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 = mcT

3. L = T

• Thermodynamics is closer to answer the following question: What makes 1-3 true?

http://en.wikipedia.org/wiki/Thermodynamics

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= mcTWork 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?

http://www.nccr-nano.org/nccr/media/nanonews/nanonews_05/highlights/highlight_13

•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.

The implications of 1st lawE = 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 efficiencyCarnot family

• From Wikipedia, Carnot is the name of a celebrated French

family in politics and science with the following members:

• Lazare Nicolas Marguerite Carnot (1753-1823), 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

Nicolas Léonard Sadi Carnot (1796-1832)

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

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.

Rudolf Clausius (1822-1888): heat cannot of itself pass from a colder to a hotter body.

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

Irreversible:S > dQ/T

S = S(B)S(A)is independent of path.

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 = m a

• 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

• 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 !

http://webphysics.davidson.edu/physlet_resources/bu_semester1/c27_process_expansion_sim.html

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 = T0

• 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 hypothesisFrom 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). ]

Major contribution of Brown: made sure non-organic particles also have Brownian motion, confirmed that Brownian motion is not a manifestation of life.Robert Brown

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 t1/2 ?

• 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>+<x2

2>+<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>=<x1

2>+<x22>+<x3

2>=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 number1. 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 =

v, : 阻泥係數3. [D] = L2/T; []=[f]/[v]=M/T4. [D] = 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 D= kT =

RT/NA

(iv) 亞佛加厥數 NA=RT/D實實實 TD, 得 NA

Perrin: NA = 7×1023

Jean B. PerrinNobel Prize for physics: 1926

Einstein relation

• Drag force: f = v

• Diffusion due to random walk: d2 = 6Dt

• To reach equilibrium: D= kT

• Random collisions (random walk) are related to the dissipation of kinetic energy to solvent molecules.

Atomic picture of thermal equilibrium

http://www.unmuseum.org/einstein.htm

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

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 = EQuestion: 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

H. Noji, R. Yasuda, M. Yoshida, K. Kinoshita Jr, Nature, 386, 299 (1997)

http://www.k2.phys.waseda.ac.jp/F1movies/F1long.htm

ATPase: 製造 ATP的蛋白質

“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.