1. Foundations of thermodynamics 1.1. Fundamental thermodynamical concepts. Introduction. Summary of contents:

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1 Introduction hermodynamics: henomenological descrition of equilibrium bulk roerties of matter in terms of only a few state variables thermodynamical laws Statistical hysics: microscoic foundation of thermodynamics degrees of freedom 2 3 state variables! Everything should be made as simle as ossible, but no simler A Einstein Summary of contents: Review of thermodynamics hermodynamical otentials Phase sace robability Quantum mechanical ensembles Equilibrium ensembles Ideal fluids Bonic systems ermionic systems Interacting systems Phase transitions critical henomena 1 oundations of thermodynamics 11 undamental thermodynamical concets System : macroscoic entity under consideration Environment : wld outside of the system infinite Oen system : can exchange matter heat with the environment Closed system : can exchange heat with the environment while keeing the number of articles fixed Ilated system : can exchange neither matter n heat with the environment Can ossibly still do wk by eg exing hermodynamical equilibrium: No macroscoic changes Uniquely described by a few external variables of state System fgets its ast: no memy effects, no hysteresis Often the term global equilibrium is used, as oosed to local equilibrium, which is not full equilibrium at all next age! Nonequilibrium: Generally much me comlicated than equilibrium state Simlest case: ilated systems each in an equilibrium state In a local thermodynamical equilibrium small regions are locally in equilibrium, but neighbour regions in different equilibria articles, heat etc will flow Examle: fluid water with non-homogeneous temerature Stronger nonequilibrium systems usually relax to a local equilibrium Degrees of freedom dof is the number of quantities needed f the exact descrition of the microscoic state Examle: classical ideal gas with N articles: 3N codinates x, y, z, 3N momenta x, y, z State variables are arameters characterizing the macroscoic thermodynamical state hese are all extensive intensive: Extensive variable: change value when the size satial volume the number of degrees of freedom is changed: volume, article number N, internal energy U, entroy S, total magnetic moment d 3 r M 1

2 > Intensive variable: indeendent of the size of the system, can be determined f every semimicroscoical volume element: eg temerature, ressure, chemical otential µ, magnetic field H, ratios of extensive variables like ρ N/, s S/N, Conjugated variables: A B aear in airs in exressions f the differential of the energy me generally, me state variable, ie in fms ±AdB ±B da; one is always extensive the other intensive Examle: ressure volume ; change in internal energy U when is changed adiabatically, at constant S is du d Process is a change in the state Reversible rocess: advances via states infinitesimally close to equilibrium, quasistatically slow rocess he direction of a reversible rocess can be reversed, obtaining the initial state f system + environment! Ithermal rocess : constant Ibaric rocess : constant Ichic rocess : constant Isentroic adiabatic rocess: S constant Irreversible rocess is a sudden ntaneous change during which the system is far from equilibrium In the intermediate stes global state variables,, are usually not well defined Cyclic rocess consists of cycles which take the system every time to its initial state 12 State variables exact differentials Let us suose that, f examle, the state of the system can be uniquely described by state variables, ja N Other state variables are then their unique functions:,, N U U,, N S S,, N By alying differential calculus, the differential of, f examle, is d d + d + dn,n, he differentials of state variables, d, d, d,, are exact differentials hese have the following roerties A heir total change evaluated over a closed ath vanishes: 1 2 d 1 2 du 0 B he total change of an exact differential is indeendent on the ath of integration: that we can write a du du 0, b U2 U1 + 2 Exact differentials Let us denote by d a differential which is not necessarily exact ie integrals can deend on the ath Assuming it deends on 2 variables x, y, the differential is exact differential if 1 du d 1 x, ydx + 2 x, ydy 1 y 2 x hen x, y that 1 x, y x,y x 2 x, y x,y y 2 1 d 2 1 is indeendent on the ath, integrable In this case x, 1 y, 2 are airs of conjugated variables with resect to Examles: are the following differentials exact? d y dx + xdy d xdx + xdy All hysical state variables are exact differentials! his will enable us to derive various identities between state variables Integrating fact If d 1 dx + 2 dy is not exact, there exists an integrating fact λx, y that in the neighbourhood of the oint x, y λ d λ 1 dx + λ 2 dy df 2

