---
title: "The Decay Constant and the Exponential Decay Law"
description: "Every unstable nucleus of one kind has the same chance of decaying each second, so the number left falls exponentially. The decay constant is that chance per nucleus per second."
canonical: https://lightmysky.com/learn/science/the-decay-constant-and-the-exponential-decay-law-mt_VIydSAys1W
source: https://lightmysky.com/learn/science/the-decay-constant-and-the-exponential-decay-law-mt_VIydSAys1W.md
retrieved: 2026-09-12
---

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# The Decay Constant and the Exponential Decay Law

Every unstable nucleus of one kind has the same chance of decaying each second, so the number left falls exponentially. The decay constant is that chance per nucleus per second.

Subject: Science · Area: Matter & Materials · Ages 17 to 18
Page: https://lightmysky.com/learn/science/the-decay-constant-and-the-exponential-decay-law-mt_VIydSAys1W

## Ready when they can

- Uses the exponential decay law to find either the number left or the time elapsed
- Explains why decay is random for a single nucleus yet predictable for a large sample
- Plots the natural log of the count against time and takes the decay constant from the gradient

## Lesson: The clockwork inside random decay

Every unstable nucleus of one kind shares the same chance of decaying each second. That chance per nucleus per second is the decay constant, written as lambda. Big lambda means fast decay. The survivors follow N equals N0 times e to the minus lambda t, linking start count, constant, and time.

**Example.** Doctors choosing a medical tracer need to know how fast it fades, and half-lives give the answer. After one half-life half the nuclei remain, after two a quarter, after three one eighth. Start with 4000 nuclei and three half-lives later only 500 are left, since one eighth of 4000 is 500.

One nucleus is pure chance, like a single dice throw you cannot call. Trillions of nuclei together average out rock steady, so the sample shrinks on a smooth predictable curve. Plot the natural log of the count against time and the points form a straight line whose slope is minus lambda.

Trade half-life and lambda with care. Lambda equals 0.693 divided by the half-life, so a short half-life means a large lambda. With a 3 s half-life, lambda is 0.693 over 3, which is 0.231 per second. Never treat the half-life as the average lifetime.

**Recap.** One random chance per nucleus builds a smooth exponential fall whose slope and half-life both reveal lambda.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Radioactive Decay: Alpha, Beta and Gamma](https://lightmysky.com/learn/science/radioactive-decay-alpha-beta-and-gamma-mt_-AmjVwkR9t)
- [Recognising Cubic, Reciprocal and Exponential Graphs](https://lightmysky.com/learn/mathematics/recognising-cubic-reciprocal-and-exponential-graphs-mt_2IJJ51rmq9)

## Opens up

- [Decay Modes, Decay Chains and Nuclear Reactions](https://lightmysky.com/learn/science/decay-modes-decay-chains-and-nuclear-reactions-mt_b_imidwJeQ)
- [Activity, Half-Life and Radioactive Dating](https://lightmysky.com/learn/science/activity-half-life-and-radioactive-dating-mt_LvZw4h5wlu)