Two radioactive materials that alpha decay, 238U and 232Th, and one that beta decays, 40K, are sufficiently abundant in granite to contribute significantly to the heating of Earth through the decay energy produced. The alpha-decay isotopes give rise to decay chains that stop when stable lead isotopes are formed. The isotope 40K has a single beta decay. (Assume this is the only possible decay of that isotope.) Here is the information: In the table Q is the total energy released in the decay of one parent nucleus to the final stable end point and ƒ is the abundance of the isotope in kilograms per kilogram of granite; ppm means parts per million.

(a) Show that these materials produce energy as heat at the rate of 1.0 x 10-9 W for each kilogram of granite.

(b) Assuming that there is 2.7 x 1022 kg of granite in a 20-km-thick spherical shell at the surface of Earth, estimate the power of this decay process over all of Earth. Compare this power with the total solar power intercepted by Earth, 1.7 x 1017W.

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Stable End Point (MeV) Decay Half-Life (y) Parent Mode (ppm) зорЬ 208Pb 4Ca 4.47 x 10° 1.41 x 10 1.28 x 10° 51.7 42.7 1.31 20Th 13