3 is an exact differential λ f are state variables Examle: find λ f the differential d xdx + xdy Legendre transfmations Legendre transfmations can be used to make changes in the set of the indeeendent state variables examle, let us look at the function fx, y of two variables We denote fx, y z f y y define the function g f yf y f yz Note: z, y is a congjugated air with resect to f! dg df y dz z dy f x dx + f y dy y dz z dy f x dx y dz hus we can take x z as indeendent variables of the function g, ie g gx, z Obviously y gx, z z Crending to the Legendre transfmation f g there is the inverse transfmation g f 13 Equations of state Encodes me of the hysical roerties of the equilibrium system Usually these relate mechanical readily observable variables, like,, N, ; not internal variables like S, internal energy U etc A tyical examle: ressure of me gas as a function of density ρ Some examles: Classical ideal gas Nk B where N number of molecules ablute temerature k B J/K Boltzmann constant Chemists use often the fm n R N 0 nr N/N 0 number of moles k B N J/K mol gas constant Avogadro s number If the gas is comosed of m different secies of molecules the equation of state is still Nk B, f g zg z g + yz Often needed identities Let x, y, x xy, z, y yx, z z zx, y If we want to give in terms of x, z, we can write where now i N m i1 N i i, i N i k B /, x, y x, yx, z Alying differential rules we obtain identities y + x x y x z z One can show that x y z x y y x y z y z x x y z x x y z 1 y x z x z y y z x z x 1 z where i is the artial ressure of the i:th comonent irial exansion of real gases When the interactions between gas molecules are taken into account, the ideal gas law receives crections which are suressed by owers of density ρ N/ : k B [ ρ + ρ 2 B 2 + ρ 3 B 3 + ] Here B n is the n:th virial coefficient an der Waals equation he molecules of real gases interact reulsively at sht distances; every article needs at least the volume b > Nb attractively otential r/r 0 at large distances due to the induced diole momenta he ressure decreases when two articles are searated by the attraction distance he robability of this is N/ 2 3

4 We imrove the ideal gas state equation where that Nk B Nb aρ 2 true ressure P D ǫ 0 electric olarization atomic total diole momentum/volume electric flux density As/m vacuum ermeability then + aρ 2 Nb Nk B In homogenous dielectric material one has P a + b E, Solid substances he thermal exansion coefficient α 1 the ithermal comressibility κ 1,N,N of lid materials are very small, the ayl series is a good aroximation yically α κ where a b are almost constant a, b 0 Curie s law When a iece of aramagnetic material is in magnetic field H we write where M B B µ 0 H + M, magnetic olarization atomic total magnetic moment/volume magnetic flux density µ 0 4π 10 7 s/am vacuum ermeability Polarization obeys roughly Curie s law Stretched wire ension [N/m 2 ] κ /Pa α 10 4 /K σ EL L 0 /L 0, M ρc H, where ρ is the number density of aramagnetic atoms C an exerimental constant related to the individual atom Note: Use as a thermometer: measure the quantity M/H where L 0 is the length of the wire when σ 0 E is the temerature deendent elasticity coefficient Surface tension t n σ σ 0 1 t t n t temerature C exerimental constants, 1 < n < 2 σ 0 surface tension when t 0 C Electric olarization When a iece of material is in an external electric field E, we define D ǫ 0 E + P, 4

