{
  "id": "half-life",
  "version": "533b16d18d25",
  "status": "published",
  "name": "Half Life Calculator",
  "question": "How much is left after its half-life?",
  "summary": "Works out radioactive decay with the half-life formula N = N₀ × (1/2)^(t ÷ T½): the amount left, the initial amount, the time passed, or the half-life, with the decay constant and mean lifetime.",
  "category": "chemistry",
  "url": "https://www.acalculator.org/chemistry/half-life-calculator",
  "markdown": "https://www.acalculator.org/chemistry/half-life-calculator.md",
  "kind": "formulas",
  "method": "N = N₀ × (1/2)^(t ÷ T½); N₀ = N × 2^(t ÷ T½); t = T½ × log₂(N₀ ÷ N); T½ = t ÷ log₂(N₀ ÷ N); λ = ln 2 ÷ T½; τ = T½ ÷ ln 2.",
  "assumptions": [
    "Decay is first order: the same fraction decays in each equal time, whatever the amount (OpenStax Chemistry 2e, 21.3).",
    "The amount left and the initial amount are in the same unit; any unit works.",
    "A year is 365 days."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "n0": {
        "title": "Initial amount (N₀)",
        "description": "The amount at the start, in any unit: grams, atoms, counts per minute, or percent.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 1000000000000000
      },
      "n": {
        "title": "Amount left (N)",
        "description": "The amount left after the time t, in the same unit as the initial amount.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 1000000000000000
      },
      "t": {
        "title": "Time passed (t)",
        "description": "How long the substance has been decaying.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "time",
        "minimum": 0,
        "maximum": 100000000000000000000
      },
      "T": {
        "title": "Half-life (T½)",
        "description": "The time it takes for half of the substance to decay.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "time",
        "minimum": 1e-12,
        "maximum": 100000000000000000000
      }
    }
  },
  "solve": {
    "fillAny": 3,
    "variables": [
      "n0",
      "n",
      "t",
      "T"
    ],
    "inputsOnly": []
  },
  "outputs": {
    "n0": {
      "label": "Initial amount (N₀)",
      "description": "The amount at the start, in any unit: grams, atoms, counts per minute, or percent.",
      "format": "number"
    },
    "n": {
      "label": "Amount left (N)",
      "description": "N₀ × (1/2)^(t ÷ T½), in the unit of the initial amount.",
      "format": "number"
    },
    "t": {
      "label": "Time passed (t)",
      "description": "How long the substance has been decaying.",
      "format": "quantity"
    },
    "half": {
      "label": "Half-life (T½)",
      "description": "The time it takes for half of the substance to decay.",
      "format": "quantity"
    },
    "percent": {
      "label": "Percent left",
      "description": "The share of the initial amount that is left: N ÷ N₀ × 100.",
      "format": "percent"
    },
    "halfLives": {
      "label": "Half-lives passed",
      "description": "How many half-lives the time passed is: t ÷ T½.",
      "format": "number"
    },
    "lambdaYear": {
      "label": "Decay constant (λ, per year)",
      "description": "ln 2 ÷ T½, with the half-life in years of 365 days.",
      "format": "number"
    },
    "mean": {
      "label": "Mean lifetime (τ)",
      "description": "The average time an atom lasts before it decays, T½ ÷ ln 2, in years of 365 days.",
      "format": "quantity",
      "unit": "yr"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "n0": 100,
      "t": 315360000000,
      "T": 180701280000
    },
    "outputs": {
      "n": 29.829243642371427,
      "percent": 29.829243642371424,
      "halfLives": 1.7452006980802792,
      "lambdaYear": 0.00012096809433855938,
      "mean": 260696840538.28802
    },
    "text": "After 10,000 yr, 29.8292 of 100 is left, with a half-life of 5,730 yr."
  },
  "examples": [
    {
      "given": {
        "n0": 1,
        "t": 473040000,
        "T": 166194720
      },
      "expect": {
        "n": 0.1390523652329965,
        "percent": 13.90523652329965,
        "lambdaYear": 0.13152697923338622
      },
      "source": "OpenStax, Chemistry 2e, section 21.3 Radioactive Decay (λ = ln 2 ÷ t½, Nₜ = N₀e^(−λt)), https://openstax.org/books/chemistry-2e/pages/21-3-radioactive-decay, retrieved 2026-10-02, example 21.6 (cobalt-60; the book rounds λ to 0.132 per year and gets 13.8%); hand calculation in content.mdx: 2^(−15 ÷ 5.27) = 0.13905",
      "tolerance": 1e-12
    },
    {
      "given": {
        "n0": 13.6,
        "n": 10.8,
        "T": 180701280000
      },
      "expect": {
        "t": 60096789469.43156,
        "halfLives": 0.33257533908687065
      },
      "source": "OpenStax, Chemistry 2e, section 21.3 Radioactive Decay (λ = ln 2 ÷ t½, Nₜ = N₀e^(−λt)), https://openstax.org/books/chemistry-2e/pages/21-3-radioactive-decay, retrieved 2026-10-02, example 21.7 (Dead Sea Scrolls, carbon-14 half-life 5,730 years, about 1,900 years old); hand calculation in content.mdx: 5,730 × log₂(13.6 ÷ 10.8) = 1,905.66 years",
      "tolerance": 1e-12
    },
    {
      "given": {
        "n0": 100,
        "n": 25,
        "t": 864000
      },
      "expect": {
        "T": 432000
      },
      "source": "hand calculation in content.mdx: 100 → 25 is 2 halvings in 10 days, so T½ = 5 days = 432,000 s",
      "tolerance": 1e-12
    },
    {
      "given": {
        "n": 10,
        "t": 94608000,
        "T": 31536000
      },
      "expect": {
        "n0": 80
      },
      "source": "hand calculation in content.mdx: 3 half-lives leave 1/8, so N₀ = 10 × 8 = 80",
      "tolerance": 1e-12
    },
    {
      "given": {
        "n0": 100,
        "t": 315360000000,
        "T": 180701280000
      },
      "expect": {
        "n": 29.829243642371427
      },
      "source": "hand calculation in content.mdx: 100 × 2^(−10,000 ÷ 5,730) = 29.83",
      "tolerance": 1e-12
    }
  ],
  "sources": [
    "OpenStax, Chemistry 2e, section 21.3 Radioactive Decay (λ = ln 2 ÷ t½, Nₜ = N₀e^(−λt); cobalt-60 and carbon-14 examples), CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/chemistry-2e/pages/21-3-radioactive-decay"
  ],
  "related": [
    "exponential-growth",
    "molarity",
    "concentration"
  ],
  "changelog": []
}