Physics 207 Lecture 23

Transcription

1 ysics 07 Lecture ysics 07, Lecture 8, Dec. Agenda:. Finis, Start. Ideal gas at te molecular level, Internal Energy Molar Specific Heat ( = m c = n ) Ideal Molar Heat apacity (and U int = + W) onstant : v = / R, onstant : p = / R + R = 5/R Adiabatic processes (no eat transfer) Heat engines and Second Law of termodynamics Reversible/irreversible processes and Entropy Lecture 8, Exercise An atom in a classical solid can be caracterized by tree independent armonic oscillators, one for te x, y and z- directions? How many degrees of freedom are tere? Assignments: roblem Set 0 (. 0 & ) due uesday, Dec., :59 M. 0:,,8,4,50,68.:,6,9,6,70 roblem Set,. : 6, 7, 7, 7, 46 (Due, Friday, Dec. 5, :59 M) Wednesday, Work on problem set ysics 07: Lecture 8, g (A) (B) () (D) 4 (E) Some oter number ysics 07: Lecture 8, g Ideal Molar Heat apacities Definition of molar eat capacities (relates cange in te internal energy to te cange in temperature) Ideal Internal Energy n lim / = nδ / δ K tot trans = U = NkB = nr n Lecture 8, Exercise An atom in a classical solid can be caracterized by tree independent armonic oscillators, one for te x, y and z- directions ( U per atom = k B )? Wat is te classical molar eat capacity ( 0!)? ere is only microscopic kinetic energy (i.e., no springs) in a monoatomic ideal gas (He, Ne, etc.) At constant : Work W is 0 so U = At constant : U = + W = - = nr = R = R + R ysics 07: Lecture 8, g (A) nr (B) nr () nr (D) 4nR (E) Some oter number ysics 07: Lecture 8, g 4 Adiabatic rocesses By definition a process in wic no eat tranfer () occurs W For an Ideal : = const Adiabatic process: is constant =nr but not isotermal Work (on system) becomes : = d = const d = const ( ysics 07: Lecture 8, g 5 ) Distribution of Molecular Speeds Maxwell-Boltzmann Distribution ery few gas molecules ave exactly / k B of energy # Molecules O at 5 O at Molecular Speed (m/s) ysics 07: Lecture 8, g 6 age

2 ysics 07 Lecture Evaporative ooling in a Bose-Einstein ondensation Granularity, Energy and te Boltzmann Statistics ere are discrete number accessible energy levels in any finite system. It can be sown tat if tere are many more levels tan particles to fill tem te probability is just (E) = exp(-e/k B ) e energy levels for a uantum Mecanical (i.e., discrete quantized states) ideal gas is sown before and after a cange (igly idealized diagram, imagine lots more levels and lots more particles). Evaporative cooling can lead to a state cange Here we increase te box size slowly and perform a quasistatic, adiabatic expansion ysics 07: Lecture 8, g 7 ysics 07: Lecture 8, g 8 apter : Heat s and te nd Law of ermodynamics A scematic representation of a eat engine. e engine receives energy from te ot reservoir, expels energy c to te cold reservoir, and does work W. If working substance is a gas ten we can use te diagram to track te W process. cycle Hot reservoir c old reservoir Area = W cycle ysics 07: Lecture 8, g 9 Heat s Example: e Stirling cycle = 4 = H = = H We can represent tis cycle on a - diagram: x start 4 a b H * reservoir: large body wose temperature does not cange wen it absorbs or gives up eat ysics 07: Lecture 8, g 0 Heat s and te nd Law of ermodynamics Hot reservoir c old reservoir W cycle A eat engine goes troug a cycle (start and stop at te same point, same state variables) st Law gives U = + W =0 Wat goes in must come out st Law gives = c + W cycle ( s > 0) So (cycle mean net work on world) net = - c = -W system = W cycle ysics 07: Lecture 8, g Efficiency of a Heat How can we define a figure of merit for a eat engine? Define te efficiency ε as: ε W cycle c = = = Observation: It is impossible to construct a eat engine tat, operating in a cycle, produces no oter effect tan te absorption of energy from a reservoir and te performance of an equal amount of work c ysics 07: Lecture 8, g age

