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Atoms
Entropy
Quanta
John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
1
The Papers of
Einstein’s Year of
Miracles, 1905
2
The Papers of 1905
1
"Light quantum/photoelectric effect paper"
2
Einstein's doctoral dissertation
3
"Brownian motion paper."
4
Special relativity
5
E=mc2
"On a heuristic viewpoint concerning the production and
transformation of light."
Annalen der Physik, 17(1905), pp. 132-148.(17 March 1905)
"A New Determination of Molecular Dimensions"
Buchdruckerei K. J. Wyss, Bern, 1905. (30 April 1905)
Also: Annalen der Physik, 19(1906), pp. 289-305.
"On the motion of small particles suspended in liquids at
rest required by the molecular-kinetic theory of heat."
Einstein inferred from the thermal properties of high
frequency heat radiation that it behaves
thermodynamically as if constituted of spatially localized,
independent quanta of energy.
Einstein used known physical properties of sugar
solution (viscosity, diffusion) to determine the size of
sugar molecules.
Einstein predicted that the thermal energy of small
particles would manifest as a jiggling motion, visible
under the microscope.
Annalen der Physik, 17(1905), pp. 549-560.(May 1905; received 11 May 1905)
“On the Electrodynamics of Moving Bodies,”
Annalen der Physik, 17 (1905), pp. 891-921. (June 1905; received 30 June, 1905)
“Does the Inertia of a Body Depend upon its Energy
Content?”
Annalen der Physik, 18(1905), pp. 639-641. (September 1905; received 27
September, 1905)
Maintaining the principle of relativity in
electrodynamics requires a new theory of space and
time.
Changing the energy of a body changes its inertia in
accord with E=mc2.
Einstein to Conrad Habicht
18th or 25th May 1905
“…and is very revolutionary”
Einstein’s assessment of his light quantum paper.
…So, what are you up to, you frozen whale, you smoked, dried, canned piece of sole…?
…Why have you still not sent me your dissertation? …Don't you know that I am one of the
1.5 fellows who would read it with interest and pleasure, you wretched man? I promise you
four papers in return…
The [first] paper deals with radiation and the energy properties of light
and is very revolutionary, as you will see if you send me your work first.
The second paper is a determination of the true sizes of atoms from the diffusion and the
viscosity of dilute solutions of neutral substances.
The third proves that, on the assumption of the molecular kinetic theory of heat, bodies on
the order of magnitude 1/1000 mm, suspended in liquids, must already perform an
observable random motion that is produced by thermal motion;…
The fourth paper is only a rough draft at this point, and is an electrodynamics of moving
bodies which employs a modification of the theory of space and time; the purely
kinematical part of this paper will surely interest you.
Why is only the light quantum
“very revolutionary”?
2
Einstein's doctoral dissertation
3
"Brownian motion paper."
4
Special relativity
5
E=mc2
"A New Determination of Molecular Dimensions"
Buchdruckerei K. J. Wyss, Bern, 1905. (30 April 1905)
Also: Annalen der Physik, 19(1906), pp. 289-305.
All the rest develop or
complete 19th century physics.
Advances the molecular kinetic
program of Maxwell and
Boltzmann.
"On the motion of small particles suspended in liquids at
rest required by the molecular-kinetic theory of heat."
Annalen der Physik, 17(1905), pp. 549-560.(May 1905; received 11 May 1905)
“On the Electrodynamics of Moving Bodies,”
Annalen der Physik, 17 (1905), pp. 891-921. (June 1905; received 30 June, 1905)
