Half-Life Definition and Examples in Chemistry, Biology, Physics


What Is Half-Life - Definition and Examples

Half-life (t½) 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) = N0e−λt

where:

  • N(t)is the quantity of the substance at time t,
  • N0 is the initial quantity,
  • λ is the decay constant,
  • e is the base of the natural logarithm.

The half-life t½​ is the time it takes for N(t) to reduce to half of N0​. The decay constant λ and the half-life relate to each other by the equation:

t½ = 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) = N0(½)t/t½ where:

  • N(t) is the quantity of the substance at time t,
  • N0​ is the initial quantity,
  • t½​ 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.
    t½ = [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: t1/2 = 1 M / 2×0.1 M/s = 5 s
  • First-Order Reactions: For first-order reactions, the rate is proportional to the concentration of one reactant. t½ = ln⁡(2) / k where ln⁡(2) ≈ 0.693.
    Example Calculation: If k = 0.2: t½ =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.
    t½ = 1 / k[A]0
    Example Calculation: If [A]0 = 1 and k=0.1 M−1s−1:
    t½ = 1 / (0.1 M−1s−1 × 1 M) =10 s

Common Misconceptions about Half-Life

People have a few common misconceptions regarding half-life:

  1. 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.
  2. Complete Decay: Some believe that after multiple half-lives, the substance completely disappears. While the amount becomes very small, it never truly reaches zero.
  3. 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.