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# Spontaneous Decay

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Polonium-214 and polonium-218 are both daughter isotopes formed from the disintegration of radon (Rn), a ubiquitous product of the spontaneous decay of uranium. Both Po-214 and Po-218 are considered carcinogens, and, when radon gas is inhaled, both polonium isotopes get deposited in the lung, and could eventually lead to lung carcinoma.

But, what exactly is spontaneous decay? Keep reading to find out!

• First, we will talk about what spontaneous decay means in nuclear chemistry.
• Then, we will explore nuclear decay and talk about how they are spontaneous and occur randomly.
• After, we will dive into the different types of nuclear decay and look at spontaneous decay equations involved.
• Lastly, we will briefly talk about the probability of spontaneous decay.

## Spontaneous Decay Chemistry

Here's a crash course in nuclear chemistry before we go into spontaneous decay. Nuclear chemistry deals with reactions that happen in the nucleus of an atom. More specifically, it focuses on the protons and neutrons, and ignores electrons since electrons are not found inside an atom's nucleus.

Nuclear reactions are reactions that happen inside the nucleus of an atom.

An important part of nuclear chemistry is the stability of isotopes. In an isotope, the nucleus has the same number of protons but a different number of neutrons. Stable isotopes, like for example, carbon-12, are isotopes that possess a stable nucleus.

Now, when an isotope has an unstable nucleus, they are called radioactive isotopes, and these isotopes undergo radioactive decay!

Radioactive decay (or nuclear decay) is the random and spontaneous decay of the nucleus of an unstable isotope (also called the parent isotopes) into an isotope with a stable nucleus (daughter isotope).

The image below shows the stable and unstable isotopes of carbon (C).

For an in-depth explanation on unstable isotopes and nuclear reactions, check out "Radioactive Isotopes"!

## Spontaneous and Random Nuclear Decay

Nuclear decay reactions are spontaneous and occur at random, meaning that one cannot know exactly when or which radioactive nucleus will decay.

During nuclear decay, the unstable nucleus of a radioisotope gives off lots of energy, and it often results in elements changing into different elements. In other words, a parent isotope will undergo nuclear decay and turn into a daughter isotope!

For example,

$$_{19}^{40}\text{K}$$ can undergo spontaneous nuclear decay and become $$_{18}^{40}\text{Ar}$$.

As a general rule, any element with an atomic number (number of protons) that is greater than 83 is considered radioactive, and atoms that possess a radioactive nucleus are called radioisotopes.

## Spontaneous Decay Equation

To be able to understand how spontaneous nuclear decay equations work, let's take a look at the different nuclear particles that might be involved in these nuclear decay reactions. These nuclear particles are:

• Proton particle
• Neutron particle
• Beta particle
• Position particle
• Alpha particle

### Alpha Particle

Let's start with alpha particles. An alpha particle ($$^{4}_{2}\alpha$$) is a nuclear particle that contains a mass number of 4 and a charge of +2. When a radioactive isotope undergoes a type of spontaneous nuclear decay called alpha decay.

Let's look at the alpha decay equation of polonium-210, a radioactive isotope of polonium (Po). Here, Polonium-210 emits an alpha particle ($$^{4}_{2}\alpha$$) to become the stable isotope lead-206.

$$_{84}^{210}\text{Po}\longrightarrow _{2}^{4}\alpha\text{ + }_{82}^{206}\text{Pb}$$

Alpha decay usually occurs in unstable isotopes that have an atomic number that surpasses 82. For example, Uranium-238 most likely undergoes alpha decay since its atomic number (92) is greater than 82.

Remember that mass number is the number of protons plus the number of neutrons.

### Beta Particle

Next, we have beta particles. A beta particle ($$^{0}_{-1}\beta$$) possesses a mass number of 0 and a -1 charge. Beta particles are emitted from a radioisotope's nucleus during beta decay.

In the case of beta decay, radioisotopes tend to undergo beta decay when its mass number is higher than the mass number for that particular element in the periodic table. For instance, the radioactive isotope of carbon (carbon-14), undergoes beta decay, emitting a beta particle and forming nitrogen-14.

• The mass number of the carbon-14 is greater than the mass number of carbon in the periodic table (14 > 12).

