths of humans and animals using carbon-14 dating.

### Storage of dangerous materials

The decay equation helps calculate how long radioactive materials need to be stored so that they no longer emit large amounts of radiation. There are three kinds of waste:

• Low-level waste from hospitals and industry. These emit low levels of ionising radiation, which are still enough to pose some environmental threat. This waste may require some combination of shielding, incineration, or compacting for shallow burial. The half life of materials of this kind can reach approximately five years.
• Intermediate-level waste, such as sludge, fuels, and chemical waste. These materials require shielding; solidification in concrete, bitumen, or silica; and burial in relatively shallow nuclear storage sites (repositories). The half life of materials of this kind range from five to 30 years.
• High-level waste, like heavy atomic elements (uranium, for example) and materials involved in nuclear fission. These products must be cooled first and then subjected to deep geological burial in concrete and metal containers for a very long time. The half lives of these kinds of materials are typically over 30 years.

### Tracers

Gamma emitters are used as tracers because their radiation is not very dangerous and can be accurately detected by specific devices. Some tracers are used to trace a substance’s distribution in a medium, like fertilisers in the soil. Others are used for exploring the human body, which means that they don’t have a very long half-life (they don’t emit radiation for a long time inside the body and damage it).

Decay calculations can also determine whether a radioisotopic tracer is fit for use. Tracers can neither be highly radioactive nor not radioactive enough because, in the latter case, radiation would not reach the measuring devices, and we would not be able to detect or “trace” them. In addition, the half-life allows us to classify them by the rate of decay.

## Half-Life - Key takeaways

• Half-life is the time it takes a sample of a certain unstable isotope to half its number of unstable nuclei.
• The process of unstable nuclei transforming into stable nuclei is called nuclear decay (or radioactive decay).
• Decay is a random process, but it is very accurately described by exponential decay when considering samples with a large number of unstable nuclei.
• The half-life of objects is a relevant quantity with many fruitful applications ranging from dating techniques to the handling of radioactive waste.

Half life is the time it takes a sample of a certain unstable isotope to half its number of unstable nuclei.

If you know the decay constant λ, you can apply the following equation to calculate the half life: τ = ln (2)/λ.

The half life of a radioactive isotope is the time it takes a sample of a certain unstable isotope to half its number of unstable nuclei.

By looking at a graph of radioactive exponential decay, you can find the half life by simply looking at the time interval passed where the number of unstable nuclei has decreased in half.

If you know the decay constant λ, you can apply the following equation to calculate the half life: τ = ln (2)/λ.

## Half Life Quiz - Teste dein Wissen

Question

What is used to date organic structures?

Carbon-14.

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Question

What type of radiation do tracers usually emit?

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Question

The rate of percentual decrease is constant in radioactive decay.

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Question

The waste from nuclear power plants has a very high half life.

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Question

Why should we consider large samples of atoms to study radioactive exponential decay?

We should consider large samples of atoms to study radioactive exponential decay because decay is a random process.

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Question

Why is uranium-236 dangerous?

It has an extremely long half life.

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Question

Can we calculate the time it takes a radioactive sample to decrease by 90%?

Yes, we can calculate the time it takes a radioactive sample to decrease by 90% with the exponential decay equation.

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Question

Does the half life depend on the initial number of unstable nuclei?

No, the half life only depends on the isotope.

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Question

Is radioactive decay a quantum effect?

Yes.

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Question

After twice the half life has passed, what is the percentage of unstable nuclei?

The percentage of unstable nuclei will be 25% of the original number of unstable nuclei.

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Question

Is radioactive waste usually stored in special places to contain the radiation?

Yes.

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Question

Is it certain that after a half life has passed, there will be exactly 50% of the original unstable nuclei?

No, it will be approximate.

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Question

Is the decay rate of the total number of unstable nuclei faster at earlier times?

Yes.

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