Exponential Decay โ Modeling Natural Decline
A(t) = Aโe^(-kt). Radioactive decay, drug clearance, cooling, battery discharge โ model any process that decreases proportionally to its current value.
Exponential Decay โ Modeling Natural Decline
A(t) = Aโe^(-kt). Radioactive decay, drug clearance, cooling, battery discharge โ model any process that decreases proportionally to its current value.
๐ Sample Examples โ Click to Load
Decay Parameters
Decay Curve Over Time (Line)
Remaining at Key Intervals (Bar)
Decayed vs Remaining (Doughnut)
โ ๏ธFor educational and informational purposes only. Verify with a qualified professional.
๐ Key Takeaways
- โข Formula: A(t) = Aโe-kt โ amount remaining after time t
- โข Half-life: tยฝ = ln(2)/k โ 0.693/k โ time to halve
- โข Always approaches zero โ exponential decay never reaches zero but gets arbitrarily close
๐ก Did You Know?
๐ How Exponential Decay Works
Exponential decay occurs when a quantity decreases at a rate proportional to its current value. The differential equation dA/dt = -kA leads to A(t) = Aโe-kt.
Main Formula
A(t) = Aโ ยท e-kt
Aโ = initial amount, k = decay constant (positive), t = time. Units of k and t must match (e.g., per year with years).
Half-Life Derivation
At half-life: A(tยฝ) = Aโ/2. So e-ktยฝ = 1/2 โ -ktยฝ = ln(1/2) = -ln(2) โ tยฝ = ln(2)/k.
Time to Reach Target
To find t when A(t) = Atarget: Atarget/Aโ = e-kt โ t = ln(Aโ/Atarget)/k.
๐ฏ Expert Tips
๐ก Half-Life to k
k = ln(2)/tยฝ โ 0.693/tยฝ. If tยฝ = 4 hours, then k โ 0.173 per hour.
๐ก Graphing Tips
On semilog plot, exponential decay appears as a straight line. Slope = -k.
๐ก Real-World Modeling
Match units: if t is in years, k must be per year. Carbon-14: k โ 1.21ร10โปโด per year.
๐ก Comparing Decay Rates
Larger k โ faster decay โ shorter half-life. Doubling k halves the half-life.
โ๏ธ This Calculator vs Manual vs Spreadsheet vs CAS
| Feature | This Calculator | Manual | Spreadsheet | CAS |
|---|---|---|---|---|
| Instant A(t) | โ | โ Slow | โ | โ |
| Half-life & time to target | โ | โ ๏ธ Manual | โ | โ |
| Charts (Line, Bar, Doughnut) | โ | โ | โ ๏ธ Manual | โ ๏ธ Code |
| Example presets | โ | โ | โ | โ |
| Copy & share | โ | โ | โ ๏ธ Limited | โ |
| Educational content | โ | โ | โ | โ |
| No setup | โ | โ | โ | โ |
โ Frequently Asked Questions
What is exponential decay?
Exponential decay describes a quantity that decreases at a rate proportional to its current value. Formula: A(t) = Aโe^(-kt). Examples: radioactive decay, drug clearance, cooling.
What is half-life?
Half-life (tยฝ) is the time for a quantity to reduce to half. tยฝ = ln(2)/k โ 0.693/k. Carbon-14 has tยฝ โ 5730 years.
Can k be negative?
For decay, k is positive. A negative k would mean growth (exponential increase). Use the Exponential Growth calculator for that.
How do I convert half-life to decay constant?
k = ln(2)/tยฝ = 0.693/tยฝ. If tยฝ = 4 hours, k โ 0.173 per hour.
Does decay ever reach zero?
Mathematically, no โ it approaches zero asymptotically. In practice, amounts become negligible after several half-lives.
What is Newton's Law of Cooling?
The rate of cooling is proportional to the temperature difference. This leads to exponential decay of (T - T_ambient) over time.
How does this differ from percentage decay?
Percentage decay per period uses A(t) = Aโ(1-r)^t. Continuous decay uses A(t) = Aโe^(-kt). They are related: k โ -ln(1-r) for small r.
What units should I use?
k and t must have matching units. If t is in years, k is per year. If t is in hours, k is per hour.
How do I find time to reach a target amount?
t = ln(Aโ/A_target)/k. Enter the target amount in the optional field to get this automatically.
๐ Exponential Decay by the Numbers
๐ Official Sources
โ ๏ธ Disclaimer: This calculator provides educational results for exponential decay modeling. For medical, nuclear, or engineering applications, verify with domain professionals. Not a substitute for professional advice.