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Objectives
0 Define the unified mass unit;0 State the meaning of the terms mass defect and binding energy
and solve related problems;0 Write nuclear reaction equations and balance the atomic and
mass numbers;0 Understand the meaning if the graph of binding energy per
nucleon versus mass number;0 State the meaning of and difference between fission and fusion;0 Understand that nuclear fusion takes place in the core of stars; 0 Solve problems of fission and fusion reactions.
The unified mass unit
0Abbreviation: u or amu (atomic mass unit)0Definition: 1/12 the mass of an atom of carbon-120A mole of is 12 g and the number of molecules is the
Avogadro constant0An atom of carbon-12 has a mass of 6.02 x1023 x M =
12 g0M = 1.99 x 10-26 kg0Hence M/12 = 1.66 x 10-27 kg
Example
0Find in units of u the masses of the proton, neutron, or electron.
Unified Mass Unit 1.6605402 x 10-27 kg 1 u
electron 9.1093897 x 10-31 kg
proton 1.6726231 x 10-27 kg
neutron 1.6749286 x 10-27 kg
The mass defect and binding energy
0To find the mass of a nucleus: Mnucleus = Matom – Zme
0Matom given in periodic table
0For helium: Mnucleus = 4.0026 – 2 x .0005486 = 4.00156 u
0Lets check it by recalling that a helium nucleus is made up of 2 protons and 2 neutrons and adding their masses: 2 mp + 2 mn = 4.0320 u
0But that’s larger than the mass of the nucleus by .0304 u! This leads to the concept mass defect.
The mass defect and binding energy
0The mass of the protons plus neutrons is larger than the mass of the nucleus. This difference is defined as mass defect: δ = total mass of nucleons – mass of nucleus = Zmp + (A-Z)mn - Mnucleus
Einstein’s mass-energy formula
0 Where is the missing mass?0 Einstein’s theory of special relativity states that mass and
energy are equivalent and can be converted into each other: E = mc2
0 The mass defect of a nucleus has been converted into energy and is stored in the nucleus. This energy is called the binding energy of the nucleus and is denoted by Eb
0 Thus Eb = δc2
0 Binding energy = the work (energy) required to completely separate the nucleons of a nucleus 0 higher binding energy, more stability
Einstein’s mass-energy formula
0How much energy corresponds to 1 u? 1 u x c2 = 1.6605402 x 10-27 x (2.9979 x 108)2 J
= 1.4923946316 x 10-10 J= 931.5 MeV
The binding energy curve
0We saw that the mass defect of helium is 0.0304 which corresponds to a binding energy of 0.0304 x 931.5 MeV = 28.32 MeV
0The alpha particle has an unusually high stability which is why unstable particles decay into them
0There are 4 nucleons in the helium nucleus so the binding energy per nucleon is 28.32 MeV/4 = 7.1 MeV
0Most nuclei have a binding energy per nucleon of approx. 8 MeV
The binding energy curve
0Maximum: 62 (Nickel) 0 If a nucleus heavier than this splits into two lighter
ones, or if two lighter nuclei fuse together, then energy is released as a result.
Energy released in a decay
0Consider the alpha decay of radium: 0For any decay, the total energy to the left of the arrow
must equal the total decay on the right. Here, that energy corresponds to each mass according to Einstein’s formula + the kinetic energies of each mass
0 If the decaying radius is at rest, then the total energy available is Mc2 where M is the mass of the radium nucleus. To the right, we have the energies of the masses of the radon and the helium nuclei. Both have KE too.
Energy released in a decay
0Thus, to be possible, the decay must be such that at the very minimum the energy corresponding to the radium mass is larger than the energies corresponding to the radon plus alpha particle masses.
0Remember that your periodic table gives atomic, not nuclear, masses. You need to subtract the mass of the electrons…unless you’re only interested in mass differences (which we are), then atomic masses are ok.