5 Laws of thermodynamics hermodynamics is based uon 4 laws these can be derived from statistical hysics, but in thermodynamics these are considered fundamental 14 0th law of thermodynamics If each of two bodies is searately in thermal equilibrium with a third body then they are al in thermal equilibrium with each other there exists a roerty called temerature thermometer which can be used to measure it 15 Wk Befe we discuss the 1st law, let us introduce the concet of Wk Wk is exchange of such noble energy as oosed to exchange of heat matter that can be comletely transfmed to me other noble fm of energy; eg mechanical electromagnetic energy Sign convention: wk W is the wk done by the system to its environment Examle: system W Note: dw is not an exact differential: the wk done by the system is not a function of the final state of the system need to know the histy! Instead 1 d dw is exact, ie, 1/ is the integrating fact f wk Examle: In general dw d σadl E dp H dm dw i f i dx i f dx, where f i is a comonent of a generalized fce X i a comonent of a generalized dislacement 1 1st law of thermodynamics otal energy is conserved In addition to wk a system can exchange heat thermal energy chemical energy, asciated with the exchange of matter, with its environment hermal energy is related to the energy of the thermal stochastic motion of microscoic articles he total energy of a system is called internal energy U Sign conventions: If the system can exchange heat articles do wk, the energy conservation law gives the change of the internal energy du dq dw + µdn, where µ is the chemical otential Me generally, du dq f dx + i U is a state variable, ie du is exact µ i dn i Cyclic rocess In a cyclic rocess the system returns to the iginal state du 0, W Q no change in thermal energy Q + is the heat abrbed by the system during one cycle Q > 0 is the heat released he total change of heat is Q + W Area d in -system Q Q W Q + + Q, he efficiency of a heat engine W > 0 is wk/heat taken: η W Q + Q+ + Q Q + 1 Q Q nd law of thermodynamics 2nd law can be stated in various histical ways: a Heat flows ntaneously from high temeratures to low temeratures b Heat cannot be transferred from a cooler heat reservoir to a warmer one without other changes System Q W µ N chemical energy c In a cyclic rocess it is not ossible to convert all Environment heat taken from the hotter heat reservoir into wk d It is not ossible to reverse the evolution of a system towards thermodynamical equilibrium without converting wk to heat e he change of the total entroy of the system its environment is ositive can be zero only in reversible rocesses 5

6 @? > f Of all the engines wking between the temeratures 1 2 the Carnot engine has the highest efficiency We consider the infinitesimal rocess y U1 dq 1 U2 2 dq du + dw du + f dx, there exists an integrating fact 1/ that 1 dq ds is exact he state variable S is entroy turns out to be temerature on an ablute scale he second law e can now be written as ds tot dt 0, where S tot is the entroy of the system + environment the entroy of the system only we have ds 1 dq, where the equality holds only f reversible rocesses he entroy of the system can decrease, but the total entroy always increases stays constant reversible rocesses the first law can be rewritten as du dq dw + µ dn ds d + µ dn 18 Carnot cycle Illustrates the concet of entroy he Carnot engine C consists of reversible rocesses x a ithermal 2 Q 2 > 0 b adiabatic 2 1 Q 0 c ithermal 1 Q 1 > 0 d adiabatic 1 2 Q 0 U 0, W Q 2 Q 1 wrong sign here, f simlicity, 3, 3 We define the efficiency as η W Q 2 1 Q 1 Q 2 Because the rocesses are reversible the cycle C can be reversed C wks as a heat um Let us consider two Carnot cycles A B, f which W A W B W A is an enegine B a heat um he efficiences are crendingly η A W Q A η B W Q B, 3, 3 9 Let us suose that, 3 * 9 *, 3 * 9 η A > η B, that Q B > Q A Q B Q A > 0 he heat would transfer from the cooler reservoir to the warmer one without any other changes, which is in contradiction with the second law fm b So we must have η A η B By running the engines backwards one can show that η B η A, that η A η B, ie all Carnot engines have the same efficiency Similarly, it follows that the Carnot engine has the highest efficiency among all engines al irreversible wking between given temeratures: assume that engine A in revious figure is me other engine than Carnot hen the argument above imlies that η A η B If A is not reversible, efficiency is not necessarily the same while running the system backwards the inequality remains in fce; if it is reversible, reversing the rocess gives η A η B Note: he efficiency does not deend on the realization of the cycle eg the wking substance Only reversibility is essential! he efficiency deends only on the temeratures of the heat reservoirs Defining temerature scale via Carnot engines: Let us consider Carnot s cycle between temeratures 3 1 η 1 f 3, 1, where we have the identity f 3, 1 Q 1 Q 3

