熱力學
陳宣毅
中央大學物理系 中央研究院物理所 國家理論科學中心
1
Outline
大綱
• Prelude, 序幕
• What is heat? 什麼是熱?
• Thermodynamics and time’s arrow. 熱力學與時間之箭
• Thermodynamics and atomic hypothesis. 熱力學與原子假說
• Thermodynamics and 21th century science. 熱力學與二十一
世紀的科學
2
References
參考資料
• 網際網路
• Wikipedia 維基百科
• The 2nd law, by Atkins. (有許多圖)
• Middle world, the restless heart of matter and
life, by Mark Haw (有許多故事)
3
Prelude
序幕
4
What is thermodynamics?
什麼是熱力學?
From Wikipedia, the free encyclopedia
• Thermodynamics is a physical science that studies the
effects on material bodies, and on radiation in regions of
space, of transfer of heat and of work done on or by the
bodies or radiation. It interrelates macroscopic variables,
such as temperature, volume and pressure, which describe
physical properties of material bodies and radiation, which
in this science are called thermodynamic systems.
什麼意思??
5
What is thermodynamics?
什麼是熱力學? (19th century)
• Historically, thermodynamics
developed out of a desire to
increase the efficiency of early
steam engines, particularly
through the work of Nicolas
Léonard Sadi Carnot (1824).
Scottish physicist Lord Kelvin was the first to formulate a concise
definition of thermodynamics in 1854:
Thermodynamics is the subject of the relation of heat to forces
acting between contiguous parts of bodies, and the relation of heat
to electrical agency
6
What is thermodynamics?
什麼是熱力學?
1.
pV = nRT
1. F = ma
2.
Q = mcT
2. F = mg
討論: 熱力學與力學有什麼不同
7
熱力學是什麼?
• 研究具有許多粒子(N>>1)的系統
8
What does thermodynamics tell us?
熱力學告訴我們什麼?
• What is heat.
熱是什麼
• The direction of time’s arrow
時間的方向
• More is different
量變造成質變
9
In thermodynamics, there are 4 laws
熱力學有四個定律
• Zeroth law: equilibrium and temperature
熱平衡與溫度
• 1st law: what is heat?
熱是什麼?
• 2nd law: time’s arrow
時間的方向
• 3rd law: low temperature limit
低溫的極限
10
1st law: what is heat?
第一定律: 什麼是熱
11
Heat as a fluid
熱的`卡路里說’
From Wikipedia:
The initial explanations of heat were thoroughly confused with
explanations of combustion. In the 17th century, phlogiston (燃素) was
thought to be the substance of heat.
In 1783 Lavoisier (拉瓦謝) argued that phlogiston theory was
inconsistent with his experimental results, and proposed a 'subtle fluid'
called caloric as the substance of heat.
According to this theory, the quantity of this substance is constant
throughout the universe, and it flows from warmer to colder bodies.
Conservation of heat was a central assumption
12
Joule’s experiment 焦耳的實驗 (1843)
Mgh = mcT
James Joule (1818-1889)
如果你相信能量守恆:
力學能可以轉變為熱能，提高系統的溫度
13
What is heat? 什麼是熱?
• Heat is a form of energy. 熱是一種能量(很抽象)
(now it is meaningful to say: “absorb heat” or “release heat”. )
• Heat goes spontaneously from high T region to low T region.
熱能會自發地由高溫的區域流向低溫的區域 (為什麼?)
• A definition in terms of mathematics is needed!
如何以數學的形式定義熱?
• Energy is …… E = mv2/2 + V(x)
Heat is …… (can you find Q=……?)
Actually it is only possible to write “Q=……”, (heat is a form of energy transfer)
我們只能定義熱能的改變量(熱是一種能量轉移的方式)
14
Heat and the 1st law of thermodynamics
熱與熱力學第一定律
• Work done by conservative force:
守恆力做功 (理想系統)
W = E, (E: energy of the system)
• In the presence of friction
當摩擦力也做功 (真實系統)
W = E  Q,
(Q: heat absorbed by the system 系統吸收的熱)
• Clausius, 1850
W = E  Q 1st law of thermodynamics
外界對系統作的功等於系統內能增加量減去系統所吸
收的熱
15
The implications of 1st law
第一定律的意義
E = W + Q
1. Heat is a form of energy transfer.
熱是能量轉移的一種形式
2. Only energy-conserving process can occur.
只有能量守恆的過程才可能發生
3. Heat can be used to perform work. (heat engine)
熱可以用來作功
16
How does “heat” look like?
如何將熱“圖像化”?
Hint: 在固定壓力下:
low T
mid T
high T
討論:
1.如何解釋什麼是熱
2.熱與溫度
3.熱、溫度、與秩序
17
2nd law: thermodynamics and time’s arrow
第二定律: 熱力學與時間之箭
18
Time’s arrow
•Why do we get older but never younger?
為何時光一去不復返?
•The answer is hidden in heat engines.
