Amongst the radioactive elements that were incorporated when the Earth was formed were:
Uranium 238U with a half-life of 4.5 × 109 y

Uranium 235U with a half-life of 7.1 × 108y

Thorium 230Th with a half-life of 8.0 × 104 y

Iron 60Fe with a half-life of 1.5 × 106 y

Manganese 53Mn with a half-life of 3.6 × 106 y
Thorium 232Th with a half-life of 13.9 × 109 y


Given that the Earth is believed to have been formed 4600 Ma ago, which of the above isotopes could still be contributing to the radioactive heating of its interior? Could you please give a brief explanation of your reasoning. Thanks.

Take the first isotope,

U_{238}\,half-life=4.5\times 10^9\,years

If you have a Kg of U238, after one half-life, 4.5 x 10^9 years, you'll have half a kilogram of U238 and half a kilogram of daughter products. After two half-lives, you'll have 1/4 Kg. After three half-lives you'll have 1/8 Kg Three half-lives is 1.35 x 10^8 years. Eventually, you'll get to where there isn't much U238 left or the amount that is left is too small to measure. That's the limit to the usefulness of the isotope for measuring times or for producing heat.

When isotopes decompose, they give off alpha, beta, or gamma rays. These radioactive rays are absorbed by nearby matter as heat. More heat is given off by radioactive particles with shorter half-lives -- because they're decomposing faster and giving off more gamma rays. So, the same analysis applies to heating the core as it does to measuring times. The question boils down to, "how much is left". You should figure out the fraction of material that will be left after the period that is specified. Then you can make conclusions from it.