“Does the Inertia of a Body Depend upon its Energy
Content?”
Annalen der Physik, 18(1905), pp. 639-641. (September 1905; received 27
September, 1905)
Establishes the real significance
of the Lorentz covariance of
Maxwell’s electrodynamics.
Light energy has momentum;
extend to all forms of energy.
6
Why is only the light quantum
“very revolutionary”?
All the rest develop or
complete 19th century physics.
Well, not always.
"Monochromatic radiation of low density
behaves--as long as Wien's radiation
formula is valid [i.e. at high values of
frequency/temperature]--in a
thermodynamic sense, as if it consisted
of mutually independent energy quanta
of magnitude [h]."
The great achievements of 19th
century physics:
•The wave theory of light; Newton’s
corpuscular theory fails.
•Maxwell’s electrodynamic and its
development and perfection by
Hertz, Lorentz…
•The synthesis: light waves just are
electromagnetic waves.
Einstein’s light quantum paper
initiated a reappraisal of the physical
constitution of light that is not
resolved over 100 years later.
7
How did Einstein find the quantum?
The content of Einstein’s
discovery was quite unanticipated:
High frequency light energy exists in
• spatially independent,
• spatially localized
points.
The method of Einstein’s discovery
was familiar and secure.
Einstein’s research program in statistical
physics from first publication of 1901:
How can we infer the microscopic
properties of matter from its macroscopic
properties?
The statistical papers of 1905: the
analysis of thermal systems consisting of
• spatially independent
• spatially localized,
points.
(Dilute sugar solutions,
Small particles in suspension)
8
If….
If we locate Einstein’s light quantum
paper against the background of his work
in statistical physics,
its methods are an inspired variation of
ones repeated used and proven effective
in other contexts on very similar
problems.
If we locate Einstein’s light
quantum paper against the
background of electrodynamic
theory, its claims are so far
beyond bold as to be foolhardy.
9
Einstein’s Early Program
in Statistical Physics
10
Einstein’s first two “worthless” papers
Einstein to Stark, 7 Dec 1907, “…I am
sending you all my publications
excepting my two worthless beginner’s
works…”
“Conclusions drawn
from the
phenomenon of
Capillarity,”
Annalen der Physik,
4(1901), pp. 513523.
“On the thermodynamic
theory of the difference
in potentials between
metals and fully
dissociated solutions of
their salts and on an
electrical method for
investigating molecular
forces,” Annalen der
Physik, 8(1902), pp.
798-814.
11
Einstein’s first two “worthless” papers
Einstein’s hypothesis:
Forces between molecules at distance
r apart are governed by a potential P
satisfying
From
macroscopic properties of
capillarity and
electrochemical potentials
P = P - cc(r)
for constants cand c characteristic
of the two molecules and universal
function (r).
infer
coefficients in the
microscopic force law.
12
Independent Discovery of the Gibbs Framework
3 papers 19021904
Einstein, Albert.
'Kinetische Theorie des
Waermegleichgewichtes und
des zweiten Hauptsatzes der
Thermodynamik'. Annalen
der Physik, 9 (1902)
13
Independent Discovery of the Gibbs Framework
3 papers 19021904
Einstein, Albert. 'Eine
Theorie der Grundlagen
der Thermodynamik'.
Annalen der Physik, 9
(1903)
14
Independent Discovery of the Gibbs Framework
3 papers 19021904
Einstein, Albert. 'Zur
allgemeinen molekularen
Theorie der Waerme'.
Annalen der Physik, 14
(1904)
15
The Hidden Gem
16
Einstein’s Fluctuation Formula
Any canonically distributed
system
 E 
p(E)  exp  
 kT 

