function calculateDecay() {
var n0 = document.getElementById('initialQty').value;
var nt = document.getElementById('remainingQty').value;
var t = document.getElementById('timeElapsed').value;
var unit = document.getElementById('timeUnit').value || 'units';
var errorDiv = document.getElementById('errorMsg');
var resultsDiv = document.getElementById('resultsArea');
// Reset display
errorDiv.style.display = 'none';
resultsDiv.style.display = 'none';
// Validation
if (n0 === " || nt === " || t === ") {
errorDiv.innerHTML = "Please fill in all numerical fields.";
errorDiv.style.display = 'block';
return;
}
var initial = parseFloat(n0);
var remaining = parseFloat(nt);
var time = parseFloat(t);
if (initial <= 0 || remaining <= 0) {
errorDiv.innerHTML = "Quantities must be positive numbers.";
errorDiv.style.display = 'block';
return;
}
if (time = initial) {
errorDiv.innerHTML = "For decay, Remaining Quantity must be less than Initial Quantity.";
errorDiv.style.display = 'block';
return;
}
// Calculation Logic
// Formula: k = -(ln(Nt/N0)) / t
var k = -(Math.log(remaining / initial)) / time;
// Percentage rate per time unit
var percent = (1 – Math.exp(-k)) * 100; // Discrete equivalent percentage loss per unit
// Or simpler continuous rate percentage: k * 100
var continuousPercent = k * 100;
// Half Life: t(1/2) = ln(2) / k
var halfLife = Math.log(2) / k;
// Display Results
document.getElementById('resK').innerText = k.toFixed(6) + " / " + unit;
document.getElementById('resPercent').innerText = continuousPercent.toFixed(4) + "% (continuous)";
document.getElementById('resHalfLife').innerText = halfLife.toFixed(4) + " " + unit;
// Update Formula Display
document.getElementById('formulaBox').innerHTML =
remaining + " = " + initial + " · e-" + k.toFixed(4) + " · " + time + "";
resultsDiv.style.display = 'block';
}
What is Continuous Decay?
Continuous decay describes a process where a quantity decreases at a rate proportional to its current value. Unlike linear decay, where a fixed amount is lost every time period, continuous decay sheds a fixed percentage of the remaining amount constantly. This model is ubiquitous in the natural sciences and finance.
Common examples include:
Radioactive Decay: Unstable isotopes break down into stable ones over time.
Pharmacokinetics: The body metabolizes drugs, reducing their concentration in the bloodstream.
Asset Depreciation: The value of machinery or vehicles often decreases exponentially.
Population Decline: Species populations can decline continuously due to environmental factors.
The Continuous Decay Formula
The mathematical model for continuous exponential decay is expressed by the differential equation dN/dt = -kN. The solution to this equation gives us the standard formula used in this calculator:
N(t) = N₀e-kt
Where:
N(t) is the remaining quantity after time t.
N₀ is the initial quantity (at time t=0).
k is the continuous decay constant (a positive number).
t is the time elapsed.
e is Euler's number (approximately 2.71828).
How to Calculate the Decay Rate (k)
Often, you know the starting amount and the ending amount after a specific period, and you need to find the rate of decay. By rearranging the formula above, we can solve for k:
Start with N(t) = N₀e-kt
Divide both sides by N₀: N(t)/N₀ = e-kt
Take the natural logarithm (ln) of both sides: ln(N(t)/N₀) = -kt
Solve for k: k = -ln(N(t)/N₀) / t
This formula allows you to determine how quickly a substance or value is diminishing based on empirical data points.
Understanding Half-Life
The "Half-Life" is a specific time period required for the quantity to reduce to exactly half of its initial value. It is strictly related to the decay constant k.
Half-Life Formula: t1/2 = ln(2) / k ≈ 0.693 / k
This metric is particularly useful in radioactive dating (like Carbon-14 dating) because it provides an intuitive way to grasp the speed of decay without dealing with small decimal rates.
Example Calculation
Imagine a medical scenario where a patient receives a dose of 100 mg of a certain medication. After 4 hours, blood tests show that only 25 mg remains in the body. What is the decay rate?