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Half Life Calculator

Enter any three values to calculate the fourth.

Result:
Formula (Half-Life):
t₁/₂ = t × ln(2) / ln(N₀ / Nₜ)

Example:
If N₀ = 100, Nₜ = 25, and t = 6 hours:
t₁/₂ = 6 × ln(2) / ln(100 / 25) = 3 hours
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What Is Half-Life?

Half-life refers to the time it takes for a substance to reduce to half of its original quantity. While it is most often associated with radioactive decay of atoms, the concept also applies to other decay processes, whether they follow an exponential pattern or not. A well-known example of half-life in action is carbon-14 dating, a method used to estimate the age of ancient organic materials.

Carbon-14 has a half-life of about 5,730 years, making it useful for dating specimens up to around 50,000 years old. This dating technique was pioneered by Willard Libby and relies on the continuous production of carbon-14 in Earth’s atmosphere. Plants absorb carbon-14 during photosynthesis, and animals acquire it by consuming plants. When the organism dies, it stops absorbing carbon-14, and the isotope begins to decay. By measuring how much carbon-14 remains, scientists can estimate when the organism died.

Half-Life Formulas for Exponential Decay

The behavior of substances undergoing exponential decay can be described using any of the following equations:

1. Nt = N0 × (1/2)^(t / t½)
2. Nt = N0 × e^(-t / τ)
3. Nt = N0 × e^(-λt)

Where:

  • N₀ = Initial amount

  • Nₜ = Remaining amount after time t

  • t½ = Half-life

  • τ = Mean lifetime

  • λ = Decay constant

For example, if a fossil sample contains 25% of the original carbon-14 level found in a living organism, you can use the first formula to calculate how much time has passed since its death. Rearranging the formula to solve for t, and knowing the values of N₀, Nₜ, and t½, would yield a result of approximately 11,460 years.

Example Calculation:

Given:

  • Half-life (t₁/₂) = 5730 years

  • Remaining amount (Nₜ / N₀) = 25% = 0.25

t = 5730 × ln(0.25) / ln(0.5)
t = 5730 × (-1.386) / -0.693
t = 11460 years

Understanding the Relationship Between Half-Life Constants

The equations above also allow us to derive a direct relationship between the different decay parameters:

  1. (1/2)^(t / t₁/₂) = e^(-t / τ) = e^(-λt)

  2. ln((1/2)^(t / t₁/₂)) = ln(e^(-t / τ)) = ln(e^(-λt))

  3. (1 / t₁/₂) × ln(1/2) = -1 / τ = -λ

  4. ln(2) × τ = t₁/₂ = λ × t₁/₂ × τ

  5. t₁/₂ = τ × ln(2) = ln(2) / λ

 

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