Exponential Decay Rate Calculator
Calculate decay constant (λ), half-life, and mean lifetime.
Decay Constant (λ):
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Percentage Decay Rate:
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Half-Life (t1/2):
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Mean Lifetime (τ):
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Understanding Decay Rate Calculation
In physics, chemistry, and finance, calculating the decay rate is essential for understanding how quickly a quantity decreases over time. This applies to radioactive decay, cooling of objects (Newton's Law of Cooling), atmospheric pressure changes, and asset depreciation.
This calculator determines the Exponential Decay Constant (λ) based on the continuous decay model. It differs from simple linear subtraction because the rate of loss is proportional to the current amount remaining.
The Formulas Used
The calculation is based on the fundamental exponential decay equation:
N(t) = N₀ * e^(-λt)
Where:
- N(t): The remaining quantity after time t.
- N₀: The initial quantity at time 0.
- λ (Lambda): The decay constant.
- t: The time elapsed.
How We Calculate the Results
By rearranging the formula above, we solve for the specific metrics:
- Decay Constant (λ): Calculated as
-ln(N(t) / N₀) / t. This represents the probability of decay per unit time. - Half-Life (t1/2): The time required for the quantity to reduce to half its initial value. Calculated as
ln(2) / λ. - Mean Lifetime (τ): The average time a particle survives before decaying. Calculated as
1 / λ. - Periodic Decay Rate (%): Calculated as
(1 - (N(t)/N₀)^(1/t)) * 100. This shows the percentage lost per single time unit.
Example Calculation
Suppose you have a radioactive isotope with an Initial Quantity of 500g. After 5 years, the remaining quantity is 300g.
- Initial (N₀): 500
- Remaining (Nₜ): 300
- Time (t): 5
- Result (λ): 0.1022 (approx)
- Half-Life: 6.78 years
This means roughly 10.22% of the remaining substance decays continuously every year.