熱力學
陳宣毅
中央大學物理系
大綱
• Prelude
• What is heat?
• Thermodynamics and time’s arrow.
• Thermodynamics and atomic hypothesis.
• Thermodynamics and 21th century science.
References
• The internet……(copy and paste!)
• Wikipedia (most of the time very reliable)
• The 2nd law, by Atkins. (good book for fun)
Prelude
What is NOT thermodynamics?
1. pV = nRT
2. Q = mcT
3. L = aT
•
http://en.wikipedia.org/wiki/Thermodynamics
Thermodynamics is closer to answer the
following question: What makes 1-3 true?
The basis of thermodynamics:
• Feynman lectures on physics, Vol 1, Ch.1:
If, in some cataclysm, all of scientific
knowledge were to be destroyed, and only
one sentence passed on to the next
generations of creatures…… I believe it is
the atomic hypothesis…… all things are
made of atoms – little particles that move
around in perpetual motion, attracting
each other when they are a little distance
apart, but repelling upon being squeezed
into one another.
When you think about atomic scale…
• Looking for basic physical laws at small
scale  quantum physics, elementary
particles.
• Connecting macroscopic phenomena with
dynamics at atomic scale 
thermodynamics and heat .
What is heat?
• From Wikipedia, the free encyclopedia
In physics, heat, symbolized by Q, is defined as a
form of energy whose absorption raises the
temperature of a body, not existing in the
transition state, and abstraction of which from the
same body lowers its temperature. Generally, heat
is a form of energy transfer, sometimes called
thermal energy, associated with the different
motions of atoms, molecules and other particles
that comprise matter when it is hot and when it is
cold.
Heat: a form of energy or a form of energy transfer?
What is heat?
• Heat is a form of energy. (Well…… now it is
meaningful to say: “absorb heat” or “release heat”. )
• Heat goes from high T region to low T region
spontaneously.
• Heat = irregular motion of atoms. (we will see more
about that……)
• A definition in terms of mathematics is needed!
• Energy is …… E = mv2/2 + V(x)
Heat is …… (can you find Q=……?)
• “Q=……”, (heat is a form of energy transfer)
Heat is a form of energy:
Joule’s experiment
James Joule (1818-1889)
Mgh = W = Q
and
Q= mcT
Work can raise temperature
of the system.
Heat
and
the 1st law of thermodynamics
• Mechanics
W = E, (E: energy of the system)
• In the presence of friction
W = E  Q,
(Q: heat absorbed by the system)
• In general
W = E  Q 1st law of thermodynamics
(heat comes from friction, heat conduction, etc)
How does “heat energy” look like?
•Atoms in the lattice are dragged by the
tip.
•The lattice begin to vibrate randomly.
•Work done by external force is
transformed into this random vibration.
•Heat = irregular motion at molecular
scale.
http://www.nccr-nano.org/nccr/media/nanonews/nanonews_05/highlights/highlight_13
The implications of 1st law
E = W + Q
1. Heat is a form of energy.
2. Only energy-conserving process can occur.
3. Heat can be used to perform work. (heat
engine)
Thermodynamics and time’s arrow
Carnot’s engine of highest possible efficiency
Carnot family
Nicolas Léonard Sadi Carnot
(1796-1832)
• From Wikipedia,
Carnot is the name of a celebrated French
family in politics and science with the
following members:
• Lazare Nicolas Marguerite Carnot (17531823), mathematician and politician.
• Nicolas Léonard Sadi Carnot (1796-1832),
mathematician and eldest son of Lazare, one
of the pioneers of thermodynamics.
• Hippolyte Carnot (1801-1888), politician and
second son of Lazare.
• Marie François Sadi Carnot (1837-1894),
son of Hippolyte, President of France, 1887–
1894.
• Marie Adolphe Carnot (1839-1920), son of
Hippolyte, mining engineer and chemist.
