use λ = 0.693 / t

🔬 Radioactive Decay

In this lesson we’ll explore how unstable nuclei lose energy by emitting particles. We’ll learn the key equations, do a quick calculation, and see how this topic appears on the A‑Level exam.

What is Radioactive Decay?

Imagine a bucket that leaks water at a constant rate. The amount of water left in the bucket decreases over time. In the same way, an unstable atom “leaks” energy by emitting particles, reducing the number of atoms that remain.

  • Atoms that are not in their lowest energy state are called unstable.

  • They release energy in the form of alpha, beta particles or gamma rays.

  • The process is random but follows a predictable statistical pattern.

Decay Constant (λ)

The decay constant λ tells us how quickly a sample decays. It is defined by the relation between the half‑life and λ:

\$\lambda = \frac{0.693}{t_{1/2}}\$

Where t1/2 is the time taken for half the atoms to decay.

Think of λ as the “leak rate” of our bucket: a larger λ means a faster leak.

Exponential Decay Formula

The number of atoms remaining after time t is given by:

\$N(t) = N_0 \, e^{-\lambda t}\$

  • N0 = initial number of atoms.

  • N(t) = number of atoms after time t.

  • e = Euler’s number (~2.718).

Because the decay is exponential, the graph of N(t) vs. t is a smooth, downward‑sloping curve.

Example Calculation

Suppose we have 1000 g of a radioactive isotope with a half‑life of 5 years. How many grams remain after 12 years?

  1. Find λ: \$\lambda = \frac{0.693}{5} = 0.1386 \text{ yr}^{-1}\$

  2. Apply the decay formula: \$N(12) = 1000 \, e^{-0.1386 \times 12}\$

  3. Compute: \$N(12) \approx 1000 \times e^{-1.6632} \approx 1000 \times 0.189 \approx 189 \text{ g}\$

After 12 years, only about 189 g of the original 1000 g remains.

Common Isotopes & Their Half‑Lives

IsotopeHalf‑LifeDecay Mode
Carbon‑145730 yrβ⁻ decay
Uranium‑2384.47 × 10⁹ yrα decay
Iodine‑1318.02 daysβ⁻ decay

Exam Tips 📚

  • Always write down the formula you are using before plugging in numbers.

  • Check units: λ is in s⁻¹ (or yr⁻¹), t in the same units.

  • Remember that t1/2 = 0.693 / λ.

  • When converting between half‑lives and λ, keep the factor 0.693 in mind.

  • For multiple‑step problems, show each step clearly; examiners look for method as well as answer.

Good luck, and remember: the key to mastering radioactive decay is practice with different isotopes and time scales!