Question

What is the Half-life of Potassium 40.

Answer 1

Hint :A quantity's half-life (symbol $t_{{\displaystyle \frac{1}{2}}}$ ) is the amount of time it takes for it to decrease to half of its original value. In nuclear physics, the phrase is used to explain how rapidly unstable atoms disintegrate radioactively and how long stable atoms survive. The word is also used to describe any sort of exponential or non-exponential decay in general. The biological half-life of medicines and other substances in the human body, for example, is discussed in medical research. Doubling time is the inverse of half-life.

Complete Step By Step Answer:
The half-life of an exponentially decaying quantity is constant during its lifespan, and it is a characteristic unit for the exponential decay equation. The reduction of a quantity as a function of the number of half-lives elapsed is shown in the table below. Potassium-40, a radioactive isotope of potassium, has two half-lives that are determined mostly by the sort of beta decay it undergoes. Potassium-40 has a half-life of $1.28\times {10}^9$ years if it decays by beta emission. The half-life of potassium-40 decays by positron emission, on the other hand, is $1.19\times {10}^{10}$ years.
The time it takes for a quantity to decline to half its value as measured at the start of the time period is called the half-life ( $t_{{\displaystyle \frac{1}{2}}}$ ). The time ( $t_{{\displaystyle \frac{1}{2}}}$ ) of K-40 in this question is 1300 million years, which implies that after 1300 million years, half of the sample will have decomposed and the other half will remain unchanged.
Let's begin with 200 gram of the sample. ${\displaystyle \frac{200}{2}}=100g$ decays after 1300 million years (first half life), leaving 100 g.
 ${\displaystyle \frac{100}{2}}=\text{ 5}0g$ decays after another 1300 million years (two half lifetimes or 2600 million years), leaving 50 g.
 ${\displaystyle \frac{50}{2}}=\text{ 25 }g$ decays after another 1300 million years (three half lifetimes or 3900 million years), leaving 25 g.
 ${\displaystyle \frac{\text{25 }}{2}}=\text{ 12}\text{.5 }g$ decays after another 1300 million years (four half lifetimes or 5200 million years), leaving 12.5 g.
There will be 12.5 g of K-40 remaining after four half lifetimes, or 5200 million years.

Note :
The decay of discrete things, such as radioactive atoms, is generally described by a half-life. In such a situation, the notion that "half-life is the time necessary for exactly half of the entities to decay" does not apply. If just one radioactive atom exists and its half-life is one second, there will be no "half of an atom" remaining after one second.