• A number of lycées, streets etc. are named
after this family throughout France
Carnot’s engine of highest possible efficiency
http://www.grc.nasa.gov/WWW/K-12/airplane/carnot.html
Efficiency of a heat engine
• Heat engine: extracts heat Q1 from reservoir TH,
releases heat Q2 to reservoir TC, performs work
W = Q1 Q2.
• Efficiency = W/Q1 = 1 – Q2/Q1
• Carnot: efficiency  1 TC/TH
• What? There is no way to transform heat 100%
into work!
Clausius and Kelvin statements of the 2nd
low of thermodynamics
Rudolf Clausius (18221888): heat cannot of
itself pass from a colder to
a hotter body.
Lord Kelvin(1824-1907): a transformation
whose only final result is to convert heat,
extracted from a source at constant
temperature, into work, is impossible.
The meaning of 2nd law
• Energy is conserved in all processes, but not all
energy conserving processes can happen!
• 100% heat  work (process A) cannot occur,
but
100% work  heat (time reversal of A) can occur
• 2nd law refers to macroscopic processes only.
• Some macroscopic processes are not reversible!
Clausius and entropy
•
Clausius:
 dQ/T  S, S: entropy
1.
For reversible processes:
S =  dQ/T
2. For thermally isolated
systems
S  0.
S > 0 if T1 > T2
Psychological barrier
S =  dQ/T for reversible processes
only……????
Q: How can you define entropy S in this
strange way???
but
V =  F·dx for conservative forces only……
We define potential energy is this strange way!
Entropy as a state function
• State function: E, V, p, T,….
Given the physical state of the system, a state function is
uniquely determined.
• Entropy is a state function:
Reversible:
S =  dQ/T
A
B
S = S(B)S(A)
is independent of path.
Irreversible:
S >  dQ/T
A lazy statement of 2nd law
• Entropy of a thermally isolated system cannot
decreases.
• Macroscopically, time’s arrow in the universe
points toward where entropy of the universe
increases.
• Can we see this “time’s arrow” from fundamental
physical laws??
Question: microscopic reversibility
• F=ma
• F = F(x,v·B); a = d2x/dt2
• Time reversal: t  t, x  x
• v  v, B  B
• F  F, a  a
• F = m a is valid under time reversal.
There is no time’s arrow in microscopic
physics.
Microscopic reversible,
macroscopic irreversible?
Raindrop splash and displacement of soil particles.
Source: USDA Natural Resources Conservation Service.
Microscopic illustration of 2nd law
• http://comp.uark.edu/~jgeabana/mol_dyn/KinThI.html
• Reverse the movie, you don’t see original lattice……
• A small change in the initial condition (due to precision of
the computer) is amplified in the future.
• The initial condition is an unlikely arrangement, therefore
you never see it again!
How to describe irreversibility quantitatively?
Example: mixing sucrose with water
Time’s arrow: Maximize “number of ways” to arrange the molecules.
“Number of ways” to arrange the
molecules in a system
Two ways to describe the physical state of a system.
1.
Mechanical state (microscopic state): momentum and
position of all particles: (r1, r2,… ,rN; p1, p2,… ,pN),
(6N variables).
2.
Thermodynamic state (macroscopic state): E,T,V,p,N,….
(much less than 6N variables)
 A macroscopic state corresponds to many microscopic
states.
W(A): number of microscopic states for a macroscopic
state A(T,V,N,…).
Boltzmann’s
idea
Ludwig Eduard Boltzmann (1844-1906)
S = k lnW
Clausius:  dQ/T  S; Boltzmann: S = k lnW?
• Example: free expansion of ideal gas
http://webphysics.davidson.edu/physlet_resources/bu_semester1/c27_process_expansion_sim.html
• E = (3/2)NkT, Q=0, W=0  E=0
• T does not change.
• S = S(T,V+V)  S(T,V) =  dQ/T
(find a reversible process to go from initial state to final state)
 Constant temperature expansion !
Entropy change in ideal gas free expansion
• S = S(T,V+V)  S(T,V) =  dQ/T of a constant temperature
expansion.
