Thermodynamics! for Environmentology!

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1 1 Thermodynamics! for Environmentology! Thermodynamics and kinetics of natural systems Susumu Fukatsu! Applied Quantum Physics Group! The University of Tokyo, Komaba Graduate School of Arts and Sciences Table of Contents Part I (Fundamentals of thermodynamics*)# 1) Thermodynamic system States, Macroscopic variables Diagram, Thermal equilibrium 2) Laws of Thermodynamics Energy, Heat, Work, Entropy, (Adiabatic, isothermal) processes 3) Thermodynamic cycle, Heat engine, Carnot cycle, Efficiency 4) Heat engines and Free energy Joule-Thomson effect 5) Statistical mechanics 6) Phase transition and Kinetics Part II (Natural systems)# 7) Natural system as a heat engine 8) Laps rate, Climate change Greenhouse effect 9) Air-water dynamics Global circulation 10) Renewable energy Heat pipe, Thermosyphon 11) Carbon cycle, Ozone depletion 12) Radiative forcing Climate forcings Natural variability *Honing the math skills that are needed? 2 Calendar 9/27, 10/4, 10/11, 10/18, 10/25, 11/1, 11/8

Physics Claire Smith Elements of Environmental

Valsaraj Environmental Physics Egbert Boeker

Calendar 9/27, 10/4, 10/11, 10/18, 10/25, 11/1, 11/8

in science, technology and engineering, including

2 3 Grading: Term-end exam. + Homework For future readings: Environmental Physics Claire Smith Elements of Environmental Engineering Kalliat T. Valsaraj Environmental Physics Egbert Boeker Thermodynamics of Natural Systems Greg Anderson Calendar 9/27, 10/4, 10/11, 10/18, 10/25, 11/1, 11/8 4 Thermodynamics One of the most important subjects in science, technology and engineering, including physics, chemistry, biology, and many relevant fields of study. We live in a world that is literally thermodynamic in many respects.

6 Water or water vapor Liquid state or gas state.

3 5 Ice or water Solid state or liquid state. Solidification (freeze) or liquefaction (thaw). Reversible? Melting ice (=water) never returns to ice spontaneously! on a warm day. 6 Water or water vapor Liquid state or gas state. Vaporization (gasification) versus liquefaction (condensation) Reversible? More or less

7 Raw egg or fried egg Irreversible Once fried, it never gets back to

Strange enough, they have failed to come to terms with quantum

Quantum Superposition IBM Almaden Stadium Corral (permission granted)

4 7 Raw egg or fried egg Irreversible Once fried, it never gets back to raw egg by chance. 8 Thermodynamics & Statistical Physics! Strange enough, they have failed to come to terms with quantum mechanics which is equally or even more important in modern physics. Quantum Superposition IBM Almaden Stadium Corral (permission granted) stm15.jpg&imgrefurl=

9 1-1. Thermodynamic System Macroscopic Water bath @ Const. temperature Gas in a He-tight container N A Microscopic details neglected = 6.02 1023 Thermal equilibrium!

5 Thermodynamic System Macroscopic Water Const. temperature Gas in a He-tight container N A Microscopic details neglected = Thermal equilibrium! Uniform T, No flow of mass or energy, No spontaneous change 1-1. Thermodynamic System Coin 10 T H T 0 Coin T Water bath (Finite size, Thermally insulated) 0 T 0 Water bath t Thermal equilibrium is established eventually.

Open Closed Isolated Mass flow Heat Q Work W No mass flow No mass flow Self-contained. No heat in/out. No work in/out. 1-2.

6 Thermodynamic System Configuration Systems of interest System {Open,Closed, Isolated, } Outside: Environment Surroundings Reservoir (source, sink) Constraints Open Closed Isolated Mass flow Heat Q Work W No mass flow No mass flow Self-contained. No heat in/out. No work in/out Thermodynamic State 12 (Thermal) Equilibrium! Uniquely specified with only a few parameters (P, V, T, N, U, S ). State quantities State quantities are macroscopic while not materials-specific. No apparent change in macroscopic parameters over time. Non-equilibrium Cf. Steady state Equilibrium Steady state Q1. Give an example of non-equilibrium steady state.!

