1. INST 240 Revolutions Lecture 11 Nuclear Energy

2. The spacetime momentum vector
Ptime mc
γmc
γmv
Pspace
Total length: mc
Length in time direction: γmc
Length in space direction: γmv = γ p

3. Non-relativistic limit
At low velocities (v << c):
Relativistic Physics
E = γmc2
p = γmv
“Classical” Physics:
E = ½mv2
p = mv
→ mc2 + ½m v2+
small corrections
→ mv + small corrections

4. Conserved Energy Etotal relativistic= Ptime c = mc2 + ½m v2
The second term is the kinetic energy!
We know that in non-relativistic processes it is (often) conserved
But here, the conserved energy has an additional term that is left even when v0
Erest= mc2

5. That’s it! So Einstein’s famous formula Erest= mc2
turns out to be the energy of an object measured by an observer at rest with respect to the object
If the object is at rest, it does not have kinetic energy (duh!), but a moving observer will not agree

6. Agreement
All observers agree on the length of the energymomentum vector (mc)
All observers agree that the total relativistic energy is conserved
Etotal relativistic= mc2 + ½m v2 = constant
Observers will disagree why it is conserved!

7. Momentum is conserved, energy is conserved
Units of momentum are units of energy divided by c
Space momentum & Energy divided by c form a vector which has Einstein’s blessing

8. Space&time and Energy&momentum diagrams

9. Spacetime and Energymomentum diagrams
c time
Energy/c
space
Momentum

10. Should energy and momentum have bigger or smaller values in a frame moving wrt the object?
A: Both have bigger values
B: Both have smaller values
C: Energy stays the same
D: Momentum stays the same
E: energy gets bigger, momentum smaller
F: momentum gets bigger, energy gets smaller

11. Implications “Objects have an intrinsic energy equivalent
to their rest mass”
“Energy is equivalent to mass”
E = mc2
(Physicists actually don’t use this form of
the equation, but it’s catchy)

12. Einstein Original Work (1906)
“Das Prinzip von der Erhaltung der Schwerpunktsenergie und die Trägheit der Energie” (Only 7 pages!)
“Schreibt man also jeglicher Energie die träge Masse E/V2 zu, so gilt (…) das Prinzip von der Erhaltung der Bewegung des Schwerpunkts auch auch für Systeme in denen electromagnetische Prozesse vorkommen.”
With träge Masse as the mass and V=c, we have m= E/c2

13. Implications Objects have mass.
Objects that store extra energy have extra mass.
Objects that have given up all their energy have less mass.

14. Which has more mass? A hot or a cold Potatoe?
A: Same
B: Hot one
C: Cold one

15. Implications ... all the bits after the explosion
A little bit heavier than...

16. Implications ... all the bits after the explosion
A little bit heavier than...
total mass = mass of component bits
+ mass due to “available energy”

17. How much heavier? Take a 1 kg block of TNT.
How much heavier is it than it’s component parts?
 Worksheet #5
Trinitrotoluene (TNT)

18. How much heavier? TNT releases 0.65 Calories of energy per gram.
c = 3x108 m/s
Trinitrotoluene (TNT)
E = mc2
energy in Joules
mass in kg

19. How much heavier? - Worksheet
TNT releases 0.65 Calories of energy per gram.
E = mc2
1 kg = 1000 g
1 Calorie = 4200 Joules
1 kg TNT releases E =2730 kJ = mc2
m = 2,730,000 J/ (300,000,000 m/s)2
= 2.73/9 x 10^6 x 10^-16 kg
= 3 x 10^-11 kg

20. How much heavier? kg kg
Note that it has the same number of atoms!
The mass comes from the bonds between the atoms

21. Why should we care? Can understand energy production in Sun, stars
Can produce power by harvesting energy stored in mass, binding energy
Can construct powerful bombs, too, unfortunately

22. How do we know how much energy the Sun produces each second?
The Sun’s energy spreads out in all directions
We can measure how much energy we receive on Earth
At a distance of 1 A.U., each square meter receives 1400 Watts of power (the solar constant)
Multiply by surface of sphere of radius bill. meter (=1 A.U.) to obtain total power output of the Sun
Why not use this directly on Earth? Not yet practical for large-scale power (small scale okay, though, e.g. solar calculators, solar heating)

