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850. Problem 54.67 (RHK) Two radioactive materials that are unstable to alpha decay,
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and
, and
one that is unstable to beta decay,
, are
sufficiently abundant in granite to contribute significantly to the heating of
the Earth through the decay energy produced. The alpha-unstable isotopes
give rise to decay chains that stop at stable lead isotopes.
is the total energy released in the decay of one parent nucleus to the final
stable endpoint and
is
the abundance of the isotope in kilograms per kilogram of granite; ppm
means parts per million. We have to show (a) that these materials
give rise to a total heat production of 988 pW for each kilogram of
granite. (b) Assuming that there is
of
granite in a 20-km thick, spherical shell around the Earth,
we have to estimate the power this will produce over the whole Earth. We
will compare this with the total solar power intercepted by the Earth,
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Solution: Click For PDF Version (a)
As the abundance of
The half-life of
The decay rate of
For this decay
We consider next the energy production in 1 kilogram of granite because of
radioactive decay of
As the abundance of
The half-life of
The decay rate of
For this decay
We consider next the energy production in 1 kilogram of granite because of
radioactive decay of
As the abundance of
The half-life of
The decay rate of
For this decay
We thus find that the total energy released per second in 1 kilogram of granite due to radioactive disintegrations will be
(b) Assuming that there is
The total solar power intercepted by the Earth is
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:
, the
number of
is
. The
disintegration constant will be
, the
number of
. Its
disintegration constant will be
:
, the
number of
. The
disintegration constant will be
.