Foundations of thermodynamics Fundamental thermodynamical concepts

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1 Foundations of thermodynamics Fundamental thermodynamical concets System is the macrohysical entity under consideration Surrounding is the wld outside of the system Oen system can exchange matter heat with the surrounding Closed system can exchange heat with the surrounding while keeing the number of articles constant Ilated system can exchange neither matter n heat with the surrounding hermodynamical equilibrium No macroscoical changes Uniquely described by external variables of state System fgets its ast; no hysteresis In global equilibrium all arts of the system are in the same state Nonequilibrium F examle, ilated systems each in an equilibrium state In a local thermodynamical equilibrium semimicroscoical regions are in an equilibrium, neighbour regions in different equilibria articles, heat will flow From stronger nonequilibria the system usually relaxes to a local equilibrium Degree of freedom is the number of quantities needed f the exact descrition of the microscoic state number of articles State variables are arameters characterizing the macroscoic state Extensive variable is rotional to the quantity of the substance; eg volume, article number N, internal energy U, entroy S, magnetization M dr m, where m is magnetic moment/volume Intensive variable is indeendent on the quantity of the substance can be determined f every semimicroscoical volume element ; eg temerature, ressure, chemical otential µ, magnetic field H, ratios of extensive varialbles like ρ N/, s S/N, Conjugated variables A B aear in airs in exressions f the differential of the energy, ie in fms ±A db ±B da; the one is always extensive the other intensive Process is a change in the state Reversibel rocess advances via states infinitesimally close to equilibrium, quasistatically he direction of a reversible rocess can be reversed by infinitesimal changes of external variables Ithermic rocess : constant Ibaric rocess : constant Ichic rocess : constant Isentroic adiabatic rocess: S constant Irreversibel rocess is a sudden ntaneous change during which the system is far from equilibria In the intermediate stes global state variables,, are not usually defined Cyclic rocess consists of cycles which take the system every time to its initial state State variables exact differentials Let us suose that, f examle,, ja N tell uniquely the state of the system State variables are then their unique functions:,, N U U,, N S S,, N In an infinitesimal change state variables transfm like d d + d + dn,n, he differentials of unique functions, d, d, d,, are exact differentials: their total change evaluated over a closed ath vanishes: 1 2 d 1 2 du 0 he total change of an exact differential is indeendent on the ath of integration 1

2 > a du du 0, b U2 U du Let us denote by df a differential which is not necessarily exact he differential is exact if df F 1 x, y dx + F 2 x, y dy F 1 y F 2 x hen F x, y that F 1 x, y F 2 x, y F x,y y 2 1 F x,y x df F 2 F 1 is indeendent on the ath We say that df df is integrable If df F 1 dx + F 2 dy is not exact, there exists an integrating fact λx, y that in the neighbourhood of the oint x, y λ df λf 1 dx + λf 2 dy df is an exact differential Legendre transfmations can be used to make changes in the set of the indeeendent state variables F 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 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 f g zg z g + yz Often needed identities Let F F x, y, x xy, z, y yx, z z zx, y hen F x z x y z 1 F x x y F y F x z z z y F + y y z x x z x y x Equations of state State variables of an equilibrium system are related by a state equation which, in most cases, is a relation between thermal variables S mechanical variables Examles: Classical ideal gas Nk B N number of molecules k B J/K Boltzmann constant Chemists use often the fm nr n N/N 0 number of moles R k B N J/K mol gas constant N Avogadro s number If the gas is comosed of m different secies of molecules the equation of state is still where now Here Nk B, N i m i1 N i i i N i k B / is the artial ressure of the i:th gas irial exansion of real gases where k B [ ρ + ρ 2 B 2 + ρ 3 B 3 + ], ρ N/ article density B n is the n:th virial coefficient an der Waals equation he molecules of real gases interact y z 2

3 reulsively at sht distances; every article needs at least the volume b > Nb attractively 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 We imrove the ideal gas state equation that then Nb Nk B aρ 2 true ressure + aρ 2 Nb Nk B 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 Stretched wire ension [N/m 2 ] α κ κ /Pa α 10 4 /K σ EtL L 0 /L 0, where L 0 is the length of the wire when σ 0 Et 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 ɛ 0 E + P, where P electric olarization atomic total diole moment/volume D electric flux density ɛ As/m vacuum ermeability In homogenous dielectric material one has P a + b E, 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 B µ 0 H + m, m magnetic olarization atomic total magnetic moment/volume B magnetic flux density µ 0 4π 10 7 s/am vacuum ermeability Polarization obeys roughly Curie s law 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 0th law 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 Wk Wk is exchange of such noble energy 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 surrounding Examle system W 3

