One-eighth of a sample of 227 Th 90 (227 90 Th) remains undecayed after 54 days. What is the half-life of the thorium isotope?

Half-Life = 18 days

Explanation:

Isotope decay follows the equation:

Ln[A] = -kt + ln[A]₀

Where [A] is amount of isotope after time t,

k is rate constant

[A]₀ is initial amount of isotope.

If we solve rate constant, we can find half-life by using:

Half-life = ln 2 / Rate constant

Replacing in isotope decay equation:

Ln[1/8] = -k*54 days + ln[1]

-2.07944 = -54k

0.0385days⁻¹ = k

Half-Life = ln 2 / 0.0385days⁻¹