Lecture 7 The Second Law of Thermodynamics Announcements Entropy Probability • Today: Give out Homework 4 Temperature Entropy S = k log W Due MONDAY Sept. 29 • Wed., Sept. 24: Start Electricity and Magnetism Order Disorder Homework 3 due 1 Atoms? / 2 m v 2 • Mon., Sept 29: Review before Exam I Irreversible Homework 4 due l e i b r s v e R e • Wed., Oct. 1: Exam I Covers material through the Review Chapters 1 – 5, 7 of March; Ch. 11-2 of Lightman “Of all the difficult concepts of classical physics, the most difficult is entropy” Korean physics student’s paper on the web Introduction Time period – starting around 1700 • Last Time: Conservation Laws • The most useful conclusions without solving equations! • Newton’s Laws show how to describe the motion • Conservation of momentum: Follows from Newton’s third law. of every object (Chapt. 2 in March) • Force of gravity obeys simple law • Conservation of energy: The most important and useful law. (Chapt. 5 in March, Chapter 1 of Lightman) • Electromagnetic forces not understood in 1700, but one its way to being described by simple laws • MORE important than Newton’s Equations! - still valid in modern physics even though Newton’s laws are not ! • Deterministic mechanical world view! • Heat – internal energy – important in establishing the law as • Entire universe is like a clock “The first law of thermodynamics” • In eternal motion – keeps ticking with every detail for all times determined by the state of the universe at one time • Today: The 2 nd Law of Thermodynamics & Entropy • Complete change from Pre-Copernican view • The second law states that entropy always increases! • Mechanical and deterministic instead of spiritual and mystical • What is entropy? (Chapt. 2 in Lightman) • Why is it so important in philospohical arguments? From the nature of time to evolution to …… William Thomson (Lord Kelvin), Development of Classical Physics 1824 - 1907 Middle “Modern” Asia, Egypt Greece, Rome Renaissance Ages Physics Mesopotamia • Conceptual ideas that lead to the second law Copernicus Al - K h awarizmi Aristotle 0 - 1 000 1000 2000 • Defined “absolute zero of temperature” and what Ptolomy Euclid Boltzmann is now known as the Kelvin scale of temperature Galileo Newton Carnot Kelvin • The key understanding of thermodynamics that 1700 1600 1800 1900 Mayer Calculus tell us what we cannot do – maximum possible Kepler • Newton’s achievements define classical physics efficiency of engines, refrigerators, …. • The fundamental underpinnnings of the inventions and changes of the industrial revolution • A gentleman that championed the work of others • The deterministic world view of the “Modern History” starting around 1600 (unlike Newton) • But something is amiss - - - - - - - - 1
Lecture 7 Ludwig Boltzmann, 1844 - 1906 Ludwig Boltzmann, 1844 - 1906 • His equation is engraved • Boltzmannn developed statistical mechanics, on his tombstone: which describes how atomic properties decide the perceptible properties of matter S = k log W • Boltzmann's equation, which expresses entropy in terms of probability Entropy • Especially interested in the second law of Number of configurations thermodynamics • At the center of philosophical debates: Boltzmann’s Constant • Do atoms exist – even though no experiments could detect them directly at the time • Committed suicide in 1906 • His equation was engraved on his headstone: http://www.math.umbc.edu/ ~gobbert/Boltzmanntomb.html Statements fo the Second Law Demonstrations • Many equivalent forms – Lightman p. 101 • In an ISOLATED system: • Pendulum • The system naturally evolves toward more probable configurations • The system evolves toward distributing its total energy equally among all its parts (conserving energy of course) v v • Heat flows from hotter to colder bodies • The system evolves toward decreasing order • Dye dropped into water • The system evolves toward increasing entropy • The system’s ability to covert work into heat is always diminishing • Conclusions: • The universe is winding down – heading toward “thermal death” • Bridge collapsing • The universe had a beginning • The direction of time is determined by this inevitable, irreversible tendency Heat Heat flow • Heat is a form of energy – internal energy of a • Heat always flows from hotter to colder bodies material made up of atoms in motion (Atoms? More about them later) Hotter Colder • Much more PROBABLE to have the faster and Hotter Colder slow molecules spread uniformly than the UNLIKELY situation of the fast ones on the left and • Heat always flows from hotter to colder bodies slow ones on the right (like the dye in water) • Two bodies in isolation tend to come to the same temperature • Observed for thousands of years – given precise definition with heat as a form of energy Equal Temperature 2