7 + Here!, 3!, 3, 3, 3 he simlest lution is, 9!, 9, 3!, 3, 9! f 3, 2 Q 2 Q 3 f 2, 1 Q 1 Q 2 f 3, 1 Q 1 Q 3 f 3, 1 f 3, 2 f 2, 1 f 2, We define the ablute temerature that is exact the entroy S the temerature are state variables Because the Carnot cycle has the highest efficiency, a cycle containing irreversible rocesses satisfies η irr 1 Q 1 Q 2 < η Carnot Q 2 2 Q 1 1 < 0 hus f an arbitrary cycle we have dq 0, where the equality holds only f reversible rocesses Because entroy S is a state variable, it cannot deend on the integration ath hus, f an arbitrary rocess 1 2 the change of the entroy can be obtained from the fmula S E H H rev ds rev dq η his theetical definition was first used by Kelvin, it gives us our familiar ablute temerature scale, u to a scale fact his is by no means unique; one could al use ft 2, t 1 [t 2 /t 1 ] a, with any a 0, this would just give us a different temerature scale t const 1/a he Carnot cycle satisfies dq 0, since, during the ithermal art a, dq Q 2 2 art c c a dq Q 1 1 Q 2 2 his is valid al f an arbitrary reversible cycle because So C + E dq i ds dq C i dq 0 irr H A L Accding to the fmula we have dq dq < 0, his is usually written as S > rev irr ds dq, dq the equality is valid only f reversible rocesses an ilated system Q 0, eg system + environment! we have S 0 hus, f an irreversible rocess, Q irr Q rev S, W irr Q irr U W rev f dx where rev is a hyothetical reversible rocess connecting the initial final states of irr -trajecty Note the sign conventions: W wk done by the system; Q heat abrbed by the system W irr W rev means less wk done by irr W > 0 me wk done to it W < rd law of thermodynamics Nernst s law 190: lim S 0 0 7

8 A less strong fm can be stated as: When the maximum temerature occuring in the rocess from a state a to a state b aroaches zero, al the entroy change S a b 0 Note: here are systems whose entroy at low temeratures is larger than true equilibria would allow his is due to very slow relaxation time 2 hermodynamic otentials 21 undamental equation Accding to the first law in a N-system, generalizes easily, f reversible rocesses du ds d + µ dn 8 S, N can be considered to be natural variables of the internal energy U, ie U US,, N urtherme, from the law one can read the relations U U U S, µ Scaling law of extensive variables: all extensive variables must be linear functions of system size each other U, S, N are extensive we have UλS, λ, λn λus,, N λ aking a derivative of wrt λ, we obtain the Euler equation f homogenous functions U U U U S + + N S, Substituting the artial derivatives in this takes the fm U S + µn S 1 U + µn his is called the fundamental equation 22 Internal energy Maxwell relations Because U U, U U Similar relations can be derived al f other artial derivatives of U we get called Maxwell s relations µ S, µ S,

9 8 In an irreversible rocess S > Q U + W, U < S + µ N If S, N stay constant in the rocess then the internal energy decreases hus we can deduce that In an equilibrium with given S, N the internal energy is at the minimum We consider a reversible rocess in an ilated system Q 0, 8 A G K E E > H E K I E J E We artition W into the comonents wk due to the d change of the total volume 0 [ wk done by the ] W free gas against the fce W free W 1 + W A L L Accding to the first law we have U Q W Q d W free Q W free Because now Q 0, we have U W free L, ie when the variables S, N are ket constant the change of the internal energy is comletely exhangeable with the wk U is then called free energy U thermodynamic otential Note: If there are irreversible rocesses in an ilated system N constants then W free U If the system does no wk, U 0, ie the system tends to minimize its internal energy 23 Enthaly Using the Legendre transfm U U H U U + We move from the variables S,, N to the variables S,, N he quantity H U + is called enthaly dh du + d + d ds d + µ dn + d + d dh ds + d + µ dn rom this we can read the artial derivatives H,N H H µ S, Crending Maxwell relations are µ S, µ S, In an irreversible rocess one has,n,n Q U + W µ N < S U H, that H < S + + µ N We see that In a rocess where S, N are constant ntaneous changes lead to the minimum of H, ie in an equilibrium of a S,, N-system the enthaly is at the minimum he enthaly is a suitable otential f an ilated system in a ressure bath is constant Let us look at an ilated system in a ressure bath dh du + d du dq dw + µ dn Again we artition the wk into two comonents: f a finite rocess H ds + dw d + dw free dh dq + d dw free + µ dn When S,, N is constant one has d W free + H W free µ dn 9