解答藏在熱機(引擎)中
19
Heat engine
熱機(引擎)
TH
Q1
W
• Heat engine::
extracts heat Q1 from reservoir TH,
releases heat Q2 to reservoir TC,
performs work W = Q1 Q2.
從高溫處吸熱，將熱轉為功，多餘的
熱傳至低溫處。
• Efficiency = W/Q1 = 1 – Q2/Q1
Q2
TC
• Can eventually efficiency  1?
效能是否可以趨近100%?
20
Efficiency of a heat engine
熱機的效能
TH
Nicolas Léonard Sadi Carnot
(1796-1832),
mathematician and one of the
pioneers of thermodynamics.
Q1
W
• Efficiency = W/Q1 = 1 – Q2/Q1
Q2
• Carnot: efficiency  1 TC/TH
TC
• In principle there is no way to
transform heat 100% into work!
21
Kelvin statement of the 2nd law of thermodynamics
克爾文對熱力學第二定律的敘述
TH
Q1
W
Q2=0
TC
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.
熱無法完全轉變為功
22
Clausius statement of the 2nd law of thermodynamics
克勞修斯對熱力學第二定律的敘述
TH
Q
TC
Rudolf Clausius (1822-1888): heat cannot of itself pass from a
colder to a hotter body.
熱不會自動由低溫處流向高溫處
23
2nd law and molecular motion
第二定律與分子運動
•可以從分子運動了解第二定律
•第二定律有例外嗎?
•甚麼是熱力學?
24
The meaning of 2nd law
第二定律的意義
• Energy is conserved in all processes, but not all energy conserving processes
can happen!
所有發生的過程都滿足能量守恆，但不是所有能量守恆的過程都會發生。
• 100% heat  work (process A) cannot occur,
but 100% work  heat (time reversal of A) can occur.
• 2nd law (as a part of thermodynamics) refers to macroscopic processes only.
第二定律只適用於巨觀系統
• Some macroscopic processes are not reversible! (Time’s arrow exists.)
有一些巨觀過程是不可逆的 (時間有一定的方向)
25
Microscopic reversibility
微觀物理的可逆性
• F = m a; F = F(x); a = d2x/dt2
• Time reversal: t  t, x  x
• v  v
• F  F, a  a
• F = m a is valid under time reversal.
There is no time’s arrow in
microscopic physics.
牛頓運動定律(基本力學定律)在時
間反演下仍然成立
26
Microscopic reversible, macroscopic irreversible?
微觀物理定律可逆，巨觀物理定律不可逆?
http://www.youtube.com/watch?v=Rxno7tTUI4c
TH
Q
Raindrop splash and displacement of soil particles.
(USDA Natural Resources Conservation Service)
TC
27
Microscopic illustration of 2nd law
從分子運動看第二定律
• http://comp.uark.edu/~jgeabana/mol_dyn/KinThI.html
• Reverse the movie, you don’t see original lattice……
倒放電影，無法回到原始的晶格
• A small change in the initial condition (due to precision of
the computer) is amplified.
由於電腦精確度的限制，倒放電影的初始態有微小的誤
差，此誤差被放大所以無法回到原始的晶格。
• The initial condition (lattice) is an unlikely arrangement,
therefore you never see it again!
原始的晶格是極難達成的排列，所以你無法在倒放的電
影看見它
28
How to describe irreversibility quantitatively?
如何定量描述巨觀不可逆現象?
Example: mixing sucrose with water
將糖溶於水
Time’s arrow: maximize “number of ways” to arrange the molecules.
時間之箭指向增加分子排列方式的方向
29
“Number of ways” to arrange the molecules
“分子排列的方式”
Two ways to describe the physical state of a system.
兩種描述一系統物理狀態的方法
1.
Mechanical state (microscopic state): momentum and position of all
particles: (r1, r2,… ,rN; p1, p2,… ,pN), (6N variables).
力學狀態: 描述所有粒子的位置與動量(速度)，共需6N個數字
2.
Thermodynamic state (macroscopic state): E,T,V,p,N,…. (much less
than 6N variables)
熱力學狀態: 描述內能、溫度、體積、壓力、粒子數等，需要用
到遠小於6N個數字

A macroscopic state corresponds to many microscopic states.
W(A): number of microscopic states for a macroscopic state
A(T,V,N,…).
一個熱力學狀態對應於許多力學狀態, W(A)>>1.

30
Boltzmann’s idea
Ludwig Eduard Boltzmann (1844-1906)
S: entropy 熵 S = k logW
31
Aside: what is log?
甚麼是對數?
• If x = ay,
then logax = y
• If the population doubles every unit time, then
n(t) = n0 2t
• If the growth rate of a population with n individuals is bn,
then n(t)=n0 ebt,
e = limx0(1+x)1/x=2.718…..
loge x= log x
32
Entropy 熵
W: 一個熱力學狀態所對應
的力學狀態數
eS/k = W
33
Clausius and entropy
克勞修斯與熵
• In mechanics 力學
V/x = F when there are only conservative forces
V/x < F when there are nonconservative forces
We define potential energy V (only depend on x).
