7.3 Mass and Energy
Recognize this guy?
It's almost impossible today to imagine Albert Einstein having trouble gaining a
foothold in the world of physics, but such was the case for him in his early 20s. Here,
he stands at a lectern in the Bern patent office in 1904.
This pic was taken shortly before Einstein released his Theory of Relativity in 1905.
Among the most well-known results of Einstein’s findings was the equation E0 = mc2
What do E, m, and c stand for? Here’s the “biggie” – how do they relate to each other?
There are lots of interpretations of Einstein’s Theory of Relativity available on line.
According to the above equation, and the interpretation of the authors of your primary
resource, a particle has a “rest” energy = mc2. This energy is intrinsic to the particle; it is
a property of the particle. Remember learning about positrons? They are equal in mass,
but opposite in charge, to an electron. They are emitted form the nucleus during Beta
decay. When a positron encounters an electron…
they mutually and
instantaneously obliterate each other! What happens to their energy? It is given off as
EM radiation.
Q: If the two particles are initially at rest, what is true of the Energy given off from the
EM radiation?
A: It = the rest energies of both combined, of course!
Q: What units are used for describing energies of nuclear particles, reactions, and / or
emissions?
A: eV = electron-volts; or MeV = mega-electron-volts
The rest energies of a system can = EU + Erest + … + Eanything that contributes to the E ☺
Q: If a system at rest absorbs energy, what can you conclude about its mass, after the
fact?
A: It has to increase, specifically by a factor of ∆M = ∆E / c2, as indicated by the formula.
If you can
another take on Einstein’s Energy Equation – Theory of Relativity,
what it implies is that if you take any mass, and accelerate it to the speed of light squared,
it will convert into pure energy!
Considering the magnitude of c where c = 3.00 x 108 m/s, even very small changes in mass
correspond to gigantic changes in energy. Think nuclear energy!
It is the
understanding of this that led Einstein to unlock the mysteries of nuclear energy.
Purportedly, Einstein was extremely disappointed to have his knowledge used for the
creation of real WMD. Morr with this later.
Now, consider two 1-kg blocks attached by a spring of force constant k, where k = 800 N/m
and x = 0.10 m
Q: If you stretch the spring a distance of x, what is the equation that represents the ∆U?
A: ∆U = ½ kx2 = ½ (800 N/m)(0.10 m)2 = 4 J. So, ∆M = ∆U / c2 = 4J / (3 x 108 m/s)2
= 4.44 x 10- 17 kg
Relatively speaking (excuse the pun!), the relative mass increase ∆M / M ~ 2 x 10 -17, is way
too itty-bitty to be seen with the
Nuclear Energy
In a nuclear reaction, the ∆ E’s are often an appreciable fraction of the rest of the energy
of the system.
Q: ‘Member from AP Chemistry…☺
☺, what is binding energy?
A: E required to break a nucleus down into its constituent parts; E contained within the
bonds between particles and / or subatomic particles.
Q: What is fission? What is fusion?
A: Fission is breaking apart a nucleus into two (or morr) smaller nuclei, and fusion is
joining nuclear particles to form a larger nucleus. During fission, the resultant mass of
the smaller nuclei is less than the mass of the original nucleus, as energy, thus mass, is
released in the process. The larger the nucleus, the more energy it takes to hold it
together.
Please form a group of 4. Take 2 minutes, and read p 202. Summarize and
discuss (or if you’re a true SF, discuss and summarize ☺) the “BIG IDEA” of the content
explained therein.
See Example 7-14, p 203
A hydrogen atom consisting of a proton and an electron has a binding energy of 13.6 eV.
By what percentage is the mass of a proton plus the mass of an electron greater than that
of a hydrogen atom?
Compare the binding energy, Eb / c2 to the masses: __Eb / c2 = 13.6 eV / c2
me + mp
me + mp
Check Table 7-1 for the masses of the e- and the p+, add them to get 938.28 MeV / c2
So, the fractional difference will be ___13.6 eV / c2
938.28 MeV / c2
YOU TRY IT! See ex 7-15, p 203
= 1.45 x 10-8 = 1.45 x 10 -6 %
Newtonian Mechanics and Relativity
Q: Considering that we’ve long had a law that explains force, mass, and acceleration
(sound familiar?) what would be the possible motivation to explore the same stuff, apart
from “knowledge for knowledge sake?”
A: At very high velocities, N2 is ineffective.
To relate N2 to Einstein’s TOR, use Ke = ½ mv2 = ½ mc2 (v2 / c2) = ½ E0 v2 / c2 where
E0 = mc2 is the rest energy of the particle.
If you take the √ of both sides, you get v/c = √KE / E0
Q: What does that really mean with respect to the KE compared to the rest energy?
A: Newtonian Laws are valid IFF the velocity of the particle is <<< c, or, if KE <<< E0
7-4 Quantization of Energy
Q: What does it mean to say E is quantized? (‘Member, I asked you that earlier?)
When you put energy into a system, the E0 increases.
Q: Can you keep adding E to any system?
A: Nope. There is a point at which you cannot add any morr E. The E of a system is
quantized, thus can increase by discrete amounts only. There comes a point when the
system cain’t take no mo. ☺
Imagine 2 blocks connected by a spring. If you pull them apart, you are doing _______ on
the system.
Q: What does that do to its PE?
Once you release them, they oscillate back and forth. The Energy of oscillation is due to
its ME, or KE + PE. Really, then, the
E of oscillation = KE of the motion + PE due to stretching of the spring = initial PE
As time goes by, the Esys decreases due to the damping effects of friction, air resistance.
Eventually, the all of the E is dissipated and the Eoscillation = 0.
Chemically speaking, if you consider the diatomic molecule O2, The force of attraction
varies linearly with the change in separation (for small changes). If a diatomic molecule is
set oscillating, with some energy, E, the energy decreases with time as the molecule either
radiates, or interacts with the environment. The decrease is NOT continuous, but in
discrete, thus finite, steps. Morrover, the lowest energy level, ground state, still has energy
associated with it. Thus, the vibrational energy of a diatomic molecule is quantized,
absorbing or releasing quanta of energy.
Whenever you have a periodic function, it is possible to calculate the period of the
function.
Q: How are the period of oscillation and the frequency of oscillation related?
A: Inversely! f = 1 / Т
NOTE: The f and the Т are independent of the energy of oscillation!
Look at Figure 7-15 in your book. What is wrong with this picture, literally?
WOW! What a big goof for an AP Physics book!
In any event, ‘member when you were a
and we studied:
• The energy of a quantum / photon, E = h f where h = 6.626 x 10-34 J • s (Note: we
used nu for frequency)
• Quantum numbers, where E = the energy level of the electron
• c = λ ν, showing the inverse relationship between speed and frequency, as well as the
fact that their product is always constant
• DeBroiglie’s equation which basically showed that any moving object oscillates to
some degree
Some new stuff to add - the energy of:
• any level can be found by En = (n + ½)h f where n = 0, 1, 2, 3, …
• ground-state can be found with E0 = ½ h f (obviously!)
• radiation, which is the result of the gain or loss of energy is Erad = Ei - Ef
All bound systems exhibit E quantization. Because the steps can be so small, they are
unobservable, but we can calculate them, nonetheless. See the next page for your HW ☺
Do p 207 – 215, #’s 1, 3, 5, 6, 13, 18, 24, 27, 32, 35, 39, 43, 49, 56, 70, 71, 73, 84, 88