7 Thermodynamic State Two systems sharing the same set of State Quantities P, V, T, N P, V, T, N No difference in the language of thermodynamics Thermodynamic State Typical diagrams/plots frequently used in thermodynamics P Isothermal expansion A Nonequilibrium States T Free Expansion (Isothermal ) Nonequilibrium States Equilibrium States B A B Equilibrium States V We must have points (states) on the diagrams. V

15 1-2. Thermodynamic State Transition from State A to B along path C P Isothermal expansion A C Slowly B V Quasi-static process Slowly changing (moving) in order to stay in equilibrium at all times?

8 Thermodynamic State Transition from State A to B along path C P Isothermal expansion A C Slowly B V Quasi-static process Slowly changing (moving) in order to stay in equilibrium at all times? Quasi-static Reversible? We must secure a reversible path to apply the appropriate math State quantities (functions of states) Dependent only on the current state (not on the path) 16 P Heat Q Work W V Heat and Work are not. (functions of path) Pressure Temperature Volume Number of moles Heat capacity Internal energy Entropy Enthalpy Gibbs free energy Free energy (Helmholtz) P! T! V! n! C! U! S! H! G! F!

9 Extensive and intensive properties not an exhaustive list Extensive properties: Additive ( amount of material) Mass m! Volume V! Number of moles n! Energy U! Entropy S! Enthalpy H! Gibbs free energy G! Intensive properties: Not additive Temperature T! Pressure P! Density ρ! Chemical potential! µ State variables and state postulate State Variables A set of state quantities that suffice to uniquely specify an equilibrium state P A e.g., P = P( T, V, n) Equation of state G = G( T, p) B V State postulate(s): The state of a simple compressible system is uniquely specified by two independent intensive quantities.

10 Thermodynamic processes Reversible! The system can be brought back to the initial state without dissipation (entropy production). P A Isothermal expansion (Quasi-static =Equilibrium) T, V L, n T, V Relaxation! (Nonequilibrium)! Slowly ( = V L + V R ), n B V Relaxation is an irreversible process Expl.1 Free expansion of gas T, V L, n V R Gas T, V L, n Vacuum T, ( V L + V R ), n Cf. Adiabatic free expansion 1-6. Thermodynamic processes 20 Expl.2 Thaw (melting) Expl.3 Friction (Mechanical) Oscillation damping ( ) x k > 0 m Expl.4 Diffusion (mixing) Expl.5 Q2. Give your own! example of an irreversible! system and explain why it is so.!! October 4, Thermodynamic processes

11 21 Reversibility Thermodynamic Reversibility Cycle Reversible Literally Reversible (getting back to where it started) but not necessarily Quasi-Static is called Cyclic A Quasi-static e.g., Joule-Thomson Cycle Quasi-static F PS F PS Compression F PS Expansion F = PS Balanced 1-7. For the researcher in you 22 Zeroth law: Thermodynamic syllogism First law: Conservation of energy (nontrivial) Second law: Entropy never decreases (conditional) Third law: Entropy must vanish at absolute zero

23 Zeroth law: Thermodynamic syllogism Two systems, A and B, are in contact so that they are in thermal equilibrium. Two systems, B and C, are likewise in thermal equilibrium.

12 23 Zeroth law: Thermodynamic syllogism Two systems, A and B, are in contact so that they are in thermal equilibrium. Two systems, B and C, are likewise in thermal equilibrium. A B C A C Then A and C must be in thermal equilibrium. To define the (absolute) temperature System System System T Temperature How can we define the absolute temperature? (Isobaric) volume expansion coefficient For ideal gas (or in the dilute-limit), Hence β = 1 V T T = V nrt P ΔT ΔV P Measure V to find T P = 1 T Charles s law 0 K Reference point(s) necessary Cf. F V β 1 V ΔT V T P ΔV 0 C = [K] = 9 5 C / T 24

25 First law: Conservation of energy Energy balance Alternatively, ΔU = Q + W du = d Q + d W The nonexistence of Perpetual Motion Machines The 1 st kind: Spontaneous motion without energy uptake.

