Calculate the Rate Constant for the Decay of Tc-99

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Tc-99 Decay Rate Constant Calculator

Calculate the radioactive decay constant (λ) for Technetium-99 or Technetium-99m.

Tc-99 (Ground State – Waste) Tc-99m (Metastable – Medical) Custom Half-Life
Years Days Hours Minutes Seconds

Calculated Results

Decay Constant (λ):

Interpretation: Approximately of the atoms decay per unit of time selected.

Understanding the Decay of Technetium-99

Technetium-99 (Tc-99) is a significant radioisotope in both nuclear medicine and nuclear waste management. To understand how quickly it transforms, we must calculate the decay rate constant (λ). This constant represents the probability of a nucleus decaying per unit of time.

λ = ln(2) / t₁/₂ ≈ 0.693 / t₁/₂

Tc-99 vs. Tc-99m

It is crucial to distinguish between the two common forms of this element:

  • Technetium-99 (Ground State): A long-lived fission product with a half-life of approximately 211,000 years. It is a major concern in long-term nuclear waste storage.
  • Technetium-99m (Metastable): Used widely in medical imaging. It has a much shorter half-life of roughly 6 hours, making it ideal for diagnostic tests without long-term radiation exposure to the patient.

Example Calculation

For Tc-99m, which has a half-life of 6.01 hours:

1. Identify t₁/₂ = 6.01 hours.
2. Apply formula: λ = 0.693147 / 6.01.
3. λ ≈ 0.1153 hr⁻¹.

This means roughly 11.53% of the remaining Tc-99m atoms decay every hour.

Why the Rate Constant Matters

The rate constant allows scientists and health physicists to predict the activity (Becquerels or Curies) of a sample at any given time. While the half-life tells us how long it takes for half the sample to vanish, the rate constant is the mathematical engine used in the exponential decay formula: N(t) = N₀e⁻λᵗ.

function updateHalfLife() { var preset = document.getElementById('isotopeType').value; var input = document.getElementById('halfLifeInput'); var unit = document.getElementById('timeUnit'); if (preset === "211000") { input.value = 211000; unit.value = "years"; input.disabled = false; } else if (preset === "6.01") { input.value = 6.01; unit.value = "hours"; input.disabled = false; } else { input.disabled = false; } } function calculateDecay() { var tHalf = parseFloat(document.getElementById('halfLifeInput').value); var unit = document.getElementById('timeUnit').value; var resultDiv = document.getElementById('decayResult'); var lambdaOut = document.getElementById('lambdaOutput'); var percentOut = document.getElementById('percentOutput'); if (isNaN(tHalf) || tHalf <= 0) { alert("Please enter a valid positive number for the half-life."); return; } // λ = ln(2) / t_1/2 var lambda = Math.log(2) / tHalf; // Formatting for display var displayLambda; if (lambda < 0.0001) { displayLambda = lambda.toExponential(6); } else { displayLambda = lambda.toFixed(8); } var unitLabel = ""; switch(unit) { case "years": unitLabel = "yr⁻¹"; break; case "days": unitLabel = "day⁻¹"; break; case "hours": unitLabel = "hr⁻¹"; break; case "minutes": unitLabel = "min⁻¹"; break; case "seconds": unitLabel = "sec⁻¹"; break; } lambdaOut.innerHTML = displayLambda + " " + unitLabel; var percentage = (lambda * 100); if (percentage < 0.000001) { percentOut.innerHTML = percentage.toExponential(4) + "%"; } else { percentOut.innerHTML = percentage.toFixed(6) + "%"; } resultDiv.style.display = "block"; }

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