Understanding Half-Life Decay Rate
The concept of half-life is fundamental in understanding the rate at which radioactive isotopes, chemical compounds, or other quantities decay over time. The half-life (T1/2) of a substance is the time required for half of the initial amount of that substance to decay or transform into another substance.
The decay process is typically exponential. This means that in each half-life period, the amount of the substance reduces by half. For instance, if you start with 100 grams of a substance with a half-life of 10 years, after 10 years, you'll have 50 grams left. After another 10 years (total of 20 years), you'll have 25 grams remaining, and so on.
This calculator helps you determine the decay rate based on the initial amount, the amount remaining, and the time that has passed. The decay rate is often expressed as a decay constant (λ), which is related to the half-life by the formula:
λ = ln(2) / T1/2
However, this calculator will directly calculate the relationship based on your inputs to show how many half-lives have passed and the resulting decay rate. The core formula used is derived from the exponential decay equation:
N(t) = N0 * (1/2)(t / T1/2)
Where:
- N(t) is the amount of substance remaining after time t.
- N0 is the initial amount of substance.
- t is the time elapsed.
- T1/2 is the half-life of the substance.
By rearranging and solving for the number of half-lives elapsed (n), we get:
n = t / T1/2
And also:
N(t) / N0 = (1/2)n
Taking the logarithm of both sides allows us to solve for n without knowing T1/2 directly. This calculator computes the effective decay rate based on the observed remaining amount after a certain time.
Example:
Suppose you start with 100 grams of Carbon-14 (N0 = 100g) and after 11460 years (t = 11460 years), you measure 25 grams remaining (N(t) = 25g). This scenario implies that two half-lives have passed (100g -> 50g -> 25g). The calculator will confirm this and provide the effective decay rate per year.
The effective decay rate per unit time can be thought of as the fraction of the substance that decays in one unit of time. If n half-lives have passed in time t, then the time elapsed is t = n * T1/2. The fraction remaining is (1/2)n.
function calculateDecayRate() {
var initialAmount = parseFloat(document.getElementById("initialAmount").value);
var remainingAmount = parseFloat(document.getElementById("remainingAmount").value);
var timeElapsed = parseFloat(document.getElementById("timeElapsed").value);
var timeUnit = document.getElementById("timeUnit").value.trim();
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = ""; // Clear previous results
if (isNaN(initialAmount) || isNaN(remainingAmount) || isNaN(timeElapsed) || timeUnit === "") {
resultDiv.innerHTML = "Please enter valid numbers for all fields and a unit of time.";
return;
}
if (initialAmount <= 0) {
resultDiv.innerHTML = "Initial amount must be greater than zero.";
return;
}
if (remainingAmount initialAmount) {
resultDiv.innerHTML = "Remaining amount must be between 0 and the initial amount.";
return;
}
if (timeElapsed <= 0) {
resultDiv.innerHTML = "Time elapsed must be greater than zero.";
return;
}
// Calculate the number of half-lives elapsed
// N(t) = N0 * (1/2)^(t / T_half)
// N(t) / N0 = (1/2)^(t / T_half)
// log2(N(t) / N0) = t / T_half
// T_half = t / log2(N(t) / N0)
// Number of half-lives (n) = t / T_half = log2(N(t) / N0)
var fractionRemaining = remainingAmount / initialAmount;
// Handle edge case where fractionRemaining is 1 (no decay)
if (fractionRemaining === 1) {
resultDiv.innerHTML = "No decay has occurred. The substance amount has not changed.";
return;
}
// Handle edge case where remainingAmount is 0
if (fractionRemaining === 0) {
resultDiv.innerHTML = "The substance has completely decayed. This implies an infinite number of half-lives have passed, or the remaining amount is effectively zero within measurement capabilities.";
return;
}
var numberOfHalfLives = Math.log(fractionRemaining) / Math.log(0.5); // Equivalent to log base 2 of fractionRemaining
// Calculate the effective decay rate constant (lambda)
// lambda = ln(2) / T_half
// Since numberOfHalfLives = timeElapsed / T_half, then T_half = timeElapsed / numberOfHalfLives
// lambda = ln(2) / (timeElapsed / numberOfHalfLives)
// lambda = (ln(2) * numberOfHalfLives) / timeElapsed
var decayConstant = Math.log(2) * numberOfHalfLives / timeElapsed;
// Calculate the half-life if needed, though the request is for decay rate
var halfLife = timeElapsed / numberOfHalfLives;
var outputHTML = "
Results:
";
outputHTML += "Initial Amount: " + initialAmount + "";
outputHTML += "Amount Remaining: " + remainingAmount + "";
outputHTML += "Time Elapsed: " + timeElapsed + " " + timeUnit + "";
outputHTML += "
Number of Half-Lives Elapsed: " + numberOfHalfLives.toFixed(4) + "";
outputHTML += "
Effective Decay Rate Constant (λ): " + decayConstant.toExponential(4) + " per " + timeUnit + "";
outputHTML += "
This means that on average, " + decayConstant.toExponential(4) + " of the substance decays per unit " + timeUnit + ".";
outputHTML += "Calculated Half-Life (if needed): " + halfLife.toFixed(4) + " " + timeUnit + "";
resultDiv.innerHTML = outputHTML;
}
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