4 Note dw is not an exact differential Instead 1 d dw is exact, ie, 1/ is the integrating fact f wk Examle dw d σa dl E dp H dm In general dw f i dx i f dx, i where f i is a comonent of a generalized fce X i a comonent of a generalized dislacement 1st law In addition to wk a system can exchange thermal energy, ie heat with its surroundings hermal energy is related to the energy of the thermal stochastic motion of microscoic articles he total energy of a body is called internal energy Sign conventions: he total change of heat is Q Q + + Q, where Q + is the heat taken by the system Q the heat released by the system he efficiency η is 2nd law η W Q + Q+ + Q Q + 1 Q Q + a Heat cannot be transferred from a cooler heat reservoir to a warmer reservoir without any other consequences b In a cyclic rocess it is not ossible to convert all heat taken from the hotter heat reservoir to wk c It is not ossible to reverse the evolution of a system towards thermodynamical equilibrium without converting wk to heat Q d he change of the total entroy of the system its surroundings is ositive can be zero only in reversible rocesses e Of all the engines wking between the temeratures 1 2 the Carnot engine has the highest efficiency W µ N chemical energy Due to the energy conservation law the change of the internal energy satisfies du dq f dx + i µ i dn i We consider the infinitesimal rocess y U1 dq 1 U2 2 U is a state variable, ie du is exact In a cyclic rocess du 0, W Q no change in chemical energy In a -system x Q + W d dq du + dw du + f dx, there exists an integrating fact 1/ that 1 dq ds Q is exact he state variable S is entroy turns out to be the called ablute temerature he second law d can now be written as ds tot dt 0 4

5 @? > F arbitrary rocesses we have ds 1 dq, where the equality holds only f reversible rocesses F reversible rocesses the first law can be rewritten as du dq dw + µ dn ds d + µ dn Carnot s cycle he Carnot cycle C consists of reversible rocesses a ithermic 2 Q 2 > 0 b adiabatic 2 1 Q 0 c ithermic 1 Q 1 > 0 d adiabatic 1 2 Q 0 U 0, W Q 2 Q 1 We define the efficiency as, 3, 3 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 engine B a heat um he efficiences are crendingly Let us suose that η A W Q A η B W Q B Q A Q B W A B QB- W Q A -W η A > η B, 2 > 2 1 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 a So we must have η A η B 1 Similarly one can show that η B η A, that η A η B, ie all Carnot engines have the same efficiency Note he efficiency does not deend on the realization of the cycle eg the wking substance he efficiency deends only on the temeratures of the heat reservoirs Similarly, one can show that the Carnot engine has the highest efficiency among all engines al irreversible wking between given temeratures Let us consider Carnot s cycle between temeratures 3 1 η 1 f 3, 1, where Here! f 3, 1 Q 1 Q 3, 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 η he Carnot cycle satisfies dq 0, since c a dq Q 2 2 dq Q 1 1 Q 2 2 his is valid al f an arbitrary reversible cycle 5

6 because So C + + E dq i C i dq 0 3rd law Nernst s law: lim S 0 0 A less strong fm can be stated as: When the maximum heat occuring in the rocess from a state a to a state b aroaches zero then al the entroy change S a b 0 Note here are systems whose entroy at low temeratures is larger than true equilibria would allow ds dq is exact the entroy S is a state variable 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 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 irr H A L Accding to the fmula we have dq dq < 0, his is usually written as rev S > irr dq ds dq the equality is valid only f reversible rocesses In an ilated system we have S 0

7 F 8 hermodynamic otentials Fundamental equation Accding to the first law du ds d + µ dn S, N are the natural variables of the internal energy U, ie U US,, N Furtherme, from the law one can read the relations S, µ U, S, N are extensive we have UλS, λ, λn λus,, N λ Let S S + ɛs, + ɛ N N + ɛn, when ɛ is infinitesimal hen US + ɛs, + ɛ, n + ɛn US,, N+ ɛs + ɛ + S, ɛn On the other h, accding to the equation we have US + ɛs, + ɛ, N + ɛn US,, N + ɛus,, N We end u with the Euler equation f homogenous functions U S + + N S, Substituting the artial derivatives this takes the fm U S + µn S 1 U + µn his is called the fundamental equation Internal energy Maxwell relations Because, Similar relations can be derived al f other artial derivatives of U we get called Maxwell s relations S, S, 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, F 8 A G K E E > H E K F I E J E We artition W into the comonents [ wk due to the d change of the [ volume wk done by the W free gas against the fce F W free W 1 + W A L F 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 F L, ] ] 7