Lecture 7 The Pendulum – in the air: Probabilty • Examples in Lightman, Ch. 2 • Casting dice • My example: Flipping a N coins Each can be head (+) or tail (-) • One way to have all heads + + + + + + + + + + + + N ways to have one tail - + + + + + + + + + + + Probability + - + + + + + + + + + + v etc. N(N-1) way to have two tails - + + + + + + + + + + - + - + + + + + + + + + - • Air molecules bouncing off the ball tend to transfer etc. energy from the ball to the MANY molecules Most Probable – equal heads and tails N heads + - + - - + + + - - + - • More PROBABLE for the energy to be distributed + - + - + - + + - - + - N/2 among many objects that concentrated in one etc. The Pendulum again The Pendulum continued v v • Because there are MANY molecules ~ 10 26 in the • Loss of energy from the pendulum to the air is a room DECREASE in order of the pendulum-air system • It unimaginably more PROBABLE for the energy be • The swinging pendulum was a very special state lost from the pendulum to the air than not that evolves toward more typical configurations Conclusions - I How can we get work from heat? • Related to statements of 2 nd law: • Simple example: (Lightman – p. 52-53) • In an ISOLATED system: • The system naturally evolves toward more probable Work = mgh configurations m • The system evolves toward distributing its total energy equally among all its parts (conserving energy of course) h m Add Heat H • Heat flows from hotter to colder bodies • The system evolves toward decreasing order • The system evolves toward increasing entropy (defined to be a measure of disorder – more later) T 1 T 2 > T 1 • What about: • Part of H goes into work, part into heating the gas • The system’s ability to covert work into heat is always diminishing • See Homework problem • Physics error in Lightman! Extra credit to analyze the error in his example. 3
Lecture 7 Heat engine Heat Engine - Schematic • Example - steam engine – see Lightman – p. 92 • Universal diagram – Lightman, p 93 Remove mass m m T 2 > T 1 Heat Work Heat Heat in out h out m m H 2 H in W H out Work W Heat T 2 ’ > T 1 T 1 T 2 ’ > T 2 > T 1 H 1 T 1 Place new mass m on piston T 1 • 1 st Law: W = H 2 – H 1 • Engine cycles back to starting configuration • Efficiency defined to be e = W/H 2 • Work is done and heat is transferred through the engine • Engines can be run forward (work output) or • Heat input is at a higher temperature than output backward (work input - like a refrigerator) Reversible Heat Engine Maximum Efficiency of a Heat Engine • An ideal engine that works perfectly – no losses • Suppose there were a more efficient engine: T 2 > T 1 T 2 > T 1 T 2 > T 1 Heat Heat Heat H 2 H 2 H 2 * Work W Work W Work W Heat Heat Heat * H 1 H 1 H 1 T 1 T 1 T 1 • Since there are no losses, use 1 st Law: W = H 2 – H 1 • If there were a more efficient engine, it could be run in reverse (right side) to have a net transfer of • Efficiency e = W/H 2 = 1 - H 1 /H 2 heat from the colder body to the hotter body! • Clausius showed H 1 /T 1 = H 2 /T 2 → e = 1- T 1 /T 2 • Maximum possible efficiency: e = 1- T 1 /T 2 Conclusions - II Can order be created • Related to statements of 2 nd law: • In a system that is NOT ISOLATED, order can increase • In an ISOLATED system: • There can be input of energy at a high temperature • The system’s ability to covert work into heat is always diminishing and output at a low temperature • What about the earth? • As temperature becomes more uniform, one cannot extract work – useful energy! • What is the external source of energy to the earth? • A guess about it temperature? • Maximum possible efficiency = e = 1- T 1 /T 2 • Since the earth does not get hotter, energy must be leaving the earth at the same rate. • To where is the energy going? • A guess about it temperature? • Maximum efficiency of earthlings? 4
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