What Is Exponential Decay Model?
An exponential decay model describes a quantity that shrinks by the same fraction in every equal time interval. The amount at time t is N0 times e to the power of minus k times t.
Choose how you describe the decay: the constant k, the half-life, or two measurements taken at different times. The calculator returns the model, the amount at your chosen time and the time needed to fall to a target.
The half-life and the constant are linked by half-life = ln 2 ÷ k, so either one determines the other.
How to Use the Exponential Decay Model Calculator
- Choose what you know.
- Enter N₀ and the half-life, k or a measurement.
- Optionally enter t or a target amount.
- Read the model and the curve.
Exponential Decay Model Formula
Decay model
Half-life
Time to reach N
Exponential Decay Model Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| N0 | Starting amount | |
| k | Decay constant | |
| t | Time | |
| t½ | Half-life = ln 2 ÷ k |
Exponential Decay Model Worked Example
A sample of 500 units has a half-life of 12. Then k = ln 2 ÷ 12 = 0.05776. After 30 time units 500 × e^(−0.05776 × 30) = 88.4 remain, and it takes ln(500 ÷ 50) ÷ 0.05776 = 39.9 time units to fall to 50.
With the example values loaded in the calculator above, the steps are:
- Model: N(t) = N₀ e^(−kt).
- k = ln 2 ÷ half-life = 0.693147 ÷ 12 = 0.057762265.
- Half-life = ln 2 ÷ k = 12.
- N(30) = 500 × e^(−0.0577623 × 30) = 88.388348.
- t = ln(N₀ ÷ target) ÷ k = ln(500 ÷ 50) ÷ 0.057762265 = 39.863137.
Result: Model = N(t) = 500·e^(−0.0577623·t); Decay constant k = 0.057762265; Half-life = 12; Amount remaining at t = 30 = 88.388348; Time to fall to 50 = 39.863137.
How to Read an Exponential Decay Model
Each half-life cuts the amount to one half, whatever the starting amount. After 2 half-lives one quarter remains, after 3 one eighth, and so on. The decay constant k and the half-life are linked by half-life = ln 2 ÷ k, so a larger k means a faster decline.
With a half-life of 12 and 500 units to start, 250 remain at t = 12, 125 at t = 24 and about 88 at t = 30. The model never reaches zero in theory. It only gets closer, which is why the calculator asks for a target amount instead.
Where Exponential Decay Model Is Used
Radioactive decay
Enter the half-life and the starting amount to see how much of an isotope remains after a given time. The model assumes a constant decay rate, which fits radioactive decay well.
Drug concentration
Estimate the level of a drug that is cleared at a rate proportional to the amount present. The half-life is then the time for the level to halve.
Cooling and depreciation
Fit k from two measurements when you do not know the half-life. Use it only if the quantity really falls by a fixed fraction per unit time.
Homework and exams
Check answers to half-life problems, including the time needed to reach a given target.
Common Exponential Decay Model Mistakes to Avoid
- Use the same time unit for k, t and the half-life. Mixing days and hours is the most common error.
- The decay constant k is positive in N(t) = N₀·e^(−kt). A negative k would be growth.
- When you build the model from two measurements, the first one must be larger than the second. A rise cannot be modelled as decay.
- The model is only as good as the constant-rate assumption. Check it against a third data point if you have one.
Exponential Decay Model Terms and Definitions
- Half-life
- The time for the amount to fall to half.
Exponential Decay Model FAQs
What is the exponential decay formula?
N(t) = N₀·e^(−kt), where N₀ is the starting amount, k is the decay constant and t is time. An equivalent form is N(t) = N₀·(1/2)^(t ÷ half-life).
How do I find k from the half-life?
Use k = ln 2 ÷ half-life, which is about 0.6931 ÷ half-life. For a half-life of 12, k = 0.6931 ÷ 12 = 0.05776 per time unit.
How do I find the time to reach a target amount?
Solve N₀·e^(−kt) = target for t, which gives t = ln(N₀ ÷ target) ÷ k. With N₀ = 500, a target of 50 and k = 0.05776, t is about 39.9 time units.
How do I find k from two measurements?
If the amount is N₁ at time t₁ and N₂ at time t₂, then k = ln(N₁ ÷ N₂) ÷ (t₂ − t₁). The starting amount then follows from N₀ = N₁·e^(k·t₁).
What is the difference between decay and growth?
Decay has a negative exponent, e^(−kt), and shrinks toward zero. Growth has a positive exponent and rises without a ceiling. The same formulas apply with the sign of k reversed.