Exponential Decay Constant Calculator

Exponential Decay Constant Formula:

\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]

Exponential Decay Constant (λ):

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1. What is Exponential Decay Constant?

The exponential decay constant (λ) represents the probability per unit time that a given atom will decay. It is a fundamental parameter in radioactive decay and other exponential decay processes, describing how quickly a quantity decreases over time.

2. How Does the Calculator Work?

The calculator uses the exponential decay constant formula:

\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]

Where:

\( \lambda \) — Exponential decay constant (per time unit)
\( \ln(2) \) — Natural logarithm of 2 (approximately 0.693147)
\( T_{1/2} \) — Half-life of the substance (time units)

Explanation: The decay constant is inversely proportional to the half-life. A smaller half-life means a larger decay constant, indicating faster decay.

3. Importance of Decay Constant Calculation

Details: The decay constant is crucial in nuclear physics, radiometric dating, medical imaging, radiation therapy, and any field involving exponential decay processes. It helps predict remaining quantities over time and determine appropriate safety measures.

4. Using the Calculator

Tips: Enter the half-life in appropriate time units (seconds, minutes, hours, years, etc.). The result will be in reciprocal time units (per second, per minute, etc.). Half-life must be greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What's the relationship between decay constant and half-life?
A: They are inversely related. Decay constant = ln(2) / half-life. A larger decay constant means shorter half-life and faster decay.

Q2: Can this calculator be used for any exponential decay process?
A: Yes, the formula applies to any process that follows exponential decay, including radioactive decay, chemical reactions, and population decline.

Q3: What are typical units for decay constant?
A: The units are reciprocal time (s⁻¹, min⁻¹, hr⁻¹, yr⁻¹, etc.), depending on the half-life units used.

Q4: How is decay constant related to mean lifetime?
A: Mean lifetime (τ) is the reciprocal of decay constant: τ = 1/λ. It represents the average time until decay.

Q5: Why is ln(2) used in the formula?
A: ln(2) comes from the definition of half-life - the time when half the original quantity remains: 1/2 = e^(-λT₁/₂), so λ = ln(2)/T₁/₂.