13 25 First law: Conservation of energy Energy balance Alternatively, ΔU = Q + W du = d Q + d W The nonexistence of Perpetual Motion Machines The 1 st kind: Spontaneous motion without energy uptake.? du = 0 ( d Q = 0) d W = 0. If, no doubt Beware: There are many fraud machines up there seemingly violating(?) the laws. 26 Heat and Work are equivalent. Thermal Mechanical A variant of Joule s experiment W = PΔV Q T 0, V 0 T 1, V 1 T 0, V 1 Mechanical equivalent of Heat 1 calorie = 4.2 J

14 Heat and/or Work increase the internal energy ΔU 27 Heat entering the system: Q The internal energy U increases by Q ΔU = U ( T 1, V 1 ) U ( T 0, V 1 ) = Q. T 0, V 1 T 1, V 1 Work on the system: The internal energy U W = PΔV (<0) increases by ΔU = U ( T 1, V 1 ) U ( T 0, V 0 ) = W. W T 0, V 0 T 1, V 1 28 Exact differential and Inexact differential A C Heat d Q C B du = d Q + d W States Exact differential Path (History) Inexact differential Exact differential is NOT path-dependent B du = U B A with prime ( ) U ( A) Work d W du = 0 conservative C+ C Net U vanishes.

15 y Exact differential Example y B 2 C 3 : y = f ( x) x 2 C 1 C 2 Path-independent integral du = 2xy dx + x 2 dy x 2 du = 2xy x 1 dx 1 + x 2 x 2 dy 1 y 1 = x 2 x y 1 2 x1 + x 2 y 2 y 2 y1 = x 2 2 y 2 x 2 1 y 1 y 2 29 y 1 A C 1 x 1 x 2 x C 2 x 2 y 2 du = x 2 x 1 dy 1 + 2xy y 2 dx 1 x 1 = x 2 y 1 y 2 y1 + x 2 x y 2 2 x1 = x 2 2 y 2 x 2 1 y 1 x 2 B du = U B A du = 0 C ( ) U ( A) C 3 x 2 d dx ( x2 f ( x) )dx = x 2 f x x 1 = x 2 2 y 2 x 2 1 y 1 ( ) x 2 x1 30 Inexact differential Path-dependent integral y A d U = xy dx + x 2 dy Example y B 2 denoted by the prime x C 2 x 2 y 2 1 y 1 C 2 B A d U C 1 x 1 x 2 x U ( B) U ( A) d U 0 C=C 1 +C 2 C 2 d U = xy x 1 dx 1 + x 2 x 2 dy 1 y 1 x = x2 2 y x 2 y 2 y 2 y1 x1 = x y 1 x y 1 + x 2 2 y 2 x 2 2 y 1 x 2 d U = x 2 1 dy + xy 2 dx x 1 y 2 y 1 = x 2 1 y 2 x 2 1 y 1 + x y x y 2 x 2 x 1

31 Second law of thermodynamics! Clausius statement:! Heat NEVER flows spontaneously from a Low-T system to a High-T system without the help of work!

Q W 100%-efficient Perpetual Motion Machines of the 2md kind are forbidden. One-way Evolution Entropy never decreases 32 Entropy!

16 31 Second law of thermodynamics! Clausius statement:! Heat NEVER flows spontaneously from a Low-T system to a High-T system without the help of work! T H T L Q Thomson (Kelvin) statement:! One CANNOT convert heat 100%-efficiently into work without influence on the surroundings! Q W 100%-efficient Perpetual Motion Machines of the 2md kind are forbidden. One-way Evolution Entropy never decreases 32 Entropy! A New State Quantity One of the most significant but elusive properties in TD. ds = d Q T Is T doing a magic? The entropy S is defined only for quasi-static processes. du = d Q + d W and d W = PΔV The first law thus reads TdS = du + PdV. This shows that S = S( U, V ). ds = 1 T du + P T dv so that S = 1 U V T, S = P V U T.