3 ysics 07 Lecture Heat s and te nd Law of ermodynamics Reservoir W eng It is impossible to construct a eat engine tat, operating in a cycle, produces no oter effect tan te absorption of energy from a reservoir and te performance of an equal amount of work. is leads to te nd Law Equivalently, eat flows from a ig temperature reservoir to a low temperature reservoir ysics 07: Lecture 8, g Lecture 8: Exercise Efficiency onsider two eat engines: I: Requires in = 00 J of eat added to system to get W=0 J of work (done on world in cycle) II: o get W=0 J of work, out = 00 J of eat is exausted to te environment ompare ε I, te efficiency of engine I, to ε II, te efficiency of engine II. Wcycle c c ε = = = (A) ε I < ε II (B) ε I > ε II () Not enoug data to determine ysics 07: Lecture 8, g 4 Reversible/irreversible processes and te best engine, ever Reversible process: Every state along some pat is an equilibrium state e system can be returned to its initial conditions along te same pat Irreversible process; rocess wic is not reversible! All real pysical processes are irreversible e.g. energy is lost troug friction and te initial conditions cannot be reaced along te same pat However, some processes are almost reversible If tey occur slowly enoug (so tat system is almost in equilibrium) arnot ycle Named for Sadi arnot (796-8) () Isotermal expansion () Adiabatic expansion () Isotermal compression (4) Adiabatic compression e arnot cycle ysics 07: Lecture 8, g 5 ysics 07: Lecture 8, g 6 e arnot (te best you can do) No real engine operating between two energy reservoirs can be more efficient tan a arnot engine operating between te same two reservoirs. A. A B, te gas expands isotermally wile in contact wit a reservoir at B. B, te gas expands adiabatically (=0, U=W B, c ), =constant. D, te gas is compressed isotermally wile in contact wit a reservoir at c D. D A, te gas compresses adiabatically (=0, U=W D A, c ) A =0 D c W cycle B =0 ysics 07: Lecture 8, g 7 arnot ycle Efficiency ε arnot = - c / A B = = W AB = nr ln( B / A ) D = c = W D = nr c ln( D / ) (work done by gas) But A A = B B =nr and = D D =nr c so B / A = A / B and / D = D / \ as well as B B = and D D = A A wit B B / A A = / D D tus ( B / A )=( D / ) c / = c / Finally ε arnot = - c / ysics 07: Lecture 8, g 8 age

4 ysics 07 Lecture e arnot arnot sowed tat te termal efficiency of a arnot engine is: ε arnot cycle = cold ot ower from ocean termal gradients oceans contain large amounts of energy arnot ycle Efficiency ε arnot = - c / = - c / All real engines are less efficient tan te arnot engine because tey operate irreversibly due to te pat and friction as tey complete a cycle in a brief time period. See: ttp:// ysics 07: Lecture 8, g 9 ysics 07: Lecture 8, g 0 Ocean onversion Efficiency ε arnot = - c / = - c / ε arnot = - c / = 75 K/00 K = 0.08 (even before internal losses and assuming a REAL cycle) Still: is potential is estimated to be about 0 watts of base load power generation, according to some experts. e cold, deep seawater used in te OE process is also ric in nutrients, and it can be used to culture bot marine organisms and plant life near te sore or on land. Energy conversion efficiencies as ig as 97% were acieved. See: ttp:// So ε =- c / always correct but ε arnot =- c / only reflects a arnot cycle Lecture 8: Exercises 4 and 5 Free Expansion and te nd Law You ave an ideal gas in a box of volume. Suddenly you remove te partition and te gas now occupies a large volume. () How muc work was done by te system? () Wat is te final temperature ( )? () an te partition be reinstalled wit all of te gas molecules back in : (A) W > 0 (B) W =0 () W < 0 : (A) > (B) = () > ysics 07: Lecture 8, g ysics 07: Lecture 8, g Entropy and te nd Law Will te atoms go back? Altoug possible, it is quite improbable e are many more ways to distribute te atoms in te larger volume tat te smaller one. Disorderly arrangements are muc more probable tan orderly ones all atoms Isolated systems tend toward greater disorder Entropy (S) is a measure of tat disorder Entropy ( S) increases in all natural processes. (e nd Law) Entropy and temperature, as defined, guarantees te proper direction of eat flow. no atoms Entropy and te nd Law In a reversible process te total entropy remains constant, S=0! In a process involving eat transfer te cange in entropy S between te starting and final state is given by te eat transferred divided by te absolute temperature of te system. S e nd Law of ermodynamics ere is a quantity known as entropy tat in a closed system always remains te same (reversible( reversible) ) or increases (irreversible). Entropy, wen constructed from a microscopic model, is a measure of disorder in a system. ysics 07: Lecture 8, g ysics 07: Lecture 8, g 4 age 4

5 ysics 07 Lecture Entropy, emperature and Heat Example: joules transfer between two termal reservoirs as sown below ompare te total cange in entropy. S = (-/ ) + (+ / ) > 0 because > > ysics 07: Lecture 8, g 5 Entropy and ermodynamic processes Examples of Entropy anges: Assume a reversible cange in volume and temperature of an ideal gas by expansion against a piston eld at constant pressure (du = d d wit = nr and du/d = v ): S = i f d/ = i f (du + d) / S = i f { v d / + nr(d/)} S = n v ln ( f / i ) + nr ln ( f / i ) Ice melting: S = i f d/= / melting = m L f / melting ysics 07: Lecture 8, g 6 e Laws of ermodynamics First Law You can t get someting for noting. Second Law You can t break even. Do not forget: Entropy, S, is a state variable ysics 07: Lecture 8, g 7 Recap, Lecture 8 Agenda:. Finis, Start. Ideal gas at te molecular level, Internal Energy Molar Specific Heat ( = m c = n ) Ideal Molar Heat apacity (and U int = + W) onstant : v = / R, onstant : p = / R + R = 5/R Adiabatic processes (no external eat transfer) Heat engines and Second Law of termodynamics Reversible/irreversible processes and Entropy Assignments: roblem Set 0 (. 0 & ) due uesday, Dec., :59 M. 0:,,8,4,50,68.:,6,9,6,70 roblem Set,. : 6, 7, 7, 7, 46 (Due, Friday, Dec. 5, :59 M) Wednesday, Start roblem Set ysics 07: Lecture 8, g 8 age 5