N(t) = n0 e^(-λt)

Definition

The equation n(t) = n0 e^(-λt) describes the exponential decay of a radioactive substance over time, where n(t) represents the remaining quantity of the substance at time t, n0 is the initial quantity, λ (lambda) is the decay constant, and e is the base of natural logarithms. This formula illustrates how the amount of radioactive material decreases as time progresses, showcasing the fundamental principles behind radioactive decay and half-life.

5 Must Know Facts For Your Next Test

1. In the equation, n0 represents the starting quantity of the radioactive substance, which diminishes as time progresses.
2. The decay constant λ is specific to each radioactive isotope and indicates how quickly it decays; a larger λ means faster decay.
3. As time increases, n(t) approaches zero, meaning that after several half-lives, only a negligible amount of the original substance remains.
4. The relationship described by this equation is crucial for applications in nuclear medicine, dating archaeological finds, and understanding nuclear reactions.
5. The concept of half-life derived from this equation allows scientists to predict how long it will take for a certain percentage of a radioactive material to decay.

Review Questions

• How does the decay constant (λ) influence the behavior of radioactive decay as described by the equation n(t) = n0 e^(-λt)?
  • The decay constant (λ) significantly impacts the rate at which a radioactive substance decays. A higher decay constant indicates that the substance will undergo radioactive decay more rapidly compared to one with a lower decay constant. This means that for substances with higher values of λ, n(t) will drop more steeply over time, leading to shorter half-lives and faster depletion of the original material.
• Explain how you can use the equation n(t) = n0 e^(-λt) to determine the half-life of a radioactive isotope.
  • To determine the half-life using the equation n(t) = n0 e^(-λt), you set n(t) equal to n0/2, indicating that half of the original amount has decayed. By substituting into the equation, you get n0/2 = n0 e^(-λt). Simplifying this leads to e^(-λt) = 1/2. Taking the natural logarithm of both sides gives you -λt = ln(1/2), and solving for t yields t = ln(2)/λ. This shows that half-life is inversely related to the decay constant.
• Evaluate how understanding the equation n(t) = n0 e^(-λt) can impact real-world applications in fields such as medicine or archaeology.
  • Understanding this equation is crucial in both medicine and archaeology because it helps professionals make predictions based on radioactive decay. In medicine, it aids in determining appropriate dosages and timing for treatments involving radioactive isotopes. In archaeology, it allows scientists to accurately date artifacts using carbon-14 dating techniques by measuring how much of a sample has decayed since its death. This knowledge leads to more informed decisions and better outcomes in these fields.

Related terms

Decay Constant (λ): A constant that represents the probability per unit time that a nucleus will decay. It is unique to each radioactive isotope.

Half-Life: The time required for half of the radioactive atoms in a sample to decay into another element or isotope. It is a measure of the stability of a radioactive isotope.

Exponential Decay:

A process where a quantity decreases at a rate proportional to its current value, characterized by a rapid initial drop that slows over time.