• Constant temperature expansion: E = (3/2)NkT
E=0  Q = W =  pdV = NkT  dV/V
 According to Clausius, S = Q/T = Nk ln[(V+ V)/V]
• Does this agree with Boltzmann’s S = k lnW?
W  VN  S = k ln[W(V+V)/W(V)] = Nk ln [(V+ V)/V]
Third law: cold world
• S = k lnW
• Lower temperature  release heat from the system
• Lowest energy = T0
• Lowest energy microscopic state: there is only one such
state! (sometimes not one, but just a few)
• S(T=0) = k ln1 = 0. 3rd law of thermodynamics.
Thermodynamics and atomic hypothesis
The importance of atomic hypothesis
From
http://physicalworld.org/restless_universe/html/ru_bolt.html
Boltzmann's contribution was vital, but had a tragic outcome.
Towards the end of the nineteenth century several puzzling
facts (which eventually led to quantum theory), triggered a
reaction against 'materialist' science, and some people even
questioned whether atoms exist. Boltzmann, whose work
was based on the concept of atoms, found himself cast as
their chief defender and the debates became increasingly
bitter. Always prone to bouts of depression, Boltzmann
came to believe that his life's work had been rejected by
the scientific community, although this was far from being
true. In 1906, he committed suicide. If despair over
rejection, or frustration over being unable to prove his
point, were contributing factors the irony would be great
indeed. Soon after Boltzmann's death, clinching evidence
was found for atoms, and few would ever doubt their
existence again
Brownian Motion and thermodynamics
400x, plastic spheres, each 913 nm in diameter
http://physics.ius.edu/~kyle/K/Brownian/Brownian.html
Robert Brown and Brownian motion
Brown (1827): observed irregular
movement of pollens in water under
microscope.
[First observation of “Brownian
motion”: S. Gray, Phil. Trans. 19,
280, (1696). ]
Robert Brown
Major contribution of Brown: made
sure non-organic particles also have
Brownian motion, confirmed that
Brownian motion is not a
manifestation of life.
Early theories of Brownian motion
• From energy of light in the microscope?
• Surface tension effect?
• (1889) temperature difference between the solution
and environment?
• Puzzle: average speed of Brownian particles V  t1/2 ?
• Why were scientists unable to explain Brownian
motion?
Einstein, Brownian motion, and
atomic hypothesis
The Miracle year:
Albert Einstein published 4 papers in
the Annalen der Physik in 1905.
– Photoelectric effect
– Brownian motion
– Special theory of relativity
Which topic is his PhD thesis?
Albert Einstein, 1905
Einstein's theory of Brownian motion
•
沒有外場的布朗運動: v ~ t-1/2
即: 運動距離 d(t) ~ t1/2
愛因斯坦:
1. 花粉在溶液中不斷因水分子的碰撞而改變運動方向。
2. 兩次碰撞(即走三步)後的位移 x=x1+x2+x3
3. 平均位移 <x>=<x1>+<x2>+<x3>=0 因為兩次碰撞間朝任何
方向移動的機率都相同。
4. 但是<x2>=<x12>+<x22>+<x32>
+2<x1x2>+2<x2x3>+2<x3x1>
5. <x1x2>=<x1><x2>=0, <x2x3>=0, <x3x1>=0. 因每一步所走的
方向與其他步無關(獨立事件)。
6. 故<x2>=<x12>+<x22>+<x32>=3 <x12>
7. 走N步: <x2>=N<x12>~t
8. 平均移動的距離 d=(<x2>)1/2~t1/2 , 平均速率 ~ t 1/2
From Brownian motion to Avogadro number
1. 花粉在水中的布朗運動: d2 = 6Dt, D:擴
散係數
2. 花粉在水中的運動受水的黏滯力: f = gv,
g:阻泥係數
3. [D] = L2/T; [g]=[f]/[v]=M/T
4. [Dg] = ML2/T2 = [E]
5. 愛因斯坦:
(i) 水分子撞花粉靜止的花粉因而獲
得能量行布朗運動。
(ii) 花粉運動受黏滯力而將能量傳回水
分子。
(iii) 要達成熱平衡需要Dg = kT = RT/NA
(iv)亞佛加厥數NA=RT/Dg
實驗:量T, g, D,得NA
Jean B. Perrin
Perrin: NA = 7×1023
Nobel Prize for physics: 1926
Einstein relation
• Drag force: f = gv
• Diffusion due to random walk: d2 = 6Dt
• To reach equilibrium: Dg = kT
• Random collisions (random walk) are related
to the dissipation of kinetic energy to solvent
molecules.
