Rate of Decay Calculator
Understanding the Rate of Decay
The rate of decay describes how a quantity decreases over time due to a natural process, such as radioactive decay, the dissipation of a drug in the body, or the cooling of an object. This process is often modeled using exponential decay, a mathematical concept where the rate of decrease is proportional to the current amount of the quantity.
The Exponential Decay Formula
The standard formula for exponential decay is:
$$N(t) = N₀ \cdot e^{-\lambda t}$$
Where:
- N(t) is the amount of the substance remaining after time t.
- N₀ (N-naught) is the initial amount of the substance at time t=0.
- λ (lambda) is the decay constant, a positive value that represents the rate of decay. A larger λ means faster decay.
- t is the time elapsed.
- e is the base of the natural logarithm, approximately 2.71828.
Key Concepts:
- Decay Constant (λ): This parameter is crucial as it dictates how quickly the quantity diminishes. It is often related to the half-life of the decaying substance.
- Half-Life: The time it takes for half of the initial amount of a substance to decay. It can be calculated from the decay constant using the formula: $T_{1/2} = \frac{\ln(2)}{\lambda}$.
Applications of Rate of Decay:
- Radiometric Dating: Used to determine the age of rocks and fossils based on the decay of radioactive isotopes like Carbon-14 or Uranium-238.
- Pharmacokinetics: Studying how drugs are absorbed, distributed, metabolized, and excreted from the body. The elimination of a drug often follows an exponential decay model.
- Physics: Describing phenomena like the discharge of a capacitor through a resistor, or the cooling of an object to room temperature.
- Population Dynamics: In some models, the decline of a population can be represented by exponential decay.
Example Calculation:
Imagine a radioactive isotope with an initial amount of 100 grams (N₀ = 100g) and a decay constant (λ) of 0.05 per year. We want to find out how much of the isotope remains after 10 years (t = 10 years).
Using the formula:
$$N(10) = 100 \cdot e^{-(0.05 \cdot 10)}$$ $$N(10) = 100 \cdot e^{-0.5}$$ $$N(10) \approx 100 \cdot 0.60653$$ $$N(10) \approx 60.653 \text{ grams}$$
Therefore, approximately 60.653 grams of the isotope would remain after 10 years. Our calculator can quickly perform such calculations for you.