Energy released in a decay
Mass of radium = 226.0254 u
Mass of radon = 222.0176 u+ Mass of helium = 4.0026 u
sum = 226.0202 u
We see that the mass of radium exceeds that of
radon + helium by 0.0052 u. Thus, there is kinetic energy released equal to 0.0052 x 931.5
MeV = 4.84 MeV
If 50 g of radium were to decay in this way, the total energy released in this way would be N x 4.84 MeV where N is the total number of nuclei in the 50 g of radium. (1.3 x 1023).The momenta of Rn and He are in opposite directions with equal magnitudes (momentum conservation). Vhelium ≈ 55 Vradon
In Summary0For the decay to take place the mass of the decaying
nucleus has to be greater than the combined masses of the products (binding energy of the decaying nucleus must be less than the binding energies of the product nuclei.
0This is why radioactive decay is possible for heavy elements lying to the right of nickel in the binding energy curve.
Nuclear reactions
0 If a nucleus cannot decay by itself, it can still do so if energy is supplied to it by a fast moving particle that collides with it
0Transmutation of nitrogen: an alpha particle colliding with nitrogen produces oxygen and hydrogen (a proton)
0Atomic and mass numbers on both sides match0Left mass = 18.0057 u, right mass = 18.0070 u (larger) so
the alpha must have enough kinetic energy to make up for the mass imbalance. KE>δ because products will also have KE
Nuclear reactions
0Generally, in a reaction , energy will be released if is positive.
0The amount of energy released is equal to 02 kinds of energy producing nuclear reactions: Fission &
Fusion
Nuclear Fission
0A heavy nucleus splits up into lighter nuclei0Uranium-235 absorbs a neutron, momentarily becoming
uranium-236, which then splits into lighter nuclei.0Many possibilities for daughter nuclei
0Production of neutrons is characteristic of fission; they can cause other fission reactions – chain reaction
0Minimum mass of uranium- 235 is needed, otherwise, neutrons escape without causing further reactions – critical mass
Nuclear fission
0The energy released can be calculated as follows:
0This energy appears as the kinetic energy of the products
0This is LOTS of energy (1 kg = 7 x 1013 J)
Mass of uranium plus neutron
= 236.0526 u
Mass of products = 143.92292 u + 88.91781 u + 3 x 1.008665 u= 235.8667250 u
Mass difference = 236.0526 u – 235.8667250 u = 0.0185875 u
Energy released = 0.185875 x 931.5 MeV = 173.14 MeV
Nuclear fission
0Nuclear reactors – controlled rate of reaction0Nuclear bomb – too much energy produced at 1 time –
uncontrolled rate of reaction0Alpha particles cannot be used to start fission because
protons would be repelled by the uranium nucleus0An electron would not sufficiently perturb the heavy
nucleus.0MUST be a neutron
Nuclear Fusion
0The joining of 2 light nuclei into a heavier one + energy
01 kg deuterium ≈ 1013 J
2 x mass of deuterium = 4.0282 u
Mass of helium + neutron
= 4.0247 u
Mass difference = 4.0282 u – 4.0247 u = 0.0035 u
Energy released = 0.0035 x 931.5 MeV = 3.26 MeV
Example
0Another fusion reaction is , where 4 h nuclei fuse into a He plus 2 positrons (electron antiparticles – same mass, opposite charge), 2 electron neutrinos and a photon. Calculate the energy released in this reaction.
4 x mass of hydrogen = 4.029104 u
Mass of helium + positrons + neutrinos
= (4.0026 – 2 x 0.0005486) + (2 x 0.0005486)
= 4.002600 u
Mass difference = 0.026504 u
Energy released= 0.026504 x 931.5 MeV
= 24.7 MeV
Nuclear Fusion
0For light nuclei to fuse, very high temperatures are required to overcome the electrostatic repulsion between nuclei.
0The enormous temperature causes the nuclei to move fast enough to approach each other sufficiently for fusion to occur.
0Temperatures > 107 K = plasma state (ionized atoms)0Plasma cannot contact other things – must be
contained by magnetic fields (tokamaks)
Tokamaks
Serious unsolved problems with prolonged confinement of
plasmas – not a viable source of commercial energy any time
soon
Fusion in Stars0The high temperatures and pressures in the interior of stars
make stars ideal places for fusion.0High temperature needed to overcome electric repulsion0High pressure ensures that sufficient numbers of nuclei are
found close to each other, increasing probability of fusion0 - common reaction in stellar cores 0Fusion = source of star’s energy, prevents it from collapsing
under its own weight and provides energy that star sends out as light and heat.
0Stars = element factories