Atomic picture of thermal equilibrium
Big ball gets kinetic energy from
small balls from random collisions.
Small balls gets kinetic energy from
big ball from viscous drag (turns
kinetic energy of the big ball into
heat).
Equilibrium: energy from big ball to
small balls = energy from small balls
to big ball
http://www.unmuseum.org/einstein.htm
Equilibrium kinetic energy of the big
ball = equilibrium kinetic energy of a
small ball = (3/2)kT.
Thermodynamics and 21th century
science
Heat conduction (19th century science)
• http://www.gcse.com/energy/conduction.htm
Electric conduction (20th century science)
http://people.deas.harvard.edu/~jones/es154/lectures/lecture_2/drude_model/drude
_model.html
J=sE
Question: s  as T ? as T ?
Entropy of the wire does not increases?
Nanomachines: Brownian motor
(21th century science)
Motor: +2
ATP: -2
1. Motor+2+ATP-2  MotorATP
2. MotorATP  Motor+2 + ADP+P-
Physics of Brownian motors
1. Symmetry
2. Thermal equilibrium
3. Time scale: on-off time b2>Dt>a2
A molecular motor at work
ATPase: 製造ATP的蛋白質
H. Noji, R. Yasuda, M. Yoshida, K. Kinoshita Jr, Nature, 386, 299 (1997)
http://www.k2.phys.waseda.ac.jp/F1movies/F1long.htm
“Nano-robotics”: 大腸菌
http://en.wikipedia.org/wiki/Escherichia_coli
Molecular motors drive E. coli swim
Rotating flagella make E. coli move
Our dream:
Make machines that move like
them!
http://www.rowland.harvard.edu/labs/bacteria/showmovie.php?mov=fluo_cell_near
Thermodynamics and the origin of
the universe
• Unsolved problems in physics:
Arrow of time : Why did the universe have such low
entropy in the past, resulting in the distinction between
past and future and the second law of thermodynamics?
http://en.wikipedia.org/wiki/Entropy_%28arrow_of_time%29
Thermodynamics of the universe
(from Wikipedia)
• The thermodynamics of the universe is dictated by which form of energy
dominates it - relativistic particles which are referred to as radiation, or
non-relativistic particles which are referred to as matter. The former are
particles whose rest mass is zero or negligible compared to their energy,
and therefore move at the speed of light or very close to it; The latter are
particles whose kinetic energy is much lower than their rest mass and
therefore move much slower than the speed of light.
• As the universe expands, both matter and radiation in it become diluted.
However, the universe also cools down, meaning that the average energy
per particle is getting smaller with time. Therefore the radiation becomes
weaker, and dilutes faster than matter.(??) Thus with the expansion of the
universe radiation becomes less dominant than matter. In the very early
universe radiation dictates the rate of deceleration of the universe's
expansion, and the universe is said to be radiation dominated. At later
times, when the average energy per photon is roughly 10 eV and lower,
matter dictates the rate of deceleration and the universe is said to be
matter dominated. The intermediate case is not treated well analytically.
As the expansion of the universe continues, matter dilutes even further
and the cosmological constant becomes dominant, leading to an
acceleration in the universe's expansion.
Epilogue: “more is different”
• Heat is a form of energy.
• Not all energy conserving processes occur
(time’s arrow).
• Biological active motion is supported by
energy source (we are not in thermal
equilibrium).
• “fundamental physical laws” may “change”,
but thermodynamic laws are there.