Variance of energy from mean
dE
  (E  E)  kT
 kT 2C
dT
2
2
2
Heat capacity is
macroscopically
measureable.
(1904)
17
Applied to an Ideal Gas
Ideal monatomic gas, n molecules
3nkT
E
2
3nk
C
2
dE
  (E  E)  kT
 kT 2C
dT
2
2
2

rms deviation of energy from mean

n=1024…. negligible
3n / 2kT
1


E (3n / 2)kT
3n / 2
n=1
1
 0.816
3/ 2

18
In 1904, no one had any solid
idea of the constitution of
heat radiation!
Applied to heat radiation
Volume V of heat radiation (Stefan-Boltzmann law)
E  VT 4 C  4VT 3
 2  (E  E)2  kT 2
dE
 kT 2C
dT

rms deviation of energy from mean


2 kVT 5/2
k 1

2
4
E
VT
V T 3/2
Hence estimate volume V in which
fluctuations are of the size of the
mean energy.
 E
2
2
(1904)
19
Einstein’s Doctoral
Dissertation
"A New Determination of
Molecular Dimensions”
20
21
How big are molecules?
= How many fit into a gram mole? = Loschmidt’s number N
Find out by determining how the
presence of sugar molecules in dilute
solutions increases the viscosity of
water. The sugar obstructs the flow and
makes the water seem thicker.
After a long and very hard calculation…
And after very many special assumptions…
Apparent viscosity m*
=viscosity m of pure water
x (1 + fraction of volume
taken by sugar )
= (r/m) N (4p/3) P 3
r= sugar density in the solution
m = molecular weight of sugar
P = radius of sugar molecule, idealized as a sphere
Well, not quite. Einstein made a
mistake in the calculation. The
correct result is
m* = m (1 + 5/2 )
The examiners did not notice.
Einstein passed and was awarded the
22
PhD. He later corrected the mistake.
Recovering N
Turning the expression for apparent viscosity inside out:
N = (3m/ 4pr) x (m*/m - 1) x 1/P 3
All
measurable
quantities
P = radius
of
molecule.
Unknown!
ONE equation in TWO unknowns.
Einstein needs another equation.
The rate of diffusion of sugar in water is fixed
by the measurable diffusion coefficient D.
Einstein shows:
N = (RT/6pmD) x 1/P
All
measurable
quantities
TWO equations in TWO unknowns.
Einstein determined
N = 2.1 x 10 23
After later correction for his calculation error
N = 6.6 x 10 23
23
The statistical physics of dilute sugar solutions
Sugar in dilute solution consists of
a fixed, large number of component
molecules that do not interact with
each other.
Hence they can be treated by exactly
the same analysis as an ideal gas!
Sugar in dilute solution exerts an
osmotic pressure P that obeys the
ideal gas law
PV = nkT
Dilute sugar solution in a gravitational field.
24
Recovering the equation for diffusion
The equilibrium sugar concentration
gradient arises from a balance of:
Sugar molecules falling
under the effect of gravity.
Stokes’ law F = 6pmPv, F = gravitational force
And
Sugar molecules diffusing
upwards because of the concentration
gradient.
density
gradient
pressure
gradient
(ideal gas law)
upward
force
The condition for perfect balance is
N = (RT/6pmD) x 1/P
25
Equilibration of pressure
by a field instead of a semi-permeable
membrane…
… was a device Einstein used repeatedly
but casually in 1905, but had been
introduced with great caution and
ceremony in his 1902 “Potentials” paper.
26
“Brownian motion paper.”
“On the motion of small particles suspended
in liquids at rest required by the molecularkinetic theory of heat.”
27
28
An easier way to estimate N?
Doctoral dissertation: Einstein determined N from TWO equations in TWO unknowns N, P.
N = (3m/ 4pr) x (m*/m - 1) x 1/P 3
N = (RT/6pmD) x 1/P
What if sugar molecules were so big that we could measure their diameter P under the microscope?
Why not just do the same analysis with microscopically visible particles?! Then P is observable.
Only ONE equation is needed.
Particles in suspension = a fixed, large number of
component that do not interact with each other.
Hence they can be treated by exactly the same
analysis as an ideal gas and dilute sugar solution!
The particles exert a pressure due to their thermal motions.
PV=nkT
…and this leads to their diffusion according to the
same relation N = (RT/6pm D) x 1/P.
Particles in suspension in a gravitational
field
Measure D and
we can find N.
29
Brownian motion
Einstein predicted thermal motions of
tiny particles visible under the
microscope and suspected that this
explained Brown’s observations of the
motion of pollen grains.
For particles of size 0.001mm, Einstein
predicted a displacement of approximately
6 microns in one minute.
“If it is really possible to observe the motion discussed here … then classical
thermodynamics can no longer be viewed as strictly valid even for
microscopically distinguishable spaces, and an exact determination of the
real size of atoms becomes possible.”
30
Estimating the coefficient of diffusion for suspended particles
…and hence determine N.
To describe the thermal motions of small particles, Einstein laid the foundations of the
modern theory of stochastic processes and solved the “random walk problem.”
Particles spread over time t,
distributed on a bell curve.
Their mean square
displacement is 2Dt.
Hence we can read D from
the observed displacement
of particles over time.
Then find N using
N = (RT/6pmD) x 1/P
31
Unexpected properties of the motion