The beta decay equation for this reaction is shown below.

$$_{6}^{14}\text{Po}\longrightarrow _{-1}^{0}\beta\text{ + }_{7}^{14}\text{N}$$

Beta particles ($$^{0}_{-1}\beta$$) are also a part of another type of nuclear decay called electron capture. During electron capture, a beta particle gets absorbed into the nucleus. Contrary to beta decay, electron capture tends to happen in radioisotopes with a mass number lower than that on the periodic table.

For example, Argon-37 has a lower mass than the mass of Argon (Ar) given in the periodic table (39.948).

$$_{-1}^{0}\beta\text{ + }_{18}^{37}\text{Ar } \longrightarrow \text{ } _{17}^{37}\text{Cl}$$

### Positron Particle

Now, while positron particles ($$^{0}_{1}\beta$$) also have a mass number of 0, they are the opposite of beta particle in that they have a +1 charge. Positron particles are involved in types of nuclear decay called positron emission.

As the name positron emission suggests, a positron particle gets emitted from the nucleus, and the radioactive isotopes that are most likely to undergo positron emission are those that have a mass number less than the one given on the periodic table.

As an example, let's look at the iodine-117. On the periodic table, the mass of iodine is 126.90. Since 117 is less than 126.90, this radioisotope can undergo positron emission to produce the daughter isotope 117Te.

$$_{53}^{117}\text{I}\longrightarrow _{1}^{0}\beta\text{ + }_{52}^{117}\text{Te}$$

Notice that, radioactive isotopes containing a mass lower than the mass of the periodic table can undergo either positron emission or electron capture!

### Proton and Neutron Particle

A proton particle ($$^{1}_{1} \text{p}$$) has a mass number of 1 and a charge of +1, whereas a neutron particle ($$^{1}_{0} \text{n}$$) has a mass number of 1 and a charge of 0.

Neutron particles ($$^{1}_{0} \text{n}$$) are commonly seen being emitted during the process of nuclear fusion and also during nuclear fission.

• Two lighter nuclei join together to form a heavier, more stable nucleus during nuclear fusion. In the process, a neutron particle is released.
• During nuclear fission, a heavy nucleus gets split up into two small nuclei, releasing two neutron particles.

Interested in learning more about nuclear fission and nuclear fusion in more detail? Check out "Nuclear Fusion and Fission"!

## Spontaneous Decay Example

Now that we know that spontaneous nuclear decay is, let's look at some examples.

What type of spontaneous nuclear decay will $$_{89}^{207}\text{Ac}$$ most likely undergo?

The first step in solving this type of problem is to look at the atomic number for actinium (Ac) given in the periodic table. Since the atomic number of Ac in the periodic table is greater than 82, then we can say that this radioactive isotope will most likely undergo alpha decay.

$$_{89}^{207}\text{Ac} \longrightarrow _{2}^{4}\alpha\text{ + }_{87}^{203}\text{Fr}$$

Now, what if the radioisotope we are dealing with does not have an atomic number greater than 82? In this case, we need to look at the mass number of that element in the periodic table.

What type of spontaneous nuclear decay will $$_{13}^{28}\text{Al}$$ most likely undergo?

Since Al-28 has an atomic number that is less than 82, we need to turn our attention to the mass number of Al. The mass of aluminum (Al) is the periodic table is given as 26.982, which is less than the mass number of Aluminum-28 (28 > 26.982).

So, $$_{13}^{28}\text{Al}$$ will most likely undergo beta decay.

$$_{13}^{28}\text{Al} \longrightarrow _{-1}^{0}\beta\text{ + }_{14}^{28}\text{Si}$$

To learn how nuclear reactions are balanced, read "Balancing Nuclear Equations"!

## Probability of Spontaneous Decay

The half-life of isotopes can be used to help indicate the probability of decay, together with the exponential decay law.

The half-life is the time is takes for half of the radioactive nuclei to decay/disintegrate.

For example, thallium-201 has a half-life of 73 hours. This means that it takes 73 hours for one-half of its nucleus to decay into the daughter isotope mercury-201. Now, the mean lifetime of Tl-201 is around 105.33 hours.

$$_{81}^{201}\text{Tl }\text{ + } _{-1}^{0}\beta \text{ }\longrightarrow \text{ }_{80}^{201}\text{Hg}$$

The exponential decay law formula is as follows:

$$N(t) = N_{0}e^{−kt}$$

Where:

• $$k$$ is equal to the decay constant.
• $$t$$ is the half-life of the radioisotope.
• $$N_{0}$$ is the initial number of nuclei

The exponential decay law tells chemists that the probability of nuclear decay per unit time is constant, and it differs between different radioisotopes.

