N(t) = n0 e^(-λt)

Definition

The equation $$n(t) = n_0 e^{-\lambda t}$$ describes the number of radioactive nuclei remaining at time $$t$$, where $$n_0$$ is the initial quantity, $$\lambda$$ is the decay constant, and $$e$$ is the base of natural logarithms. This equation is fundamental in understanding how radioactive decay occurs over time, illustrating that the rate of decay is exponential and dependent on the decay constant, which ties directly into concepts like half-life.

5 Must Know Facts For Your Next Test

1. In this equation, $$n(t)$$ represents the remaining amount of a radioactive isotope at time $$t$$, which decreases over time due to decay.
2. The initial quantity $$n_0$$ is crucial as it sets the starting point for understanding how much of the substance remains after a certain period.
3. The term $$e^{-\lambda t}$$ indicates that as time increases, the number of remaining nuclei decreases exponentially.
4. The decay constant $$\lambda$$ is unique to each radioactive isotope and reflects how quickly it decays; a higher value means faster decay.
5. This equation can be rearranged to derive half-life, allowing for quick calculations regarding how long it takes for a sample to reduce to half its original amount.

Review Questions

• How does the equation n(t) = n0 e^(-λt) illustrate the concept of exponential decay in radioactive materials?
  • The equation $$n(t) = n_0 e^{-\lambda t}$$ shows that the number of remaining radioactive nuclei decreases exponentially as time progresses. This means that rather than a linear reduction, the rate at which the material decays slows down over time; initially, a large number of atoms decay quickly, but as fewer remain, the total number decaying per unit time becomes smaller. This relationship between time and remaining quantity is essential in understanding how isotopes behave over extended periods.
• Explain how the decay constant (λ) relates to both the equation n(t) = n0 e^(-λt) and the concept of half-life.
  • The decay constant $$\lambda$$ is a critical component of the equation $$n(t) = n_0 e^{-\lambda t}$$ as it directly influences how rapidly an isotope decays. A higher decay constant means a shorter half-life, indicating that a given quantity will decrease more rapidly. The relationship allows us to connect both concepts: while the equation gives us a specific snapshot of decay at any point in time, understanding $$\lambda$$ helps us predict when exactly half of a sample will remain.
• Evaluate how the understanding of n(t) = n0 e^(-λt) impacts fields like geology or archaeology in determining ages of materials.
  • Understanding the equation $$n(t) = n_0 e^{-\lambda t}$$ allows scientists in fields such as geology and archaeology to date ancient materials through techniques like radiocarbon dating. By measuring the remaining amount of a radioactive isotope within a sample and knowing its decay constant, researchers can accurately calculate how long it has been since the material was formed. This application not only provides insights into historical timelines but also enhances our comprehension of geological processes and material preservation.

Related terms

Decay Constant (λ): A probability rate that indicates how quickly a radioactive substance will decay, unique to each isotope.

Half-life: The time required for half of the radioactive nuclei in a sample to decay, a key concept that relates to the decay constant.

Radioactive Isotope: An isotope of an element that has an unstable nucleus and emits radiation as it decays to a stable form.