10 ie W free is the minimum wk required f the change H Note: Another name of enthaly is heat function in constant ressure Joule-homn henomenon low of gas through a ous wall choke :, 3 8? D A 1 2 are constant in time, 1 > 2 the rocess irreversible When a differential amount of matter asses through the choke the wk done by the system is dw 2 d d Initial state init 0 inal state 0 final he wk done by the system is W dw 2 final 1 init Accding to the first law we have that U U final U init Q W W, U init + 1 init U final + 2 final hus in this rocess the enthaly H U + is constant, ie the rocess is isenthalic, H H final H initial 0 We consider now a reversible isenthalic dn 0 rocess init final Here dh ds + d 0, ds d S,, that d ds + d S On the other h, C where C is the ibaric heat caacity see thermodynamical renses Using the Maxwell relation S the artial derivative relation we can write d ds + d C C Substituting into this the differential ds in constant enthaly we get called Joule-homn coefficients [ ] H C Defining the heat exansion coefficient α that α 1, we can rewrite the Joule-homn coefficient as α 1 C H We see that when the ressure decreases the gas cools down, if α > 1 warms u, if α < 1 ideal gases H 0 holds real gases is below the inversion temerature ositive, the gas cools down 24 ree energy he Legendre transfm U U S U S U defines the Helmholtz free energy d S d d + µ dn, the natural variables of are, N We can read the artial derivateves S,N µ, rom these we obtain the Maxwell relations,n µ, µ,,n H 10

11 In an irreversible change we have < S + µ N, ie when the variables, N are constant the system drifts to the minimum of the free energy Crendingly W free, when,, N is constant ree energy is suitable f systems where the exchange of heat is allowed; ie the control variables are tyically at constant N ery useful quantity in hysics! 25 Gibbs free energy he Legendre transfmation U U G U S U defines the Gibbs function the Gibbs free energy Its differential is G U S + dg S d + d + µ dn, the natural variables are, N the artial derivatives we can read the exressions G S,N G,N G µ, rom these we obtain the Maxwell relations,n,n µ,,n µ, In an irreversible rocess,n G < S + + µ N, holds, ie when the variables, N stay constant the system drifts to the minimum of G Crendingly W free G, when,, N is constant he Gibbs function is suitable f systems which are allowed to exchange mechanical energy heat in heat ressure baths 2 Gr otential he Legendre transfm U U Ω U S defines the gr otential Its differential is N Ω U S µn dω S d d N dµ, the natural variables are, µ he artial derivatives are now Ω S,µ Ω,µ Ω N µ We get the Maxwell relations,µ µ, µ, In an irreversible rocess,,µ U S,,µ,µ Ω < S N µ, holds, ie when the variables, µ are ket constant the system moves to the minimum of Ω Crendingly W free Ω, when,, µ is constant he gr otential is suitable f systems that are allowed to exchange heat articles Bath A bath is an equilibrium system, much larger than the system under consideration, which can exchange given extensive roerty with our system Pressure bath, 8 he exchanged roerty is the volume a crending generalized dislacement; f examle magnetization in a magnetic field Heat bath 11

12 Particle bath Baths can al be combined; f examle a suitable otential f a ressure heat bath is the Gibbs function G 27 hermodynamic rense functions Rense functions are thermodynamic quantities most accessible to exeriment hey give us infmation about how a secific state variable changes as other indeendent state variables are changed hey can be classfied as mechanical comressbility, suscetibility thermal heat caacity renses 1 Heat exansion coefficient α 1 where ρ N/ α 1 ρ, 5,, 3, 5,N ρ,,n 2 Ithermal comressibility κ 1 1 ρ,n 3 Adiabatic comressibility κ S 1 1 ρ ρ,n ρ he velocity of und deends on the adiabatic comressibility like 1 c S, mρκ S where m the article mass One can show that κ κ S + α2 C 4 Ichic heat caacity Heat caacity C is a measure of the amount of heat needed to raise the temerature of a system by a given amount In a reversible rocess we have Q S he heat caacity C is defined that C Q S Keeing volume N constant, we define C, accding to the first law at constant N, du ds d + µ dn ds Using S /, we can write C U Ibaric heat caacity C Because,N dh ds + d + µ dn, using S G/,N, one can write C H,N 2 G 2,N Relating rense functions,, + a Maxwell relation Since C C +, 1 C C + α2 κ α κ, 12

