Half-Life Calculator

Determine the half-life, initial quantity, remaining quantity, or time elapsed for a decaying substance

Initial Quantity (N₀):

Remaining Quantity (N):

Half-Life (t₁/₂):

Time Elapsed (t):

Calculate Reset

Result:

Understanding Half-Life

Half-life is the time required for a quantity to reduce to half its initial value. It’s commonly used in nuclear physics to describe the rate at which unstable atoms undergo radioactive decay, but it also applies to other exponential decay processes.

Half-Life Formula

The relationship between the initial quantity (N₀), remaining quantity (N), half-life (t₁/₂), and time elapsed (t) is given by:

N = N₀ × (1/2)^(t / t₁/₂)

This formula allows you to calculate any one of the variables if the other three are known.

Applications

• Radiocarbon Dating: Estimating the age of archaeological samples.
• Pharmacology: Determining how long a drug remains active in the body.
• Environmental Science: Understanding the decay of pollutants.

Half-Life Calculator

Half-life is defined as the amount of time it takes a given quantity to decrease to half of its initial value. The term is most commonly used in relation to atoms undergoing radioactive decay, but can be used to describe other types of decay, whether exponential or not. One of the most well-known applications of half-life is carbon-14 dating. The half-life of carbon-14 is approximately 5,730 years, and it can be reliably used to measure dates up to around 50,000 years ago. The process of carbon-14 dating was developed by William Libby, and is based on the fact that carbon-14 is constantly being made in the atmosphere. It is incorporated into plants through photosynthesis, and then into animals when they consume plants. The carbon-14 undergoes radioactive decay once the plant or animal dies, and measuring the amount of carbon-14 in a sample conveys information about when the plant or animal died.

Below are shown three equivalent formulas describing exponential decay:

N(t)=N0(12)tt1/2N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}}N(t)=N0​(21​)t1/2​t​

N(t)=N0e−tτN(t) = N_0 e^{-\frac{t}{\tau}}N(t)=N0​e−τt​

N(t)=N0e−λtN

(t) = N_0 e^{-\lambda t}

N(t)=N0​e−λt

If an archaeologist found a fossil sample that contained 25% carbon-14 in comparison to a living sample, the time of the fossil sample’s death could be determined by rearranging equation 1, since N_t, N₀, and t_1/2 are known.

N(t)=N0​(21​)t1/2​t​⇒

t=−ln2t1/2​ln(N0​Nt​​)​

t=5730ln⁡(25100

)−0.693=11460t = \frac{5730 \ln\left

(\frac{25}{100}\right)}{-0.693} = 11460

t=−0.6935730ln(10025​)​=11460

This means that the fossil is 11,460 years old.

Derivation of the Relationship Between Half-Life Constants

Using the above equations, it is also possible for a relationship to be derived between t_1/2, τ, and λ. This relationship enables the determination of all values, as long as at least one is known.

(1) (12)tt1/2=e−tτ=e−λt\left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} = e^{-\frac{t}{\tau}} = e^{-\lambda t}(21​)t1/2​t​=e−τt​=e−λt

(2) ln⁡((12)tt1/2)=ln⁡(e−tτ)=ln⁡(e−λt)\ln\left( \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \right) = \ln\left( e^{-\frac{t}{\tau}} \right) = \ln(e^{-\lambda t})ln((21​)t1/2​t​)=ln(e−τt​)=ln(e−λt)

(3) (1t1/2×ln⁡(12)=−1τ=−λ)×t1/2×τ×−1\left( \frac{1}{t_{1/2}} \times \ln\left( \frac{1}{2} \right) = -\frac{1}{\tau} = -\lambda \right) \times t_{1/2} \times \tau \times -1(t1/2​1​×ln(21​)=−τ1​=−λ)×t1/2​×τ×−1

(4) ln⁡(2) τ=t1/2=λt1/2\ln(2)\, \tau = t_{1/2} = \lambda t_{1/2}ln(2)τ=t1/2​=λt1/2​

(5) t1/2=τln⁡2=ln⁡2λt_{1/2} = \tau \ln 2 = \frac{\ln 2}{\lambda}t1/2​=τln2=λln2​