606.53066
393.46934
0.60653066
60.653066
%
13.862944
39.34693
606.53066
393.46934
0.60653066
60.653066
%
13.862944
39.34693
The Exponential Decay Calculator models processes where a quantity decreases at a rate proportional to its current value, following the formula $$P(t) = P_0 \cdot e^{-rt}$$. This ubiquitous model describes radioactive decay, drug metabolism, capacitor discharge, Newton's law of cooling, and depreciation.
Exponential decay is characterized by its half-life — the constant time interval required for any quantity to reduce by half, regardless of the starting amount. This calculator computes remaining values, decayed amounts, and half-lives for any decay scenario.
The continuous exponential decay formula is:
$$P(t) = P_0 \cdot e^{-rt}$$
where $$P_0$$ is the initial quantity, $$r$$ is the decay rate (as a positive decimal), and $$t$$ is elapsed time. The negative sign in the exponent ensures the quantity decreases over time.
Half-life is found by setting $$P(t) = P_0/2$$:
$$\frac{P_0}{2} = P_0 \cdot e^{-rt_{1/2}} \implies t_{1/2} = \frac{\ln 2}{r}$$
The decay factor $$e^{-rt}$$ gives the fraction remaining at time $$t$$. The remaining percentage is $$e^{-rt} \times 100\%$$, and the decayed amount is $$P_0(1 - e^{-rt})$$.
The Remaining Value shows how much is left after time $$t$$. The Decayed Amount is what has been lost. The Remaining Percentage expresses survival as a percent of the original. The Half-Life is the time for any quantity at this rate to halve. The Decay Factor is the multiplicative fraction remaining — multiply any starting quantity by this factor to find what remains after time $$t$$.
Inputs
Results
After 7 years (~1 half-life), roughly half the material remains: 496.6 out of 1000. Half-life is 6.93 years.
Inputs
Results
500 mg of a drug with 15%/hour elimination rate leaves ~274 mg after 4 hours. Half-life is ~4.62 hours.
Half-life ($$t_{1/2}$$) is the time required for a quantity to reduce to half its initial value. It is constant for exponential decay — after one half-life, 50% remains; after two half-lives, 25%; after three, 12.5%; and so on.
Mathematically, no — $$e^{-rt}$$ approaches zero asymptotically but never equals it. In practice, the quantity becomes negligible. After 10 half-lives, only $$\frac{1}{1024} \approx 0.1\%$$ remains.
They are inversely related: $$r = \frac{\ln 2}{t_{1/2}}$$ and $$t_{1/2} = \frac{\ln 2}{r}$$. A larger decay rate means a shorter half-life, and vice versa.
In this calculator, the decay rate $$r$$ is entered as a percentage and converted to a decimal (the decay constant $$\lambda$$). In physics literature, $$\lambda$$ typically refers to the decimal rate directly: $$P(t) = P_0 e^{-\lambda t}$$.
Yes. Carbon-14 has a half-life of 5,730 years, so the decay rate is $$r = \ln(2)/5730 \approx 0.0121\%$$ per year. Enter the initial amount and time to find the remaining C-14.
Linear decay loses a fixed amount per period (e.g., -50/year). Exponential decay loses a fixed percentage per period (e.g., -5%/year). Exponential decay is faster initially but slows as the quantity diminishes.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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