Authoritative Data Source and Methodology
Primary reference: IUPAC Compendium of Chemical Terminology (the “Gold Book”), entry “half-life; t1/2”, 3rd edition (2014). DOI: 10.1351/goldbook. Link: https://goldbook.iupac.org/terms/view/H02723. All formulas employ first-order exponential decay, consistent with the IUPAC definition and standard kinetics.
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.
The Formula Explained
Exponential decay (first order):
$$N(t) = N_0 \, e^{-\lambda t} \quad\text{and}\quad N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}$$
Relationship between half-life and decay constant:
$$\lambda = \frac{\ln 2}{t_{1/2}} \quad\Longleftrightarrow\quad t_{1/2} = \frac{\ln 2}{\lambda}$$
Solving for time to reach a target:
$$t = \frac{\ln\!\left(\frac{N_0}{N(t)}\right)}{\lambda} = t_{1/2}\,\log_2\!\left(\frac{N_0}{N(t)}\right)$$
Glossary of Variables
| Symbol | Name | Units | Notes |
|---|---|---|---|
| N0 | Initial quantity | Any (e.g., mg, Bq) | Starting amount at t = 0 |
| N(t) | Remaining quantity | Same as N0 | Amount after time t |
| t | Time elapsed | s, min, h, d, y | Must be ≥ 0 |
| t½ | Half-life | s, min, h, d, y | Positive |
| λ | Decay constant | 1/s, 1/min, 1/h, 1/d, 1/y | Positive |
| n | Number of half-lives | — | n = t / t½ |
Worked Example
How It Works: A Step-by-Step Example
Suppose a drug has a half-life of 8 hours. If the initial dose is N0 = 120 mg, how much remains after t = 24 hours?
- Number of half-lives: n = t / t½ = 24 / 8 = 3.
- Apply decay: N = N0 × (1/2)^n = 120 × (1/2)^3 = 120 × 0.125 = 15 mg.
- Cross-check using λ: λ = ln 2 / t½ = 0.693147/8 h = 0.086643 h⁻1; N = 120 × e^(−0.086643×24) ≈ 15 mg.
Therefore, 15 mg remain after 24 hours. Fraction remaining is 0.125; 87.5% has decayed.
Frequently Asked Questions (FAQ)
Is this calculator valid for any decay process?
It models first-order exponential decay, which covers radioactive decay and many pharmacokinetic cases. Processes that are zero-order or mixed-order require different models.
Which should I enter: half-life or decay constant?
Either one is sufficient. The tool converts them via λ = ln(2)/t½ and uses whichever you provide.
Can I solve for time to reach a target amount?
Yes. Select “Time t” as the unknown, then provide N0, N(t), and either t½ or λ.
Do the units have to match?
Time units are selectable and internally converted. N0 and N(t) must use the same unit (e.g., both mg or both Bq).
What precision do you use?
Internally we use double precision. Results are rounded for readability, but the underlying calculation retains higher precision.
Why do I get an error when N(t) ≥ N0?
For a decaying process, N(t) must be less than or equal to N0. If greater, check your inputs or whether your process involves growth instead of decay.
How can I cite this tool?
You can cite the IUPAC Gold Book entry for half-life (2014) as the methodological reference and include the URL of this calculator.