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Half Life and Mean Life

Half Life and Mean Life

This lesson aligns with NGSS PS1.C

Introduction
The term Half-Life characterizes the radioactive decay of a specific sample or element within a designated time frame. This terminology is commonly encountered by experts studying nuclear Physics. Beyond its application in Physics, the concept of Half-Life is also extensively utilized to elucidate diverse decay processes, encompassing both exponential and non-exponential decay. Moreover, this term finds relevance in the field of medical sciences, where it is employed to signify the biological half-life of specific chemicals within the human body or in pharmaceutical drugs. This article aims to learn the concept of half-life and mean life, exploring their definitions, mathematical formulations, and real-world implications.

Understanding Half-Life
The half-life of a radioactive isotope is defined as the time it takes for half of a sample of the substance to undergo decay. The fundamental principle was initially identified in 1907 by Ernest Rutherford and is commonly denoted by the symbol Ug or [math]t_1/2[/math].
It is important to note that the half-life does not indicate the total duration during which a material will retain its radioactive properties. Rather, it signifies the period required for the material's radioactivity to decrease by half.

Experiment
Consider a radioactive element with a half-life of one hour. In such a scenario, half of the atoms would undergo decay within the initial hour, leaving the remaining half to decay within the subsequent hour. This prompts the question: why didn't the remaining material decay during the first hour?
The explanation lies in the understanding that when half of the atoms of a radioactive element have decayed after one half-life, it is appropriate to infer that they possess a life expectancy at an average level, which is distinctly characterized by atoms with a mean life significantly longer than their half-life. Consequently, the mean life can be determined as the half-life divided by 2, representing the natural logarithm. It is worth noting that half-life is frequently defined in terms of probability.

Half-life Formulas
Mathematically, the relationship between the initial quantity of a radioactive substance (N₀), the remaining quantity after a certain time (N(t)), and the half-life (t₁/₂) is expressed as follows:

Mean Life
Mean life, also known as the average life or the expectation of life, is another important parameter associated with radioactive decay. It represents the average time it takes for a radioactive substance to decay completely.
This parameter is computed by summing the lifetimes of all specific unstable nuclei within a given sample and subsequently dividing this sum by the total number of present unstable nuclei.
It is noteworthy that the mean life of a specific element with an unstable nucleus is 1.443 times greater than its half-life. The half-life denotes the time interval required for the decay of half of the unstable nuclei within a sample.
For example, consider the decay of Lead-209 to Bismuth-209. In this case, the mean life of Bismuth-209 is measured at 4.69 hours, while its half-life is approximately 3.25 hours.

Mean Life Formula
Mathematically, the mean life (τ) is related to the half-life by the equation:
The half-life and the mean life of substances are related to each other.

Comparison between Half life and Mean life
The two parameters exhibit significant variation across different substances. For instance, the half-life of Polonium-212 is less than 1 microsecond, whereas, for Thorium-232, the half-life extends beyond 1 billion years.


Summary
  • The half-life of a radioactive isotope is defined as the time it takes for half of a sample of the substance to undergo decay.
  • Mean life represents the average time it takes for a radioactive substance to decay completely.
  • It is noteworthy that the mean life of a specific element with an unstable nucleus is 1.443 times greater than its half-life.

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