Please enter a valid positive number for Half-Life.
Decay Constant (λ):–
Mean Lifetime (τ):–
Decay Per Unit Time:–
*Values represent the probability of decay per Year.
function calculateDecay() {
// Clear previous error
var errorDiv = document.getElementById("error-message");
var resultArea = document.getElementById("result-area");
errorDiv.style.display = "none";
resultArea.style.display = "none";
// Get inputs
var tHalf = parseFloat(document.getElementById("halfLifeInput").value);
var unit = document.getElementById("timeUnit").value;
var precision = parseInt(document.getElementById("precision").value);
// Validation
if (isNaN(tHalf) || tHalf <= 0) {
errorDiv.innerText = "Please enter a valid positive number for Half-Life.";
errorDiv.style.display = "block";
return;
}
// Logic
// Decay Constant (lambda) = ln(2) / t_half
var ln2 = Math.log(2); // approx 0.693147
var lambda = ln2 / tHalf;
// Mean Lifetime (tau) = 1 / lambda = t_half / ln(2)
var tau = tHalf / ln2;
// Decay Percentage per 1 unit of time = (1 – e^-lambda) * 100
var decayPercent = (1 – Math.exp(-lambda)) * 100;
// Display results
resultArea.style.display = "block";
// Format Unit string for display (singular vs plural)
var unitSingular = unit.endsWith('s') ? unit.slice(0, -1) : unit;
document.getElementById("unitDisplay").innerText = unitSingular;
// Set Values
// Lambda unit is time^-1
document.getElementById("decayConstantResult").innerHTML = lambda.toFixed(precision) + " /" + unitSingular + "";
// Tau unit is time
document.getElementById("meanLifetimeResult").innerHTML = tau.toFixed(precision) + " " + unit + "";
// Percent
document.getElementById("decayPercentResult").innerHTML = decayPercent.toFixed(precision) + "%";
}
Understanding Decay Rate and Half-Life
In physics and chemistry, the relationship between the decay rate (specifically the decay constant, symbolized as λ) and the half-life (symbolized as t1/2) is fundamental to understanding radioactive decay, chemical kinetics, and pharmacokinetics. This calculator helps you instantly convert a known half-life into a decay constant.
The Formula
For a quantity undergoing exponential decay (first-order kinetics), the rate of decay is proportional to the amount remaining. The constant of proportionality is called the decay constant (λ). The formula connecting half-life and the decay constant is derived from the natural logarithm of 2:
λ = ln(2) / t1/2 ≈ 0.693 / t1/2
Where:
λ (Lambda) is the decay constant (probability of decay per unit time).
t1/2 is the half-life (time required for the quantity to reduce to half its initial value).
ln(2) is the natural logarithm of 2, approximately 0.693147.
Mean Lifetime (τ)
Another useful metric provided by this calculator is the Mean Lifetime (τ), which represents the average time a particle survives before decaying. It is mathematically the reciprocal of the decay constant:
τ = 1 / λ = t1/2 / ln(2)
The mean lifetime is always longer than the half-life by a factor of approximately 1.443 (which is 1/ln(2)).
Real-World Example: Carbon-14
Scenario: Carbon-14 is used in radiocarbon dating and has a half-life of approximately 5,730 years.
Calculation:
Step 1: Identify t1/2 = 5,730.
Step 2: Calculate λ = 0.693147 / 5730.
Result: The decay constant is approximately 0.000121 per year.
This means that in any given year, approximately 0.0121% of the remaining Carbon-14 atoms in a sample will decay.
Applications
While often associated with nuclear physics, calculating decay rates from half-lives is essential in several fields:
Medicine: Calculating drug elimination rates to determine dosage schedules.
Environmental Science: Estimating how long pollutants persist in an ecosystem.
Geology: Dating rocks and fossils using isotopic ratios.
Finance: While termed differently, exponential depreciation uses similar mathematics.
Frequently Asked Questions
What unit is the decay constant in?
The decay constant uses units of inverse time (e.g., s-1, min-1, year-1). If your half-life is entered in years, the resulting rate represents the probability of decay per year.
Why is ln(2) used in the formula?
Exponential decay is described by the function N(t) = N0e-λt. When half the substance remains, N(t)/N0 = 0.5. Solving 0.5 = e-λt involves taking the natural log, where ln(0.5) equals -ln(2). This results in the derivation λ = ln(2)/t1/2.