23. Energy Output of the Sun
Total power output: 4  1026 Watts
The same as
100 billion 1 megaton nuclear bombs per second
4 trillion trillion 100 W light bulbs
$10 quintillion (10 billion billion) worth of energy per 9¢/kWh
The source of virtually all our energy (fossil fuels, wind, waterfalls, …)
Exceptions: nuclear power, geothermal

24. Where does the Energy come from?
Anaxagoras ( BC): Sun a large hot rock – No, it would cool down too fast
Combustion?
No, it could last a few thousand years
19th Century – gravitational contraction?
No! Even though the lifetime of sun would be about 100 million years, geological evidence showed that Earth was much older than this
Grav contraction -- Kelvin and Helmholtz. There probably do exist bodies that give off heat from this process, called brown dwarfs, but none has been decisively observed
By the end of the 19th century, geological evidence showed that the Earth was much older; and with the discovery of radioactivity, the age of the Earth was determined to be 4.5 billion years.
So, what is the process that powers the Sun?

25. What process can produce so much power?
For the longest time we did not know
Only in the 1930’s had science advanced to the point where we could answer this question
Needed to develop very advanced physics: quantum mechanics and nuclear physics
Virtually the only process that can do it is nuclear fusion

26. Nuclear Fusion Atoms: electrons orbiting nuclei
Chemistry deals only with electron orbits (electron exchange glues atoms together to from molecules)
Nuclear power comes from the nucleus
Nuclei are very small
If electrons would orbit the statehouse on I-270, the nucleus would be a soccer ball in Gov. Kasic’s office
Nuclei: made out of protons (el. positive) and neutrons (neutral)
Nuclear Fusion
Nuclei ~ 10^-15 m, atom ~ 10^-10 m; nucleus discovered 1905 (Rutherford)
Nucleus = protons+neutrons in the 1930’s
The mass of the total products of the reaction is less than the original mass. Where does the mass go? Mass is not conserved: it can be converted into energy: E=mc2 (Einstein, 1905).
the speed of light c is very large, so a small amount of mass yields a lot of energy when it is converted!
to produce the amount of power emitted by the Sun, 600 million tons of hydrogen must be combined to form helium per second: a lot of mass, but small compared to the Sun
at this rate, the Sun has enough fuel to last another 5 billion years

27. The Structure of Matter
Atom: Nucleus and Electrons
The Structure of Matter
Nucleus: Protons and Neutrons (Nucleons)
Nucleon: 3 Quarks
| m |
| 10-14m |
|10-15m|

28. Basic Nuclear Physics
The strong force between protons and neutrons is short ranged – think velcro!
Each particle “sticks” only to its neighbors
The electrical repulsion between protons is weaker but long ranged
Each proton repels every other one
Bigger nuclei have more trouble holding together – repulsion eventually wins!

29. Nuclear fusion reaction
In essence, 4 hydrogen nuclei combine (fuse) to form a helium nucleus, plus some byproducts (actually, a total of 6 nuclei are involved)
Mass of products is less than the original mass
The missing mass is emitted in the form of energy, according to Einstein’s famous formula:
E = mc2
(the speed of light is very large, so there is a lot of energy in even a tiny mass)

30. Nuclear Fusion 4 1H (protons) 4H  He + 2e+ + 2γ + 2ν
Small nuclei (like hydrogen) can “fuse” to form larger nuclei (helium, etc.), releasing energy
Basic reaction:
4H  He + 2e+ + 2γ + 2ν
where
e+ is a positron (anti-particle of the electron)
γ is a gamma-ray photon
ν is a “neutrino”
Most of the energy released is carried by the positrons and gamma rays
4 1H (protons)
4H e

31. Hydrogen fuses to Helium
Heavier elements as well…
Start: protons  End: Helium nucleus + neutrinos
Hydrogen fuses to Helium

32. Why does a ball roll downhill?
Why should the fuse?
Why does a ball roll downhill?
To minimize energy!
Iron (Fe) weighs less
per proton than anything else
Each proton in hydrogen
weighs more
Each proton in Uranium weighs more