• Clausius found (before Boltzmann) 在玻茲曼之前，克勞
修斯就發現了
TS = Q for reversible processes
TS > Q for irreversible processes
We define entropy S (only depends on macroscopic state of
the system).
熵是狀態函數
34
“S” statement of 2nd law
第二定律的“熵”敘述
S = klnW, TS  Q
• Entropy of a thermally isolated system cannot decreases.
獨立系統的熵必須增加或不變
• Microscopically, time’s arrow: maximize “number of ways”
to arrange the molecules.
微觀: 時間之箭指向增加分子排列方式的方向
• Macroscopically, time’s arrow in the universe points
toward where entropy of the universe increases.
巨觀:時間之箭指向宇宙總熵增加的方向
35
Thermodynamics and the origin of the universe
熱力學與宇宙的起源
• It currently seems that the ultimate reason for a preferred
time direction is that the universe as a whole was in a
highly ordered state at its very early stages, shortly after
the big bang, and that any fluctuations in it were
uncorrelated. The question of why this highly ordered state
existed, and how to describe it, remains an area of research.
http://en.wikipedia.org/wiki/Entropy_%28arrow_of_time%29
36
第二定律與開放系統
老掉牙的陷阱:
•熵永遠增加嗎?
•水凝結成冰時 水的熵有增加嗎?
新問題: 開放系統的時間之箭指向何處?
37
定溫系統的時間之箭
U = W+Q (1st law)
Q  TS (Clausius)
 U-TS  W
Under constant T, Q  (TS),
 (U-TS)  W
If you leave a constant T system alone, W=0,
 (U-TS)  0
F  U-TS (free energy 自由能)cannot increase in a constant T
system.
38
相變與自由能
F = U-TS
low T
mid T
high T
39
“降低能量”
• 我們常聽到:”一個系統喜歡往能量較低的
狀態演化”
• 到底是什麼意思??
40
Thermodynamics and atomic hypothesis
熱力學與原子假說
41
Atomic hypothesis
原子假說
• 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.
42
When you think about atomic scale…
當我們想到原子尺度時……
• Looking for basic physical laws at small scale 基本物理定律
 quantum physics, elementary particles. 量子力學與基本粒子
• Connecting macroscopic phenomena with dynamics at atomic
scale 從原分子的運動來了解巨觀現象
 thermodynamics and heat. 熱力學
43
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
44
What did people think about atomic hypothesis?
• Newton (1642-1727): accept it with no exp evidence.
• Boscovich (1711-1787): 1st one who attempted to construct
inter-atom forces, and describing natural phenomena from
this constuction.
• Immanuel Kant (1724-1804): atoms, if they exist at all, are
merely centers of force and have no separate existence!
45
What made people accept atomic hypothesis?
大家如何接受原子假說?
• 什麼現象? 布朗運動
• 什麼理論? 愛因思坦
• 什麼物理量 ? 亞佛加厥數
46
Robert Brown and Brownian motion
布朗運動與布朗
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.
47
Brownian motion
布朗運動
plastic spheres, each 913 nm in diameter
http://physics.ius.edu/~kyle/K/Brownian/Brownian.html
48
Early theories of Brownian motion
早期的理論解釋
• 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?
49
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
50
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
51
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
Jean B. Perrin
實驗:量T, g, D,得NA
Nobel Prize for physics: 1926
Perrin: NA = 7×1023
52
Einstein relation
• Drag force: f = gv 黏滯力
• Diffusion due to random walk: d2 = 6Dt
隨機行走造成的擴散現象
• To reach equilibrium: Dg = kT 熱平衡
• Random collisions (random walks) are related to the
dissipation of kinetic energy to solvent molecules.
物體與溶劑分子的隨機碰撞與物體在溶劑裏運動時受到
的黏滯力有直接的關聯
53
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.
54
Thermodynamics and the 21th
century science
熱力學與二十一世紀的科學
55
Motor proteins
蛋白質馬達
56
Brownian motor: a nanomachine
布朗馬達: 一個奈米機器
Motor: +2
ATP: -2
1. Motor+2+ATP-2  MotorATP
2. MotorATP  Motor+2 + ADP+P-
57
Physics of Brownian motors
布朗馬達的物理
1. Symmetry
2. Thermal equilibrium
3. Time scale: on-off time b2 > Dt > a2
58
A molecular motor at work
工作中的分子馬達
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
59
“Nano-robotics”: 大腸菌
http://en.wikipedia.org/wiki/Escherichia_coli
60
Molecular motors drive E. coli swim
推動大腸菌的分子馬達
61
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
62
New development in the 21th century
63
Epilogue: “more is different”
結語: 量變造成質變
• Heat is a form of energy (transform).
• 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.
64