13 hus, C > C 28 hermodynamical equilibrium state Accding to the 2nd law, the entroy of an ilated equilibrium system must be at maximum hus, any local fluctuation must cause the entroy to decrease; if it were not, the system could move to a new higher entroy state, which cannot haen in an equilibrium system by definition We divide the system into fictitious arts: Extensive variables satisfy 8, 7, 8, E 29 Stability conditions of matter In a steady equilibrium the entroy has the true maximum that small variations can only reduce the entroy We again consider an ilated system divided into arts, denote the equilibrium values common f all fictitious arts by the symbols, {µ j } the equilibrium values of other variables by the suerscrit 0 In der to study the maximality of the entroy, the entroy of the artial system α, S α, needs to be exed into second der in { U α, α, N jα } We write S α close to an equilibrium as the ayl series S α U α, α, {N jα } Sα 0 U0 α, α 0, {N0 jα } U α + α U U,N 0 S α α U α N j α S α α U α N jα N jα U, 0 + j j { + 1 α α U α + 2 U α α + } 0 α N jα j jα U, + 0 U,N α Since each element is in equlibrium the state variables are defined in each element, eg S α S α U α, α, {N jα } S α 1 α U α + α α α µ jα α N jα Let us assume that the system is comosed of two arts: α {A, B} hen U B U A, B A N jb N ja S S α α 1 1 A j B µja A µ jb B A U A + A B B N ja A In an equilibrium S is at its maximum, S 0 A B A B µ ja µ jb his is valid al when the system consists of several hases Here U α U α U 0 α crendingly f other quantities he variations of artial derivatives st f 0 α U α 2 S U 2 + j 0 [ j [ ] 0 U α + U ] 0 N jα U U, similarly f other artial derivatives he 1st der differentials dro out, because α U α 0 hus, S α { 0 1 α α U α + 2 U α α + 0 α N jα } j jα U, his can be rewritten as 0 U,N U,N S α { 1 1 α U α + α 2 α α } N jα j µjα α α α 13

14 8 5 Using the first law we get { S 1 2 α α S α + α α j µ jα N jα } his can be further written as { S 1 C 2 α κ α [ α 2 N α ] 0 µ + N α }, 2 where α Nα Since S 0, we must have, 0 0 α + α N, N, C 0, κ 0, µ 0 he condition C 0 is a condition f thermal stability: if a small excess of heat energy is added to a volume element of fluid, the temerature of the volume element must increase he condition κ 0 is a condition f mechanical stability: if a volume of a small fluid element fixed N increases, the ressure must go down, that the larger ressure from the environment halts reverses the growth Likewise µ 0 is a condition f chemical stability Recall that C 2 2 > 0 hus, is a concave function of Similarly, 1 2 κ,n 2,N is a convex function of 3 Alications of thermodynamics 31 Classical ideal gas a full descrition of the thermodynamics of a system we need to know both the equation of state me thermodynamic otential EOS gives us mechanical rense functions, but f thermal rense functions we need al me otential rom the ideal gas equation of state Nk B we obtain directly the mechanical rense functions α 1 Nk B,N 1 κ 1 Nk B 2 1,N hermal rense functions cannot be derived from the equation of state Emirically it has been observed C 1 2 fk BN Here 1 2 fk B is the secific heat caacity/molecule f is the number of degrees of freedom of the molecule Atoms/molecule f translations rotations olyatomic 3 3 real gases f f,, which is different from ideal gas because of internal degrees of freedom vibrations, interactions between the molecules quantum mechanical effects Entroy Entroy can be obtained from thermal mechanical rense functions by integrating along the trajecty 8 8 he differential is ds d + 1 C d + since accding to Maxwell relations Integrating we get 5 d, d S S d C + d Nk B 0 S 0 + C ln 0 + Nk B ln 0 14

