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Calculate the total binding energy of
^{20}_{10} Ne.
Solution:
By Table 29.3 in the text, we see that
^{20}_{10} Ne has
a mass of 19.992439 u. Since
Neon has 10 protons and 10 neutrons, the mass deficiency is
10 x 1.007825 + 10 x 1.008665 - 19.992439 = 0.172461 u .
This corresponds to an energy ofE = mc^{ 2} = 0.172461 u x 931.5 c ^{2} = 160.6 MeV .
Thus, the binding energy is 160.6 MeV, or about 8.0 MeV per nucleon.
Suppose you begin with
1.0 x 10^{- 2} g of a pure
radioactive substance and 4 h later determine that
0.25 x 10^{- 2} g remain. What is the half-life
of the substance?
Solution:
For this we will use the relation
N = N_{0}e^{ - t} = ^{t/T1/2},
where T_{1/2} = ln 2/ has been used. We then have= 0.25 = ^{t/T1/2} ln 0.25 = ln - ln 2,
from which we find T_{1/2} = 2.0 h.
The ^{14}C content decreases after the death of a living
system with a half-life of 5739 years. If the ^{14}C content
of an old piece of wood is found to be 12.5% of that of
an equivalent present-day sample, how old is the piece of
wood?
Solution:
This will also use the relation
N = N_{0}e^{ - t} = ^{t/T1/2}.
With the data given, we find0.125 = ^{t/5739 yr} ln 0.125 = ln - ln 2,
from which we deduce that t = 17,217 yr.
Find the energy released in the alpha-decay
^{238}_{92} U ^{234}_{90} Th + ^{4}_{2} He
Solution:
For this we shall need the masses
M238.050786 u , | |||
M234.043583 u , | |||
M4.002603 u , |
238.050786 - (234.043583 + 4.002603) = 0.0046 u .
This corresponds to an energy ofE = mc^{ 2} = 0.0046 u x 931.5 c ^{2} = 4.29 MeV .
Thus, the reaction releases an energy of 4.29 MeV.
Suppose that the sun consists entirely of hydrogen and that
the dominant energy-releasing reaction is
4^{4}_{2} He + 22 + .
If the total power output of the sun is assumed to remain constant at 3.9 x 10^{26} W, how long will it take for all of the hydrogen to be burned up? Take the mass of the sun as 1.99 x 10^{30} kg.Solution:
We first find the total number of hydrogen atoms in the sun by
calculating
= 1.192 x 10^{57} atoms .
Now, the reaction quoted has a mass difference of4 x 1.007825 - (4.002603 + 2 x 0.000549) = 0.027599 u ,
which corresponds to an energy ofE = mc^{ 2} = 0.027599 u x 931.5 c ^{2} = 25.71 MeV .
Since each individual reaction consumes 4 Hydrogen atoms, the total energy available in the sun is1.192 x 10^{57} atoms x 25.71 x 10^{6} x = 1.225 x 10^{45} J .
The lifetime of the sun can then be estimated asx x x = 9.96 x 10^{10} yr .
Thus, this gives an estimated lifetime of about 99.6 billion years.www-admin@theory.uwinnipeg.ca