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Calculate decomposition rate, exponential decay, remaining mass, decomposition percentage, and half-life with our easy online calculator.
| Formula | Purpose | Notes |
|---|---|---|
| M = M₀ · e^(−kt) | First-order decomposition model | Core USDA/EPA-style equation |
| k = −ln(M/M₀) ÷ t | Decomposition rate constant | Unit of k matches unit of t (e.g. day⁻¹) |
| Decomposition % = (1 − M/M₀) × 100 | Percentage decomposed | — |
| Half-life = ln(2) ÷ k ≈ 0.693 ÷ k | Time for mass to halve | Same time unit as k |
The Decomposition Rate calculator helps you calculate how fast a material breaks down over time. It uses the first-order exponential decay model. You can find the decomposition rate constant, predict remaining mass, calculate decomposition percentage, and find half-life.
This calculator is useful for decomposition rate math, decay rate problems, physics, chemistry, environmental studies, and other processes that follow exponential decay. We developed this calculator so users can easily calculate Decomposition Rate without doing complex logarithm and exponential calculations by hand.
The main decomposition rate equation is:
M = M₀ × e^(−kt)
Here, M is the remaining mass, M₀ is the initial mass, k is the decomposition rate constant, t is elapsed time, and e is the natural exponential constant.
If you know the initial mass and remaining mass, you can calculate the decay rate with:
k = −ln(M / M₀) / t
The calculator also finds the amount decomposed:
Decomposition % = (1 − M / M₀) × 100
The half-life formula is:
Half-life = ln(2) / k
or:
Half-life = 0.693147 / k
The time unit must match the unit used for the rate constant. For example, a rate in day⁻¹ requires time in days.
Suppose a material starts with 100 kg and has 70 kg remaining after 30 days.
First, calculate the remaining fraction:
M / M₀ = 70 / 100 = 0.70
Now calculate the decomposition rate:
k = −ln(0.70) / 30
k = 0.0118892 day⁻¹
Next, calculate the decomposed percentage:
Decomposition % = (1 − 0.70) × 100
Decomposition % = 30%
Now find the half-life:
Half-life = 0.693147 / 0.0118892
Half-life ≈ 58.31 days
So, the decomposition rate constant is about 0.0119 day⁻¹, 30% of the original mass has decomposed, and the estimated half-life is 58.31 days.
If you use the Predict mode with M₀ = 100 kg, k = 0.0118892 day⁻¹, and t = 30 days:
M = 100 × e^(−0.0118892 × 30)
M ≈ 70 kg
This confirms the exponential decay calculation.
The Decomposition Rate calculator provides a simple way to solve exponential decay and decomposition problems. Its core equation is M = M₀ × e^(−kt). You can use it to find the decay rate, predict remaining mass, calculate decomposition percentage, and determine half-life. For accurate results, always use matching time units for k and t.
Use k = −ln(M/M₀) / t when you know the initial mass, remaining mass, and elapsed time.
For first-order exponential decay, divide the negative natural logarithm of the remaining fraction by elapsed time: k = −ln(M/M₀) / t.
Use Half-life = ln(2) / k, or approximately 0.693147 / k.
Use M = M₀ × e^(−kt). Enter the initial amount, decay rate, and elapsed time to find the remaining amount.
The main equation is M = M₀ × e^(−kt). The rearranged equation for finding k is k = −ln(M/M₀) / t.
They can use the same mathematical model when the process follows first-order exponential decay. The exact physical process determines whether this model is appropriate.