Displacement is proportional to square
root of time. So an average velocity
cannot be usefully defined.
displacement  0 as time gets large.
time
The “jiggles” are not the visible result
of single collisions with water
molecules, but each jiggle is the
accumulated effect of many collision.
32
Thermodynamics of
Systems of Independent
Components
(ideal gas law)
33
The simplest case: an ideal gas
(e.g. most ordinary gases at ordinary temperatures and low pressures)
Pressure P x volume V =
number of molecules n
x Boltzmann’s constant k
x temperature T
PV = nkT
What we shall see:
The law depends only the gases
having a very few simple
properties: they consist of very
many spatially localized,
independent components, fixed in
number.
The remarkable fact: Very
few micro-properties of the
gas enter into the law. Only n
and k.
Boltzmann’s constant k
= ideal gas constant R
/Loschmidt’s number N
k = R/N
34
From micro to macro…
An ideal gas in a gravitational field.
h
Micro-structure
A dilute gas is a fixed, large number of
component molecules that do not interact with
each other.
Because of their thermal energy at temperature T,
the molecules spread to fill the volume V,
exerting a pressure
We can infer the ideal gas law
from the micro-structure.
P on the vessel walls.
Observed property
Ideal gas law
PV = nkT
35
…and back
An ideal gas in a gravitational field.
Micro-structure
A dilute gas is a fixed, large number of
component molecules that do not interact with
each other.
h
The ideal gas law
is the macroscopic
signature of…
…And we can invert the inference.
Observed property
Ideal gas law
PV = nkT
36
A much simpler derivation
Very many,
independent, small
particles at
equilibrium in a
gravitational field.
Pull of gravity
equilibrated by
pressure P.
Independence expressed:
energy E(h) of each particle
is a function of height h only.
37
A much simpler derivation
Boltzmann
distribution of
energies
Probability of one molecule at height h
P(h) = const. exp(-E(h)/kT)
Density of gas at height h
r = r0 exp(-E(h)/kT)
Density gradient due to gravitational field
dr/dh = -1/kT (dE/dh) r = 1/kT f = 1/kT dP/dh
where f = - (dE/dh) r is the gravitational force
density, which is balanced by a pressure gradient P
for which f = dP/dh.
Rearrange
Ideal gas
law
So that
Reverse inference
possible, but
messy. Easier with
Einstein’s 1905
derivation.
d/dh(P - rkT) = 0
P = rkT
PV = nkT
since r= n/V
38
“Light
quantum/photoelectric
effect paper”
"On a heuristic viewpoint concerning the
production and transformation of light."
39
Einstein’s astonishing idea
The light quantum hypothesis
“Monochromatic radiation of low density (within
the range of validity of Wien’s radiation formula)
behaves thermodynamically as if it consisted of
mutually independent energy quanta of magnitude
[h].”
i.e. Electromagnetic radiation does not always
behave like waves, contrary to the most
successful science of the nineteenth century.
Sometimes it behaves like little lumps of
energy of size h.
Einstein’s reasons:
Certain experimental effects are only
plausibly explained by light consisting
of quanta: especially the photoelectric
effect.
There is an atomistic signature
in the thermodynamics of high
frequency radiation.
40
Why the atomistic signature of radiation is hard to
see
Atomistic signature of systems with
• fixed, large number
High frequency radiation is a system
with
• spatially localized
• independent
components is the ideal gas law.
• variable, large number
Constant temperature expansion of
an ideal gas. The number of
molecules is constant and the
pressure drops, as the ideal gas law
demands.
Constant temperature expansion of
high frequency radiation. The
number of quanta increases and the
pressure stays constant.
• spatially localized
• independent
components, so the ideal gas law is hard to see.
41
Einstein noticed a new signature of independent
atoms…
An ideal gas can spontaneously compress to half its volume (with very small
probability).
Probability one molecule
is on the left is (1/2).
Probability all n
molecules are on the left
n
is (1/2) .
Entropy S =
- k log (probability)
Entropy difference
DS = - kn log 2
The signature of
independent atoms:
entropy depends on the
logarithm of volume
ratios.
42
…which he found in high frequency radiation
Spontaneous volume fluctuations in high frequency radiation are also possible.
Radiation of energy E may spontaneously halve in volume with very small probability.
Probability of spontaneous
compression is (1/2)E/h
Just as is the radiation consisted
of n = E/h spatially localized,
independent quanta of energy.
Entropy difference
DS = - k (E/h) log 2
log (probability)
= - Entropy s/k
Determined from
measured
properties of
radiation.
43
Conclusion
44
Unity in Einstein’s statistical physics of
1905
Einstein investigated many different thermal systems in 1905.
What made these investigations tractable was a common feature:
Each consisted of a large number of spatially localized component that do not
interact with each other. And so:
Ideal
gas
is
just
like
Dilute
sugar
solution
is
just
like
Particles
in
suspension
is
just
like
High
frequency
radiation
45