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SOLUTIONS TO EXERCISE 16

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(a) Water weighs 1000 kg/m^3.  One mole of water weighs 0.018 kg, and contains
6 * 10^23 water molecules.  Each water molecule contains one oxygen atom (which
has 8 protons) and two hydrogen atoms (one proton each) – a total of 10 protons
per molecule.  Thus, 1000 m^3 of water contains

  (1000 m^3 * 1000 kg/m^3)
  ------------------------ * (6 * 10^23 molecules/mole) * (10 protons/molecule)
      (0.018 kg/mole)

  = 3.3 * 10^32 protons.

If the half-life of a proton is 10^31 years, then each year there should be
an average of  (3.3 * 10^32)/(10^31) = 33 proton decays per year.


(b) If after seven years, no proton decays have occurred in these 1000 m^3 of
water, then there have been

  (3.3 * 10^32 protons) * (7 years) = 2.3 * 10^33 "proton years."

If no single decay has occurred in that time, then it could be that protons
never decay.  However, it's possible that a decay could occur tomorrow; in that
case, the half life would probably be about

   2.3 * 10^33 proton years
   ------------------------  =  2.3 * 10^33 years.
            1 proton