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Energy of a nucleus
Ch. 14
The mass of a helium nucleus is slightly smaller (<1%)
than the combined masses of its four nucleons. This
mass difference is converted to energy via E = mc2 .
2 protons
2 neutrons
6.695·10-27 kg
6.645·10-27 kg
Conversion of mass to energy
5 ·10-29 kg of mass is converted to energy when 2 protons
and 2 neutrons are combined to form the helium nucleus:
E = m c2 = (5·10-29 kg) · (3·108 m/s)2 = 4.5 ·10-12 J = 28 MeV
Each of the four nucleons releases 28 MeV / 4 = 7 MeV
1 J = 6.24  1018 eV
Energy per nucleon
Energy per
nucleon in
· 1H
Principle of nuclear fusion
Energy is released when
combining two light nuclei
into one heavier nucleus.
Nucleon Number A
Fe and Ni have are the most stable nuclei (lowest energy per nucleon).
Fusion vs. fission
• Energy is released either
by combining two small
nuclei (fusion) or by splitting a large nucleus into
two pieces (fission).
• The energy is released as
radiation and as kinetic
energy. Both eventually
turn into heat (the fireball from a nuclear bomb
and the steam generated
in a nuclear reactor).
Fe Ni
Nucleon Number A
Fusion vs. fission in bombs, reactors
• Fusion powers hydrogen bombs.
Fission powers atomic bombs.
• Fusion has not yet been tamed for peaceful purposes.
Fission generates energy in nuclear reactors.
Fusion in stars
• Stars convert hydrogen to helium and heavier elements.
When Fe and Ni are reached, fusion stops. The star has
burnt its nuclear fuel and collapses under its own gravity.
• In massive stars, this collapse releases a huge amount of
gravitational energy that leads to a supernova.The outer
90% of the star is ejected, and the center becomes either
a black hole (> 3 solar masses) or a neutron star (between
1.4 and 3 solar masses), where the atoms collapse into a
single huge nucleus. Lighter stars become white dwarfs .
• All elements heavier than iron/nickel are created during
a supernova explosion, which has enough thermal energy
to form nuclei with higher energy per nucleon.
Stable nuclei
Red dots = stable nuclei.
The gray region contains
unstable nuclei, created
in the laboratory.
Stable nuclei have about
equal neutron and proton
numbers N and Z (dashed).
At high Z, there are more
neutrons than protons, because protons are charged
and repel each other.
Radioactive decay
If the ratio of protons to neutrons
gets too far off-balance, a nucleus
will spontaneously transform itself
into another nucleus with a better
ratio by emitting , ,  particles.
 particle = 2p 2n = He nucleus
 particle = electron
 particle = photon
Marie Curie,
Nobel prizes in
physics, chemistry
Isotopes are different versions
of the same element (same Z).
They have the same number of
electrons and protons , but a
different neutron number N.
Their chemical behavior is the
same, since that is determined
by the electron number (= Z).
Stable isotopes are shown as
red dots.The gray region contains unstable isotopes which
are radioactive.
Different isotopes of the same element
Isotopes of hydrogen
One proton
One proton
one neutron
One proton
two neutrons
These three isotopes play a central role in various fusion reactions.
Isotopes of carbon
• Carbon has 6 protons and 6 electrons (Z=6). Its outer shell contains
4 electrons, which determine the chemical properties of carbon.
• The most common isotope of carbon has 6 neutrons, 12 nucleons.
It is commonly labeled 12C (“C twelve”).
is another isotope of carbon containing 8 neutrons, 14 nucleons.
is unstable and decays radioactively.
The decay of 14C is exponential (Lect. 4, Slides 5,6). After 6000 years,
half of the 14C has decayed (= half-life). After another 6000 years, one
loses another half, and so on every 6000 years.
Carbon-dating question
The 14C/12C ratio in a fossil bone is found
to be ⅛ of the ratio in a living animal.
What is the approximate age of the fossil ?
A. 6 000 years
B. 18 000 years
C. 32 000 years
D. 48 000 years
Since the ratio has been reduced
by a factor of ⅛ = ½½½ = (½)3,
three half-lives have passed, i.e.
3 · 6000 years = 18 000 years
Radioactive dating
• Radioactive 14C is created continuously by cosmic rays (next slide).
oxidizes to CO2 and is converted by plants into organic matter.
Particularly durable are wood and charcoal generated from wood.
• Animals and humans eat plants and incorporate 14C into the bones.
• Decaying 14C is replenished as long as plants and animals are alive.
• Once a plant or animal dies, its 14C content decreases and thereby
starts the clock for radiocarbon dating.
• By measuring the 14C/12C ratio of a sample from an archaeological
site one can determine its age. (Willard Libby, 1969 Nobel Prize)
• This can be done up to an age of about 60 000 years, when the 14C
concentration has been reduced by a factor of (½)10 = 1/1024 .
Production of carbon 14C
A cosmic ray proton shatters the
nucleus of an atom in the upper
atmosphere, creating neutrons n
plus other debris.
A 14N nucleus absorbs a neutron
and emits a proton, becoming 14C .
Concentration of
• A balance between the production and decay rates determines
the equilibrium ratio:
• Such an extremely low ratio of one part in a trillion requires a
very sensitive detector which can detect single 14C atoms.
• It helps to have a large number of C atoms from a macroscopic
sample (compare Avogadro’s number, 1024 ).
Geological dating
• For older specimens one uses isotopes with longer half-life,
for example 235U (uranium). Its half-life is 0.7 billion years.
• The oldest rocks on Earth have been dated this way. These
are 4.4 billion years old.
• A focused ion beam removes a small amount of material from
several spots on one of the tiny red zircon crystals.