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Interpretation:
The energy released (in kJ) for the given decays and the heat released by the decay of
are to be calculated.
Concept introduction:
The amount of a particular radioactive isotope left after time t is given as:
Here,
is the rate constant for the radioactive decay,
Nearly all radioactive decays are of first order and the rate constant is given as:
Here,
is the half -life of the radioactive substance.
The energy during the decay of a radioactive substance is given as:
Here,
is the amount of energy released,
is the mass defect which occurs during the course of decay and c is the speed of the light.
The mass defect (
) is given as the difference of total atomic mass of the product and the reactant in the balanced radioactive reaction.
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Answer to Problem 100AP
Solution:
Explanation of Solution
a) The energy released (in joules) in each of the following two decays
The decay of
The decay of
Mass of
is
Mass of
is
Mass of electron is
Mass of
is
The balanced reaction for
decay is given as follows:
Now, the mass defect
is calculated as follows:
Substitute the values of masses in the above expression.
Now, it is known that
Thus, the conversion factor for this is
Hence, the mass defect can be converted to kilogram unit by using the above conversion factor as follows:
Now, the energy during the decay of a radioactive substance is given as:
Substitute the values of speed of light and energy in the above expression.
Thus, the energy released during the decay of
is
The balanced reaction for
decay is given as follows:
Now, the mass defect
is calculated as follows:
Substitute the values of masses in the above expression.
Now, it is known that
Thus, the conversion factor is
Hence, the mass defect can be converted to kilogram unit by using the above conversion factor as follows:
The energy during the decay of a radioactive substance is given as follows:
Substitute the values of energy and mass defect in the above expression.
Thus, the energy released during the decay of
is
b) The number of moles of
that will decay in a year, starting with 1 mole of
Initial time,
Final time,
It is also known that the rate constant for the radioactive decay is given as follows:
Substitute the value of half-life.
Now, number of moles of
decaying in one year can be calculated as follows:
Here,
is the final number of moles.
Substitute the values of
in the above expression.
Thus, the number of moles that has decayed in one year is calculated as follows:
So, moles of
decayed
c) The amount of heat released (in kJ) corresponding to the
It is known that half- life of
is much shorter than that of
formed from
is converted to
Now, it is known that one mole of particles is equivalent to Avogadro’s number.
Hence, the conversion factor is
Thus, by using the above conversion, the number of nuclei decayed in one year can be calculated as follows:
Now, the total energy released during the decay given in part (a) is as follows:
This energy corresponds to the energy released by one nuclei.
Hence, the conversion factor is
Moreover, it is also known that one kilojoules is equivalent to thousand joules.
Hence, the conversion factor is
Thus, by using the above conversion factor, the energy released in one year can be calculated as follows:
Hence, the heat released by the decay of
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Chapter 20 Solutions
Chemistry
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