This law also implies that the probability of survival (P) is equal to the final number of nuclei divided by the initial number of nuclei. Therefore, the probability of decay is 1 - P.

$$\text{P }= \frac{N}{N_{0}}$$

Now, I hope that you were able to gain a better understanding of what spontaneous decay is!

## Spontaneous Decay - Key takeaways

• Nuclear reactions are reactions that happen inside the nucleus of an atom.
• Radioactive decay (or nuclear decay) is the random and spontaneously decay of the nucleus of an unstable isotope (also called the parent isotopes) into an isotope with a stable nucleus (daughter isotope).
• The four types of spontaneous nuclear decay are alpha decay, beta decay, electron capture, and positron emission.

## References

1. Zumdahl, S. S., Zumdahl, S. A., & Decoste, D. J. (2019). Chemistry. Cengage Learning Asia Pte Ltd.
2. Theodore Lawrence Brown, Eugene, H., Bursten, B. E., Murphy, C. J., Woodward, P. M., Stoltzfus, M. W., & Lufaso, M. W. (2018). Chemistry : the central science (14th ed.). Pearson.
3. Swanson, J. (2021). Everything you need to ace chemistry in one big fat notebook. Workman.
4. House, J. E., & Kathleen Ann House. (2016a). Descriptive inorganic chemistry. Amsterdam ; Boston ; Heidelberg ; London ; New York ; Oxford ; Paris ; San Diego ; Singapore ; Sydney ; Tokyo Elsevier.
5. Wolfram|Alpha: Making the world’s knowledge computable. (n.d.). Www.wolframalpha.com. https://wolframalpha.com
6. Kumar, V., Abbas, A. K., Aster, J. C., Cotran, R. S., & Robbins, S. L. (2021). Robbins & Cotran pathologic basis of disease. Elsevier.

## Frequently Asked Questions about Spontaneous Decay

Spontaneous decay refers to the spontaneous radioactive decay of the unstable nucleus of a parent isotope into a daughter isotope with a stable nucleus.

Spontaneous decay is caused by the radioactivity of the nucleus of some isotopes.

In chemistry,  spontaneous decay is simply the radioactive decay of unstable isotopes into an isotope of stable nucleus.

The spontaneous decay of potassium-40 into argon-40 is an example of a balanced radioactive decay equation.

A nuclear equation that shows the disintegration of an unstable nucleus represents spontaneous decay.

## Spontaneous Decay Quiz - Teste dein Wissen

Question

Nuclear reactions are reactions that happen ____ the nucleus of an atom.

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Answer

inside

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Question

Isotopes have the number of ____ but a different number of neutrons.

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Answer

protons

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Question

Carbon-12 is an example of a _____ isotope of carbon.

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Answer

stable

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Question

When an isotope has an unstable nucleus, they are called _______.

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Answer

Radioactive isotopes

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Question

True or false: Radioactive isotopes undergo nuclear decay

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Answer

True

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Question

Radioactive decay (or nuclear decay) is the random and _____ decay of the nucleus of an unstable isotope (also called the parent isotopes) into an isotope with a stable nucleus (daughter isotope).

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Answer

spontaneus

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Question

Radioactive isotopes decay into stable isotopes also known as ______.

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Answer

daughter isotopes

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Question

What is the mass number of an alpha particle?

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Answer

4

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Question

Polonium-210 undergoes _____ decay to form lead-206.

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Answer

Alpha decay

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Question

_____ is the number of protons plus the number of neutrons.

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Answer

mass number

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Question

A ____ particle  has a mass number of 0 and a -1 charge

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beta

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Question

A ____ particle  has a mass number of 0 and a +1 charge

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Answer

positron

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Question

During electron capture, a beta particle gets _____.

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Answer

absorbed

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Question

True or false: radioactive isotopes containing a mass lower than the mass of the periodic table can undergo either positron emission or electron capture.

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Answer

True

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Question

What is the definition of half-life?

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Answer

The half-life is the time is takes for half of the radioactive nuclei to decay/disintegrate.

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