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Decay Constant (λ):
How to Calculate Decay Rate from Half-Life
In nuclear physics and chemistry, the decay rate (often called the decay constant, λ) represents the probability that a particular nucleus will undergo radioactive decay per unit of time. Understanding the relationship between half-life and the decay constant is essential for dating archaeological finds, managing medical isotopes, and calculating nuclear safety protocols.
The Mathematical Formula
The relationship between half-life (T1/2) and the decay constant (λ) is derived from the exponential decay law. The formula used for this calculation is:
λ = ln(2) / T1/2
λ ≈ 0.693147 / T1/2
Step-by-Step Calculation Example
Let's say you have a substance with a half-life of 5.27 years (Cobalt-60). Here is how you find the decay rate:
Identify the half-life: T1/2 = 5.27 years.
Apply the formula: λ = 0.693147 / 5.27.
Solve: λ ≈ 0.1315 per year.
This result means that approximately 13.15% of the remaining Cobalt-60 atoms will decay every year.
Important Considerations
Inverse Relationship: As the half-life increases, the decay rate decreases. A substance that lasts a long time has a very small decay probability per second.
Units: The units of the decay constant are always the inverse of the time unit used for the half-life (e.g., s⁻¹, min⁻¹, yr⁻¹).
Statistical Nature: Radioactive decay is a random process. The decay constant applies to a large population of atoms, not a single individual atom.
function calculateDecayRate() {
var halfLifeValue = document.getElementById('halfLifeInput').value;
var unit = document.getElementById('timeUnit').value;
var resultDiv = document.getElementById('resultDisplay');
var lambdaSpan = document.getElementById('lambdaResult');
var unitSpan = document.getElementById('unitResult');
var formulaSpan = document.getElementById('formulaExplanation');
if (halfLifeValue === "" || parseFloat(halfLifeValue) <= 0) {
alert("Please enter a valid positive half-life value.");
return;
}
var halfLife = parseFloat(halfLifeValue);
var ln2 = Math.log(2);
var lambda = ln2 / halfLife;
// Formatting the result
var displayLambda;
if (lambda 10000) {
displayLambda = lambda.toExponential(6);
} else {
displayLambda = lambda.toFixed(8);
// Remove trailing zeros
displayLambda = parseFloat(displayLambda).toString();
}
var unitLabel = "";
if (unit === "seconds") unitLabel = "s⁻¹ (per second)";
if (unit === "minutes") unitLabel = "min⁻¹ (per minute)";
if (unit === "hours") unitLabel = "h⁻¹ (per hour)";
if (unit === "days") unitLabel = "d⁻¹ (per day)";
if (unit === "years") unitLabel = "yr⁻¹ (per year)";
lambdaSpan.innerHTML = displayLambda;
unitSpan.innerHTML = "Units: " + unitLabel;
formulaSpan.innerHTML = "Calculation logic: λ = ln(2) / " + halfLife + " = 0.693147 / " + halfLife + " = " + displayLambda;
resultDiv.style.display = "block";
}