Second Law Of Thermo. - University Of California, San Diego

Second Law of Thermo.Ken Intriligator’s week 6 lectures, Nov. 4, 2013Engine / refrigerator efficiency andEntropyMonday, November 4, 2013

Engine efficiency goal: Extract heat, and convert it into work.Want maximal work output per heat input. Consider a cyclic engine: same initial andfinal state, so no change in internal energy.First law: net heat in equals net work out. Second law: “Heat can not be convertedentirely into work, with no other systemchange!” Related: “Heat can’t flow, byitself, from colder to hotter body.”Monday, November 4, 2013

EnginesTake heat out of a hot region, do some work,and waste some heat, into colder region.Hot region2nd law: can’tentirely eliminatethe wasted heat!does workMonday, November 4, 2013

Maximum efficiency:TCW2nd law: e Q eR 1 THHWe’ll explain it on thechalkboard shortly.Here W is the work done, which we want to be asbig as possible, per “what we pay for,” which is theheat appearing in the denominator.Efficiency e output work done per heat comingin, from the hot region. As we’ll discuss, the 2ndlaw tells us what the maximum efficiency is.Monday, November 4, 2013

Related“Heat always flows from hotter object into colder, notvice versa.” For example, heat flows out from the warmkids out into the cold ocean. Why? Note that theocean is huge, and contains much more thermal energythan the little kids. As we’ll discuss, this is an equivalentstatement of the 2nd law of thermodynamics to thestatement about maximal engine efficiency.Monday, November 4, 2013

Entropy SA state variable, determined entirely by thestate of the system, not by the history (path)of how the system got in that state. Likeenergy E. Recall that heat Q and work W areNOT state variables, they depend on the path.Entropy is a number (with units of Boltzmann’sconstant k) which measures the disorder ofthe system. If w is the number of microscopicstates with the same macroscopic state variables(e.g. pressure and temperature), thenS kB ln wMonday, November 4, 2013

Entropy, qualitativelySmaller # for S,more ordered.Larger # for S,more disordered.Irreversible: Sbef ore Saf terMonday, November 4, 2013

Arrow of timeNewton’s law, F ma, is symmetric if we change tto -t. Microscopic collisions look the same if wefilm them and then play the movie backwards (asidefrom some subtle, tiny effects at the subatomic level).But if you watch a film of scrambling an egg, it’d be easyto tell if it’s being played forwards or backwards! Why?Large numbers (like Avogadro’s) of particles are muchmore likely to be in disorganized vs organized states,because there are so many more possible disorganizedones. More scrambled vs unscrambled states. This is the.Monday, November 4, 2013

Second law of thermo:Saf ter (universe) Sbef ore (universe)With the equal sign if, and only if the process isReversible. The egg scramble is an exampleof an irreversible process. A reversible one is wherethe system undergoes tiny changes, which could havegone either way by slightly adjusting the conditions.Maximal engine or refrigerator efficiency is achievedonly when every process is reversible. An idealization.The above inequality gives the arrow of time: “after” isdifferent from “before” because the entropy is bigger.Monday, November 4, 2013

First law, more precised/W pdV Equality iff process is reversible.d/Q T dSEquality iff process is reversible.dE d/Q d/W T dS pdVThese equalities are whether or not it’sreversible.Monday, November 4, 2013

Entropy change Suniverse Ssystem Ssurroundings 0 S Sf Si dS reversibled/ QRTSince entropy, S, is a state variable, this is independentof the path. Can pick any reversible path one likes.Sometimes, computationally, some paths are moreconvenient than others. So pick a convenient one.Monday, November 4, 2013

Examples (more, soon) S Reversible adiabatic process:Reversible isotherm process:Ideal gas isotherm:QW1 S TTTMonday, November 4, 2013 pdV N k S d/Q 0.Td/QQ TTdV N k ln(Vf /Vi )V

Ideal gas, cont.soso1dE T dS pdV f N kdT21dTdVdS f N k Nk2TV1T2V2S2 S1 f N k ln( ) N k ln( )2T1V1Now, move to chalkboard for further discussionand examples.Monday, November 4, 2013

Cyclic and/or reversibleCyclic process (same initial and final states) Ssystem 0, Suniverse 0.Reversible process: Suniverse 0.Cyclic and Reversible process: Ssystem Suniverse 0.This is the case with maximal efficiency.Monday, November 4, 2013

Carnot engineMaximal efficiency if every step is reversible.The Carnot engine has 4 reversible stages:A B : isotherm expansion at THB C : adiabatic expansion from TH TCC D : isotherm compression at TCD A : adiabatic compression from TC THCyclic total process, so state variables Eand S have no net change after full cycle.Monday, November 4, 2013

Carnot engine,cont.W Monday, November 4, 2013 pdVW Q cyclic! T dS

Carnot engine, contWill show (on chalkboard) that:Wtotal N kB (TH TC ) ln(VB /VA )QH QA B N kB TH ln(VB /VA )WTCQCe 1 1 QHTHQHMonday, November 4, 2013

Boltzmann formulaS kB ln wE.g. ideal gas with N molecules in volume V:w VNS N kB ln VE.g. free expansion of ideal gas: S N kB ln(Vf /Vi )Monday, November 4, 2013as seen earlier.

Reversible. The egg scramble is an example of an irreversible process. A reversible one is where the system undergoes tiny changes, which could have gone either way by slightly adjusting the conditions. Maximal engine or refrigerator efficiency is achieved only when every process is reversible. An idealization.