{
    "name": "Half-Life Calculator",
    "url": "https://calculators.geekmidst.com/calculators/half-life-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/half-life-calculator.md",
    "category": "Mathematics",
    "description": "Solve exponential decay: enter three of initial amount, remaining amount, half-life and time elapsed, and find the fourth plus the decay constant.",
    "summary": "Fill in any three of the four boxes and leave the one you want blank. Each half-life halves what is left, whatever the substance.",
    "also_called": [
        "Radioactive decay calculator",
        "Exponential decay calculator"
    ],
    "applies_to": "Worldwide",
    "last_updated": "2026-10-29",
    "computation": "Results are computed in the visitor’s browser on the HTML page. This API describes the calculator; it does not compute results.",
    "inputs": [
        {
            "name": "n0",
            "label": "Initial quantity (N₀)",
            "type": "number",
            "default": "100",
            "optional": true
        },
        {
            "name": "nt",
            "label": "Remaining quantity (N)",
            "type": "number",
            "default": "",
            "optional": true
        },
        {
            "name": "half_life",
            "label": "Half-life (T½)",
            "type": "number",
            "default": "5730",
            "optional": true
        },
        {
            "name": "time",
            "label": "Time elapsed (t)",
            "type": "number",
            "default": "11460",
            "optional": true
        }
    ],
    "outputs": [
        {
            "key": "answer",
            "label": "Answer",
            "format": "text"
        },
        {
            "key": "remaining_pct",
            "label": "Fraction remaining",
            "format": "percent"
        },
        {
            "key": "half_lives",
            "label": "Number of half-lives elapsed",
            "format": "number"
        },
        {
            "key": "n0",
            "label": "Initial quantity",
            "format": "number"
        },
        {
            "key": "nt",
            "label": "Remaining quantity",
            "format": "number"
        },
        {
            "key": "half_life",
            "label": "Half-life",
            "format": "number"
        },
        {
            "key": "time",
            "label": "Time elapsed",
            "format": "number"
        },
        {
            "key": "decay_constant",
            "label": "Decay constant λ (per time unit)",
            "format": "text"
        },
        {
            "key": "mean_life",
            "label": "Mean lifetime τ",
            "format": "number"
        }
    ],
    "formula": {
        "text": "N = N₀ × (½)^(t ÷ T½);  T½ = t × ln 2 ÷ ln(N₀ ÷ N);  t = T½ × log₂(N₀ ÷ N);  λ = ln 2 ÷ T½;  τ = 1 ÷ λ",
        "variables": {
            "N₀": "starting quantity",
            "N": "quantity remaining after time t",
            "T½": "half-life",
            "λ": "decay constant",
            "τ": "mean lifetime"
        }
    },
    "worked_example": {
        "inputs": {
            "n0": "100",
            "nt": "",
            "half_life": "5730",
            "time": "11460"
        },
        "steps": [
            "t ÷ T½ = 11,460 ÷ 5,730 = 2 half-lives.",
            "N = 100 × (½)² = 100 × ¼ = 25.",
            "λ = ln 2 ÷ 5,730 ≈ 0.000120968 per year."
        ],
        "result": "After 11,460 years, two carbon-14 half-lives, 25 of the original 100 units remain."
    },
    "interpretation": "After 7 half-lives less than 1% remains, and after 10 about 0.1%, which is why “ten half-lives” is a common rule of thumb for “essentially gone”.",
    "assumptions": [
        "Simple first-order (exponential) decay with a constant half-life.",
        "All quantities use the same unit, and the half-life and time use the same time unit."
    ],
    "limitations": [
        "Real samples can contain several isotopes or decay chains, and drug elimination often departs from a single half-life; those need more detailed models."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "Does the amount ever reach zero?",
            "answer": "In the model it never quite does: each half-life halves what is left. In practice a sample becomes undetectable, or for atoms, the last one decays at a random moment."
        },
        {
            "question": "How do I find a half-life from two measurements?",
            "answer": "Enter the starting and remaining amounts and the time between them, and leave the half-life blank. For 100 → 12.5 in 30 hours, that is three halvings, so T½ = 10 hours."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Exponent Calculator",
            "url": "https://calculators.geekmidst.com/calculators/exponent-calculator"
        },
        {
            "name": "Log Calculator",
            "url": "https://calculators.geekmidst.com/calculators/log-calculator"
        },
        {
            "name": "Compound Interest Calculator",
            "url": "https://calculators.geekmidst.com/calculators/compound-interest-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}