17 (Total) differential, partial derivative Partial derivatives 33 df( x, y, z) = F dx + F dy + F dz x y,z y x,z z x,y y,z fixed x,z fixed x, y fixed Relation between infinitesimals E.g. Δy F( x, y) = xy x is fixed xδy 2 nd order to be neglected ΔxΔy ΔF( x, y) = ( x + Δx) ( y + Δy) xy = xy + xδy + yδx + ΔxΔy xy = xδy + yδx + O(2) y xy x yδx Δx y is fixed df( x, y) = y dx + x dy F x y = y, F y x = x 34 Entropy never decreases in an isolated system! Statistical mechanics provides a plausible explanation for this. The energy is conserved:! The entropy adds up:! The first derivative! ds = 0 U = U A +U B S( U A,U B ) = S( U A ) + S( U B ) with Entropy reaches its maximum value in thermal equilibrium U A S( U A ) U B S( U B ) Constant-volume vessels! S S ds = U A du A + U V B = 1 1 T A du A = 0 T B T A = T B Thermal equilibrium du A + du B = 0. S U A,B V V du B = 1 T A,B

35 Entropy never decreases in an isolated system! An alternative route leading to the same conclusion!

ds = S A U V = 1 T A du + 1 T B T A = T B S du + B U 0 U ( ) ( du ) = 0 S( U A,U B ) = S A ( U ) + S B ( U 0 U ).

Even without ceiling? The 2 nd law of thermodynamics! Clausius statement:! Heat flows from High to Low!

18 35 Entropy never decreases in an isolated system! An alternative route leading to the same conclusion! The total energy is fixed:! U 0 = U A +U B The entropy is additive:! The first derivative must vanish! ds = 0. ds = S A U V = 1 T A du + 1 T B T A = T B S du + B U 0 U ( ) ( du ) = 0 S( U A,U B ) = S A ( U ) + S B ( U 0 U ). V d( U 0 U ) S A,B U V = 1 T A,B du Thermal equilibrium is reached ( Thermalized ). du Entropy increases, why? Even without ceiling? The 2 nd law of thermodynamics! Clausius statement:! Heat flows from High to Low! The entropy change associated with the transfer of heat:! dq T H dq dq T L Isolated system dq ds = 1 T L dq + 1 T H = 1 1 T L T H simply because! ( dq) dq > 0 T H > T L S reaches its maximum eventually.! Thermal equilibrium 36

19 37 More about Exact differential For a state quantity A = A( X, Y ), the ED exists when and only when Y A = X Y X A Y X Clairaut-Schwarz so that da = A dx + X Y Maxwell s relations A dy. Y X du = TdS PdV df = PdV SdT dg = VdP SdT T = P V S S V P = S T V V T V = S T P P T 38 Specific heat of an ideal gas Starting from TdS = PdV + du, (Defined along the QS path) C P = Since d Q dt Specific heat at constant pressure U T P P = C P = nr + = T S T U T U T V = P V P T P + Specific heat at constant volume U T and P V = P T P T V = nr + C V P. nrt P Mayer s relation P = nr,

V 1 V nrt W = 2 V dv = nrt ln V 2 V 1 V 1 V 2 V V 2 > V 1 V 2 < V 1 Expansion Work done by the system Compression Work done on the system 40

20 39 Quasi-static isothermal processes! P The internal energy does not change. dt = 0 TdS = PdV For an ideal gas, Upon integration Isotherm d W = nrt W V dv. V 1 V nrt W = 2 V dv = nrt ln V 2 V 1 V 1 V 2 V V 2 > V 1 V 2 < V 1 Expansion Work done by the system Compression Work done on the system 40 Quasi-static adiabatic processes! Heat does not enter or leave the system. ( T )ds = 0 du = PdV For an ideal gas, du = nrt V dv. Table of α values Monoatomic Diatomic On the other hand, du = C V dt = αnrdt dv V 1 α V + α dt T = 0 T = const. PV 1+α α = const. Cf. PV = nrt

21 41 1) Prove that the Clausius and Thomson statements are equivalent. 2) Show that U = 0, so U = U ( T ) for an ideal gas*. V T 3) Show that U = U T P T V for an ideal gas*. *The EOS for an ideal gas: PV = nrt

Chemistry. Lecture 10 Maxwell Relations. NC State University

Chemistry. Lecture 10 Maxwell Relations. NC State University Chemistry Lecture 10 Maxwell Relations NC State University Thermodynamic state functions expressed in differential form We have seen that the internal energy is conserved and depends on mechanical (dw)