**Half-life** (t_{½}) is a fundamental concept in various scientific fields, particularly in physics, chemistry, and biology. It represents the time for half of a given quantity of a substance to transform into something else. Examples of processes that transform with a half-life include radioactive decay, a chemical reaction, or the elimination of a substance from a biological system. However, the principle also applies to other disciplines, such as advertising campaigns, the emission of electromagnetic radiation, and atmospheric pressure decrease with altitude.

### Historical Context and Etymology

Ernest Rutherford introduced the term “half-life period” in 1907. Rutherford developed the concept while studying the decay rates of radioactive elements. The word shortened to “half-life” in the 1950s.

### Relation to Probability

The concept of half-life is inherently probabilistic. It does not imply that exactly half of the atoms in a radioactive sample decay after one half-life, but rather that there is a 50% probability for any given atom to decay during that period.

**Example:** Consider a radioactive isotope with a half-life of 10 years. If we start with 1000 atoms, after 10 years, we expect 500 atoms (50%) to have decayed. However, it’s probabilistic, so in a real-world scenario, the exact number might slightly vary but be close to this average.

### Exponential Decay

Half-life relates to exponential decay, a process where the quantity of a substance decreases at a rate proportional to its current value. This exponential decay formula mathematically describes the relationship:

**N(t) = N _{0}e^{−λt}**

where:

- N(t)is the quantity of the substance at time t,
- N
_{0}is the initial quantity, - λ is the decay constant,
- e is the base of the natural logarithm.

The half-life t_{½} is the time it takes for N(t) to reduce to half of N_{0}. The decay constant λ and the half-life relate to each other by the equation:

**t _{½} = ln(2) / λ**

Here, ln(2) ≈ 0.693. This shows that half-life is a measure of the time scale over which the exponential decay occurs.

### Applications of Half-Life

#### Radioactive Decay

Radioactive decay is a random process at the level of single atoms, governed by the half-life. The decay has the equation:

N(t) = N_{0}(½)^{t/t½} where:

- N(t) is the quantity of the substance at time t,
- N
_{0} is the initial quantity, - t
_{½} is the half-life.

**Example Calculation:**

Suppose we have 200 grams of a radioactive substance with a half-life of 5 years. Find the amount remaining after 15 years:

N(15) = 200 (½)^{15/5} = 200 (½)^{3} = 200 × ⅛ = 25 grams

#### Biological Half-Life

In pharmacology and toxicology, the biological half-life is the time it takes for a substance (such as a drug) to reduce to half its initial concentration in the body. This concept is more complex due to factors like metabolism, excretion, and interactions with other substances. The biological half-life varies between individuals, depending on organ function, age, sex, and overall health. Unlike the predictable nature of radioactive decay, the biological half-life requires careful empirical determination.

#### Chemical Reaction Kinetics

In chemistry, half-life describes the kinetics of chemical reactions. The half-life depends on the order of the reaction:

**Zero-Order Reactions:**For zero-order reactions, the rate of reaction is constant.

t_{½}= [A]_{0}/ 2kt where:- [A]
_{0}is the initial concentration

- k is the rate constant.

**Example Calculation:**If [A]_{0}= 1 M and k=0.1 M/s: t_{1/2}= 1 M / 2×0.1 M/s = 5 s- [A]
**First-Order Reactions:**For first-order reactions, the rate is proportional to the concentration of one reactant. t_{½}= ln(2) / k where ln(2) ≈ 0.693.**Example Calculation:**If k = 0.2: t_{½}=0.693 / 0.2 s^{−1}≈ 3.47 s**Second-Order Reactions:**For second-order reactions, the rate is proportional to the square of the concentration of one reactant or the product of two concentrations.

t_{½}= 1 / k[A]_{0}**Example Calculation:**If [A]_{0}= 1 and k=0.1 M^{−1}s^{−1}:

t_{½}= 1 / (0.1 M^{−1}s^{−1}× 1 M) =10 s

### Common Misconceptions about Half-Life

People have a few common misconceptions regarding half-life:

**Exact Halving:**One common misconception is that exactly half of the substance decays after one half-life. The process is probabilistic, meaning the actual amount varies slightly but averages out over many measurements.**Complete Decay:**Some believe that after multiple half-lives, the substance completely disappears. While the amount becomes very small, it never truly reaches zero.**Constant Rate Misinterpretation:**The half-life often gets confused with a constant decay rate. While the decay rate decreases as the amount of the substance decreases, the half-life remains constant for a given substance.

### References

- Crowe, Jonathan; Bradshaw, Tony (2014).
*Chemistry for the Biosciences: The Essential Concepts*. OUP Oxford. ISBN 9780199662883. - Fantke, Peter; Gillespie, Brenda W.; Juraske, Ronnie; Jolliet, Olivier (2014). “Estimating Half-Lives for Pesticide Dissipation from Plants”. Environmental Science & Technology. 48 (15): 8588–8602. doi:10.1021/es500434p
- Serway, Raymond A.; Moses, Clement J.; Moyer, Curt A. (1989).
*Modern Physics*. Fort Worth: Harcourt Brace Jovanovich. ISBN 0-03-004844-3. - Rösch, Frank (2014).
*Nuclear and Radiochemistry: Introduction*. Vol. 1. Walter de Gruyter. ISBN 978-3-11-022191-6.