Calculate the decay constant (λ) and remaining quantity based on Half-Life.
The time required for a quantity to reduce to half.
Seconds
Minutes
Hours
Days
Years
Must be in the same units as Half-Life.
Decay Constant (λ):–
Mean Lifetime (τ):–
Decay Percentage (per unit):–
Remaining Quantity (Nt):–
How to Calculate Rate of Decay from Half-Life
In nuclear physics, chemistry, and pharmacology, the "rate of decay" describes how quickly an unstable substance transforms into a stable one. This rate is governed by the substance's half-life ($t_{1/2}$), which is the time it takes for half of the initial quantity to decay.
The Decay Constant Formula
The rate of decay is most often mathematically expressed as the Decay Constant, denoted by the Greek letter Lambda ($\lambda$). It represents the probability of decay per unit time. The relationship between the half-life and the decay constant is fixed and can be calculated using the natural logarithm of 2.
λ = ln(2) / t1/2 ≈ 0.693 / t1/2
Where:
λ (Lambda): The decay constant (inverse time, e.g., $s^{-1}$, $year^{-1}$).
ln(2): The natural logarithm of 2, approximately equal to 0.693147.
t1/2: The half-life of the substance.
Calculating Remaining Quantity
Once you have the decay constant or the half-life, you can determine how much of a substance remains after a specific period of time ($t$) using the exponential decay formula:
N(t) = N0 × e-λt
Alternatively, using half-life directly:
N(t) = N0 × (1/2)(t / t1/2)
N(t): Quantity remaining after time $t$.
N0: Initial quantity.
t: Time elapsed.
Real-World Example: Carbon-14
Carbon-14 is commonly used for radiocarbon dating and has a half-life of approximately 5,730 years. To find its rate of decay (decay constant):
$$ \lambda = \frac{0.693}{5730} \approx 0.000121 \text{ per year} $$
This implies that roughly 0.0121% of a sample of Carbon-14 decays every year. By inputting these figures into the calculator above, you can verify the decay constant and project the remaining amount after any number of years.
Mean Lifetime
Another useful metric is the Mean Lifetime ($\tau$), which is the average time a radioactive particle survives before decaying. It is simply the reciprocal of the decay constant:
τ = 1 / λ = t1/2 / ln(2)
function calculateDecayRate() {
// 1. Get Inputs
var halfLifeVal = document.getElementById('halfLifeInput').value;
var unit = document.getElementById('timeUnitSelect').value;
var initialQty = document.getElementById('initialQtyInput').value;
var timeElapsed = document.getElementById('elapsedTimeInput').value;
// 2. Validate Half Life
if (halfLifeVal === "" || isNaN(halfLifeVal) || Number(halfLifeVal) <= 0) {
alert("Please enter a valid positive Half-Life value.");
return;
}
var tHalf = parseFloat(halfLifeVal);
// 3. Calculate Lambda (Decay Constant)
// Formula: Lambda = ln(2) / HalfLife
var ln2 = 0.69314718056;
var lambda = ln2 / tHalf;
// 4. Calculate Mean Lifetime
// Formula: Tau = 1 / Lambda
var meanLife = 1 / lambda;
// 5. Calculate Decay Percentage per Unit Time
// Percentage lost in 1 unit of time = (1 – e^-lambda) * 100
var decayPerUnit = (1 – Math.exp(-lambda)) * 100;
// 6. Handle Optional Calculation (Remaining Quantity)
var remainingResult = null;
var showRemaining = false;
if (initialQty !== "" && timeElapsed !== "") {
var n0 = parseFloat(initialQty);
var t = parseFloat(timeElapsed);
if (!isNaN(n0) && !isNaN(t)) {
// Formula: N(t) = N0 * (0.5)^(t / tHalf)
var ratio = t / tHalf;
remainingResult = n0 * Math.pow(0.5, ratio);
showRemaining = true;
}
}
// 7. Format and Display Results
var unitSuffix = unit.toLowerCase().replace(/s$/, ''); // Remove trailing 's' for singular notation if desired, though usually inverse unit is kept simple.
// Scientific notation checks for very small numbers
var displayLambda = lambda < 0.0001 ? lambda.toExponential(4) : lambda.toFixed(6);
var displayMean = meanLife.toFixed(2);
var displayPercent = decayPerUnit < 0.01 ? decayPerUnit.toExponential(4) : decayPerUnit.toFixed(4);
document.getElementById('resLambda').innerHTML = displayLambda + " / " + unitSuffix;
document.getElementById('resMeanLife').innerHTML = displayMean + " " + unit;
document.getElementById('resPercent').innerHTML = displayPercent + "%";
var remainingRow = document.getElementById('remainingQtyRow');
if (showRemaining) {
var displayRem = remainingResult 0 ? remainingResult.toExponential(4) : remainingResult.toFixed(4);
document.getElementById('resRemaining').innerHTML = displayRem;
remainingRow.style.display = "flex";
} else {
remainingRow.style.display = "none";
}
document.getElementById('decayResults').style.display = "block";
}
function resetDecayCalculator() {
document.getElementById('halfLifeInput').value = "";
document.getElementById('initialQtyInput').value = "";
document.getElementById('elapsedTimeInput').value = "";
document.getElementById('timeUnitSelect').value = "Years";
document.getElementById('decayResults').style.display = "none";
document.getElementById('remainingQtyRow').style.display = "none";
}