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

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 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