15 * [ ] f/2 S S 0 + Nk B ln Note: A contradiction with the 3rd law: S, when 0 3rd law relies on the quantum nature of real matter! 0 Internal energy We substitute into the first law N const the differential get ds du ds d d + [ du C d + 0 d, ] d Accding to Maxwell relations to the equation of state we have Nk B, du C d U U 0 + C 0 U fk BN 0 If we choose U 0 C 0, we get f the internal energy U 1 2 fnk B C C + α2 κ C Nk B 1 2 f + 1 C γc, where γ is the adiabatic constant γ C /C f + 2/f 32 ree exansion of gas 8, 3, 9 In the rocess 1 2 Q W 0, U 0 Process is irreversible a Ideal gas U 1 2 fk BN, 1 2, because U 1 U 2 he cange in the entroy is thus S Nk B ln 2 1 b Real gas equation of state he internal energy the number of articles are constant: du U d + U d 0 he Joule coefficient characterizes the behaviour U,N of the gas during free exansion cf Joule-homn coefficient: U,N U U 1 C α κ Note that this is differential fm ; with a finite rocess one must integrate over differential changes 33 Mixing entroy Conside different gases A B, searated by a artition: We assume that initially A B A B he artition is removed the gases mix he rocess is irreversible is adiabatic Q 0 In a mixture of ideal gases every comonent satisfies the state equation j N j k B * * he concentration of the comonent j is where the total ressure is x j N j N j, j Method 1: Each constituent gas exs to volume Since A B A B, we have j x j he change in the entroy is see the free exansion of a gas j S j N j k B ln j 15

16 S mix Nk B x j lnx j S mix 0, since 0 x j 1 Method 2: a rocess taking lace in constant ressure temerature the Gibbs function is the suitable otential: where G Befe mixing U S fk BN S + Nk B [φ + ln] Nµ,, φ µ0 k B ξ f 2 + 1ln G b j j N j k B [φ j + ln] Examle: j A B C D M j H 2 S O 2 H 2 O SO 2 ν j We define the degree of reaction ξ that dn j ν j dξ When ξ increments by one, one reaction of the reaction fmula from left to right takes lace Convention: When ξ 0 the reaction is as far left as it can be hen ξ 0 Let us assume that remain constant during the reaction hen a suitable otential is the Gibbs function G j µ j N j after mixing G a j N j k B [φ j + ln j ], Its differential is dg j µ j dn j dξ j ν j µ j the change in the Gibbs function is Because G mix G a G b j j S we get f the mixing entroy N j k B lnx j G, P,{N j} S mix S a S b j N j k B ln j N j k B lnx j Gibbs aradox: If A B, ie the gases are identical no changes take lace in the rocess However, accding to the fmer discussion, S > 0 he rean is that in classical ensemble the articles are distinguishable, mixing of A B really haens In quantum mechanics this aarent contradiction is removed by emloying quantum statitics of identical articles 34 Chemical reactions Consider f examle the chemical reaction 2 H 2 S + 3 O 2 2 H 2 O + 2 SO 2 In general the chemical reaction fmula is written as 0 j ν j M j We define r G G ξ, j ν j µ j r G is thus the change in the Gibbs function er one reaction often called the affinity Since, is constant G has a minimum at equilibrium he equilibrium condition is thus r G eq j ν j µ eq j 0 In a nonequilibrium dg/dt < 0, if r G > 0 we must have dξ/dt < 0, ie the reaction roceeds to left vice versa Let us assume that the comonents obey the ideal gas equation of state hen µ j k B [φ j + ln + lnx j ], where j x j is the artial ressure of comonent j So r G k B j φ j µ0 j k B η j f jln ν j φ j + k B ln νj x ν j he equilibrium condition can now be written as j x νj j j νj K, j Here ν j I are the stochiometric coefficients M j st f the molecular secies where K e j νjφj 1

