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**Answer**

**Radioactive dating**

Radioactive dating is also called radiometric dating. This is an age-old method for determining the ages of geological materials (rocks and minerals) as well as fossils and ancient artefacts using radioactive isotopes.

Radiometric dating has also been used to date not only natural but aslo man-made materials. The basic logic here is that a acomparisson is made between the natural abundance of a radioactive isotope on earth, that of its daughter product in a sample (decayed product) and its half-life i.e its decay rates.

For accurate determination of ages of materials by radioactive dating, the following conditions are necessary;

- The half-life of the parent nuclide should be long enough so that the parent nuclide will be of significant amount at the time of measurement
- The unique half0life of the parent nuclide should be accurately known
- Sufficient amount of daughter product is produced so as to enable its accurate measurement and there should be clear distinction between nuclide produced and the initial amount of the daughter known to be present in the material

The number of atoms $N$ at any given time t, can be expressed in terms of atoms at time t=0, ${N}_{o}$ and decay constant ($\lambda $) as in the equation below

$N$ = ${N}_{o}$ ${e}^{-\lambda t}$ 1

We can also derive the equation for determining the population of daughter nuclide. Now by letting the number of daughter nuclides at t=0 be ${D}_{o}$, the population of decay product (daughter nuclide) $D$ is given by;

$D$ = ${D}_{o}$ + ${N}_{o}$- $N$, where ${N}_{o}$- $N$ is the parent nuclides decayed, now substitute $N$ in equation 1 into this equation, we obtain,

$D$ = ${D}_{o}$ + ${N}_{o}$ - ${N}_{o}$${e}^{-\lambda t}$

= ${D}_{o}$ + ${N}_{o}$ (1-${e}^{-\lambda t}$) 2

The above equation gives the population of daughter nuclide base on ${N}_{o}$, we can equivalently give the equation in terms of $N$.

$D$ = ${D}_{o}$+ $\frac{N}{{e}^{-\lambda t}}$(1-${e}^{-\lambda t}$)

= ${D}_{o}$ + $N$${e}^{\lambda t}$(1-${e}^{-\lambda t}$)

= ${D}_{o}$ + $N$( ${e}^{\lambda t}$ - 1) 3

Where t = age of the sample, $D$=number of atoms of daughter nuclide, ${D}_{o}$is the number of atoms of daugter nuclide in the original sample i.e at t=0, N= number of atoms in the parent sample at time t which is given by equation 1.

**Carbon-14 and Radiocarbon dating**

Carbon-14 commonly referred to as radiocarbon is an unstable and weakly radioactive isotope of carbon with half-life of 5730 years.Other isotopes of carbon i.e carbon 12 and carbon 13 are stable and therefore are not radioactive.

Carbon-14 dating also known as radiocarbon (${}_{14}\mathrm{C}$) dating was developed by the American physicist Willard F. Libby in the year 1946, it is a versatile method of age determination that is commonly used by archaeologists because it can date relatively recent archaeological/organic materials and fossils but not applicable to metals.

Age of materials is estimated by determining the amount of radiocarbon in a sample and campare it against international reference standard. It is suitable for determining the ages of organic materials between 500 to 50,000 years old, beyond 50000 years, the amount of ${}_{14}\mathrm{C}$ becomes insignificantly small to be measurable.

Carbon-14 dating is based on carbon-14 decaying to nitrogen. Carbon-14 is continually replenished in the atmosphere when neutrons in cosmic rays bombard nitrogen-14. Living organisms generally take in carbon from the atmosphere in form of carbon dioxide, in most instances, green plants absorb carbon-dioxide and pass the radiocarbon to animals along the food chain.

Carbon-14 is continually added to living organisms as long as they take in air and food. Absorption of carbon-14 stops immediately the organism dies and the amount in their tissues start to decay. The rate of decay is constant, thus, the date the organism died can be determined by measuring the amount of residual radiocarbon carbon.

**Other dating methods**

**Potassium-argon dating**- used to determine ages of rocks by measuring the ratio of radioactive argon to radioactive potassium within it.it is based on decay of solid ${}_{40}\mathrm{K}$ to gaseous ${}_{40}\mathrm{Ar}$. Estimates ages of rocks between 100000 to 4.3 billion years ago

**Uranium-lead dating - **A dating technique that measures the ratio of uranium isotopes ${}_{238}\mathrm{U}$ or ${}_{235}\mathrm{U}$to non-radioactive lead isotopes *${}_{206}\mathrm{Pb}$*, ${}_{207}\mathrm{Pb}$and ${}_{208}\mathrm{Pb}$. the method is used to date very old rocks whose ages are 1 million to 4.5 billion years ago. The method is reputted for its accuracy given that it has an error margin of 2 million years.

**Chlorine-36 dating**- method used to date old ground water (100000-1 million years old), this method involves calculating the amount of ${}_{36}\mathrm{Cl}$ produced in the atmosphere as a result of the interraction of cosmic rays with argon atoms.

**Luminescence dating**-Dates crystalline materials to the time they were heated by sun or human-made fires.

There are other methods such fission-track, samarium-neodymium, rubidium-strontium, and uranium-thorium, all these methods depend on aspects of radioactive decay to estimate ages of rocks

**REFERENCE**

1.Strunk, A., Olsen, J., Sanei, H., Rudra, A., & Larsen, N. K. (2020). Improving the reliability of bulk sediment radiocarbon dating. Quaternary Science Reviews, 242, 106442.

2. Goh, K. M. (1991). Carbon dating. Carbon Isotope Techniques, 1, 125.

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