---
title: "N = N₀e^(-λt) — AP Physics 2 Definition & Exam Guide"
description: "N = N₀e^(-λt) is the radioactive decay equation in AP Physics 2 Topic 15.7. Learn how λ relates to half-life and how to solve decay problems on the exam."
canonical: "https://fiveable.me/ap-physics-2-revised/key-terms/n-n0e-t"
type: "key-term"
subject: "AP Physics 2"
unit: "Unit 15"
---

# N = N₀e^(-λt) — AP Physics 2 Definition & Exam Guide

## Definition

N = N₀e^(-λt) is the exponential decay equation in AP Physics 2 (Topic 15.7) that gives the number of radioactive nuclei N remaining after time t, where N₀ is the starting number and λ is the decay constant, the probability per unit time that any single nucleus decays.

## What It Is

N = N₀e^(-λt) describes how a sample of radioactive material shrinks over time. You start with N₀ unstable nuclei, and after time t, only N of them are still radioactive. The [decay constant](/ap-physics-2-revised/unit-15/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI "fv-autolink") λ sets the pace. A bigger λ means a steeper, faster decay curve.

Here's the physics behind the math. You can never predict when one specific [nucleus](/ap-physics-2-revised/key-terms/nucleus "fv-autolink") will decay (the CED says this time is indeterminable), but with a huge number of nuclei, probability takes over and the population as a whole follows a smooth exponential curve. That's why the same fraction of the sample decays in every equal time interval, instead of the same raw number. The decay constant connects directly to [half-life](/ap-physics-2-revised/key-terms/half-life "fv-autolink") through λ = ln(2)/t₁/₂, so the equation and the half-life concept are two views of the same behavior.

## Why It Matters

This equation lives in Topic 15.7 (Fission, Fusion, and Nuclear Decay) in [Unit 15](/ap-physics-2-revised/unit-15 "fv-autolink"): Modern Physics, and it's the mathematical heart of learning objective 15.7.B, which asks you to describe the [radioactive decay](/ap-physics-2-revised/key-terms/radioactive-decay "fv-autolink") of a sample with a finite number of nuclei. The essential knowledge for 15.7.B says radioactive decay is characterized by an exponential decrease in the number of radioactive nuclei over time. N = N₀e^(-λt) is exactly that statement written as math. It also captures a big modern-physics idea, that individual quantum events are random but large populations are predictable. You can't say when one nucleus decays, yet you can say precisely how many remain after 30 years.

## Connections

### [Half-life (Unit 15)](/ap-physics-2-revised/key-terms/half-life)

Half-life is the time for half the radioactive nuclei to decay, and it's baked into this equation through λ = ln(2)/t₁/₂. They're the same physics in two costumes. Half-life is great for quick mental math at nice intervals, while N = N₀e^(-λt) handles any time t, including awkward ones like 7.3 years.

### [E=mc² (Unit 15)](/ap-physics-2-revised/key-terms/e=mc²)

The decay equation tells you how many nuclei transform, and [E=mc²](/ap-physics-2-revised/key-terms/e=mc² "fv-autolink") tells you where the energy comes from when each one does. Every decay event converts a tiny bit of mass into kinetic energy of the products or photons, so the two equations together describe both the timing and the energetics of nuclear decay.

### Nucleon conservation (Unit 15)

N = N₀e^(-λt) counts how many nuclei have decayed, but conservation of [nucleon number](/ap-physics-2-revised/unit-15/8-types-of-radioactive-decay/study-guide/yQriAHMZEcNvi6kE "fv-autolink") constrains what each decay can produce. The total count of protons plus neutrons never changes in a nuclear reaction, which is why decay transforms a nucleus into different nuclei rather than making nucleons vanish.

## On the AP Exam

Expect multiple-choice questions in two flavors. The first is straight calculation, like a sample with λ = 0.05 yr⁻¹ and 8.0 × 10²⁰ initial nuclei, asking how many remain after 30 years (plug in, get N₀e^(-1.5) ≈ 1.8 × 10²⁰). The second is conceptual model identification, like data showing Cobalt-60 dropping from 8,000 to 4,000 to 2,000 nuclei in equal 5.27-year intervals, and asking which mathematical model fits. The giveaway is that the sample halves each interval, so it's exponential, not linear. On free-response questions, decay supports the kind of reasoning 15.7.B rewards, like explaining why decay is probabilistic for one nucleus but predictable for a sample, or sketching and interpreting an exponential decay curve.

## N = N₀e^(-λt) vs N = N₀(1/2)^(t/t₁/₂)

These are the same law written two ways. The half-life form N = N₀(1/2)^(t/t₁/₂) counts in halvings and is perfect when t is a whole number of half-lives. The exponential form N = N₀e^(-λt) uses the decay constant and works cleanly for any time. They're linked by λ = ln(2)/t₁/₂, so converting between them is one step. Pick whichever matches the information the problem gives you.

## Key Takeaways

- N = N₀e^(-λt) gives the number of radioactive nuclei remaining after time t, starting from N₀ nuclei with decay constant λ.
- The decay constant λ and half-life are linked by λ = ln(2)/t₁/₂, so a large λ means a short half-life and fast decay.
- You can never predict when an individual nucleus will decay, but the population of a large sample follows this smooth exponential curve.
- Exponential decay means the same fraction of nuclei decays in every equal time interval, which is why a sample halves over and over instead of decreasing by a fixed amount.
- If data shows a quantity halving in equal time intervals, like Cobalt-60 going 8,000 to 4,000 to 2,000 every 5.27 years, the exponential model is the answer.

## FAQs

### What is N = N₀e^(-λt) in AP Physics 2?

It's the radioactive decay equation from Topic 15.7. It tells you how many radioactive nuclei (N) remain in a sample after time t, given the initial number N₀ and the decay constant λ.

### Can N = N₀e^(-λt) predict when a single nucleus will decay?

No. The CED is explicit that the decay time of an individual nucleus is indeterminable. The equation only describes the statistical behavior of a large sample, where probability makes the overall decay predictable.

### What's the difference between the decay constant λ and the half-life?

The decay constant λ is the probability per unit time that a nucleus decays, while half-life t₁/₂ is the time for half the sample to decay. They're inversely related through λ = ln(2)/t₁/₂, so you can convert one to the other in a single step.

### How do I solve a radioactive decay problem with N = N₀e^(-λt)?

Plug in N₀, λ, and t, then evaluate the exponential. For example, with N₀ = 8.0 × 10²⁰ nuclei, λ = 0.05 yr⁻¹, and t = 30 years, the exponent is -1.5 and N ≈ 1.8 × 10²⁰ nuclei. If you're given half-life instead of λ, convert first using λ = ln(2)/t₁/₂.

### Does radioactive decay violate conservation of mass?

No, but mass alone isn't conserved. Through [mass-energy equivalence](/ap-physics-2-revised/key-terms/mass-energy-equivalence "fv-autolink") (E=mc²), some mass converts into the kinetic energy of decay products or into photons. Nucleon number is conserved in every nuclear reaction, so the total count of protons and neutrons never changes.

## Related Study Guides

- [15.7 Fission, Fusion, and Nuclear Decay](/ap-physics-2-revised/unit-15/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI)

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