17 is the equilibrium constant of the reaction, which deends only on he equilibrium condition is histically called the law of mass action the reaction above x 2 C x2 D x 2 A x3 C K he heat of reaction is the change of heat energy r Q er one reaction to right A reaction is Endothermic, if r Q > 0 ie the reaction takes heat Exothermic, if r Q < 0 ie the reaction releases heat We write r G as r G k B lnk + k B j ν j lnx j Q U + W U + U + H, since 0 When the total amount matter is constant dg S d + d holds in a reversible rocess G d 1 dg G G 2 d 2 + S d + d H 2 d + d, because G H S We see that [ ] G H 2 r G,N k B d d lnk, that r H k B 2 d d lnk his exression is known as the heat of reaction 35 Phase equilibrium In a system consisting of several hases the equilibrium conditions f each air A B of hases are A B A B µ ja µ jb, j 1,,H, where H is the number of article secies in the system Let us assume that the number of hases is, f each secies there are 1 indeendent conditions µ iα µ iα,, {N jα } Because the chemical otential is an intensive quantity it deends only on relative fractions, µ jα µ jα,, x 1α,,x H 1,α, the conditions take the fm here are µ 1A,, x 1A,, x H 1,A µ 1B,, x 1B,, x H 1,B µ HA,, x 1A,,x H 1,A µ HB,, x 1B,,x H 1,B M H variables, Y H 1 equations he simultaneous equations can have a lution only if M Y H + 2 his condition is know as the Gibbs hase rule ure matter the equilibrium condition µ A, µ B, defines in the, -lane a coexistence curve If the hase B is in equilibrium with the hase C we get another curve µ B, µ C, he hases A, B can C can be simultaneously in equilibrium in a crossing oint, called trile oint, of these curves 3 Phase transitions In a hase transition the chemical otential G µ is continuous Instead S, G G are not necessarily continuous A transition is of first der, if the first der derivatives S, of G are discontinuous, of second der, if the 1st der derivatives are continuous but 2nd der discontinuous Otherwise the transition is continuous 17

18 + In a first der transition from a hase 1 to a hase 2 S G G 2 2 G + G 1 1 When we cross a coexistence curve stay constant, Q S U + U + H Q is called the hase transition heat the latent heat Note: irst der transitions have non-zero latent heat but not the higher der ones yical hase diagram of lid-liquid-gas -system: lines are 1st der transitions, critical oint is 2nd der trile oint is 1st der Water: 10Pa, 001 C; c 22 MPa, c C lid trile oint lid liquid gas liquid trile oint critical oint critical oint gas coexistence const itherm Because of hase coexistence, hase diagrams are simlest in fce -tye codinates,, µ, H, 37 Phase coexistence On the coexistence curve? A N E I J A? A? K H L A G 1,, N G 2,, N dg S d + d when the number of articles N is constant Along the curve G 1 + d, + d, N G 2 + d, + d, N, that on the curve S 1 d + 1 d S 2 d + 2 d d d S 2 S 1 S H we end u with the Clausius-Claeyron equation d 1 d coex H Here H H 2 H Examles a aour ressure curve We consider the transition Assuming ideal gas we have B K I E? K H L A? H E J E? E J I B K I K > E J E L H? K H L A L H H A I I K H A J H E A E J? K H L A liquid vaour v Nk B because liquid va, d H lv d Nk B 2 coex If the vaourization heat the latent heat H lv is roughly constant on the vaour ressure curve we can integrate 0 e H lv /NkB this assumtion is not true near crit oint! b usion curve ls liquid lid can be ositive negative f examle H 2 O Accding to the Clausius-Claeyron equation we have d d H ls ls d d > 0, if ls > 0 1 d d < 0, if ls < 0 2 @ B K @ B K 18

19 + I 8 8 We see that when the ressure is increased in constant temerature the system 1 drifts deeer into the lid hase, 2 can go from the lid hase to the liquid hase c Sublimation curve dh ds + d C d + 1 α d, because S S, using Maxwell relations definitions of thermodynamic rense functions ds d + d d + C d he vaour ressure is small d 0, H s Hs 0 + C s d H v Hv 0 + C v d 0 0 lid hase vaour gas Let us suose that the vaour satisfies the ideal gas state equation hen vs Nk B s Nk B d d H vs H vs vs Nk B 2, where H vs H s H v a monatomic ideal gas C 5 2 k BN,, E I J D A H * In many equations of state the hase transition haens when there is aarent instability d/d > 0 f examle, van der Waals In this case, we can use Maxwell s construction: he oints A B have to be chosen that the area I area II ln 0 H0 vs Nk B +5 2 ln 1 k B N 0 Cs d d+constant 2 Here H 0 vs is the sublimation heat at vanishing temerature ressure Coexistence range E I J D A H A H? * L A H D A J Matter is mechanically stable only if d d < 0 hus the range of stability lies outside of the oints A B Overheated liquid undercooled vaour are metastable suercooling, -heating Accding to the Gibbs-Duheim relation consider dg! we have along itherms dµ S N d + N d dµ N d hus, when the hases A B are in equilibrium, µ A µ B B A d 0 N 19