8 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 Enthaly W free U Using the Legendre transfm U H U U + We move from the variables S,, N to the variables S,, N he quantity is called enthaly H U + dh du + d + d ds d + µ dn + d + d dh ds + d + µ dn From this we can read the artial derivatives H,N H H µ S, Crending Maxwell relations are S, S, In an irreversible rocess one has,n,n Q U + W µ N < S U H, that We see that H < S + + µ N 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 finete 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 ie W free is the minimum wk required f the change H Note An other name of enthaly is heat function in constant ressure Joule-homn henomenon P 1 1 choke Q0 P are temal constants, 1 > 2 the rocess irreversible When an infinitesimal amount of matter asses through the choke the wk done by the system is dw 2 d d Initial state init 0 Final 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 8

9 hus in this rocess the enthaly H U + is constant, ie the rocess is isenthalic, H H final H init 0 We consider now a reversibel isenthalic dn 0 rocess init final Here S,, that d On the other h dh ds + d 0, ds d ds +, C d S 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 F ideal gases H 0 holds F real gases is below the inversion temerature ositive, the gas cools down Free energy he Legendre transfm U F U S F U S defines the Helmholtz free energy df S d d + µ dn, the natural variables of F are, N We can read the artial derivateves F S F,N F µ, From these we obtain the Maxwell relations,n,, In an irreversible change we have,n F < S + µ N, ie when the variables, N are constant the system drifts to the minimum of the free energy Crendingly W free F, when,, N is constant Free energy is suitable f systems where the exchange of heat is allowed Gibbs function he Legendre transfmation U G U S defines the Gibbs function the Gibbs free energy G U S + H 9

10 Its differential is dg S d + d + µ dn, the natural variables are, N F the artial derivatives we can read the exressions G S,N G,N G µ, From these we obtain the Maxwell relations,n,n,,n, 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 Gr otential he Legendre transfm 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, S, We get the Maxwell relations,µ,, In an irreversible rocess,µ,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 heat, 8 he exchanged roerty is the volume a crending generalized dislacement; f examle magnetization in a magnetic field Heat bath Particle ath Baths can al be combined; f examle a suitable otential f a ressure heat bath is the Gibbs function G, 5,, 3, 5 hermodynamic renses 1 olume heat exansion coefficient α 1 α 1 ρ,n ρ,,n 10

11 where ρ N/ 2 Ithermic 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 In a reversible rocess we have Q S he heat caacity C is defined that C Q S In constant volume we define C In constant volume the number articles being fixed, accding to the first law we can write du ds d + µ dn ds, C 5 Ibaric heat caacity C Because one can write,n dh ds + d + µ dn, C H,N a Maxwell relation Since,, +, C C + 1 C C + α2 κ α κ, hermodynamic equilibrium conditions We divide the system into fictitious semimicroscoic arts: Extensive variables satisfy F 8, 7, 8, E S α α U α N j α S α α U α N jα 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 We suose that the system is comosed of two arts: α {A, B} hen U B U A, B A N jb N ja 11

12 S S α α 1 1 A j B µja A A U A + µ jb B 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 Stability conditions of matter In a steady equilibrium the entroy has the true maximum that small variations can only reduce the entroy We denote the equilibrium values common f all fictitious arts by the symbols, {µ j } the equilibrium values of other variables by the suerscrit 0 We write the entroy S α of the fictious artial system α close to an equilibrium as the ayl series S α U α, α, {N jα } SαU 0 α, 0 α 0, {Njα} U α + α α α U,N + 0 N jα j jα U, { U α + 2 α α + } 0 N jα j jα U, + U,N α Here U α U α U 0 α crendingly f other quantities he variations of artial derivatives st f 0 α 2 S 2 + j 0 [ j [ ] 0 U α + ] 0 N jα U, U,N α In an equilibrium , j S α { 0 1 U α + 2 α α + 0 N jα } j jα U, his can be rewritten as 0 U,N S α { 1 1 α U α + α 2 α α } N jα j Using the first law we get { S 1 2 α µjα α α S α + α α j µ jα N jα } α his can be further written as { S 1 C 2 α κ α [ α 2 N α ] 0 + N α }, 2 where α Nα Since S 0, we must have, 0 0 α + α N, N, C 0, κ 0, 0 similarly f other artial derivatives 12

13 8 8 5 Alication of thermodynamics Classical ideal gas From the equation of state Nk B we obtain 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 we have 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 oly 3 3 F real gases f f, Entroy ds d + 1 C d + since accding to Maxwell relations Integrating we get 8 S S d, d d C + d Nk B 0 Internal energy We substitute into the firs law the differential get ds du ds d d + [ du C d + d, ] d Accding to a 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 fk BN C C + α2 κ 1 C Nk B 2 f + 1 C γc, where γ is the adiabatic constant γ C /C f + 2/f Free exansion of gas 8, 3, 9 S 0 + C ln 0 + Nk B ln 0 [ ] f/2 S S 0 + Nk B ln Note A contradiction with the third law: S, when In the rocess 1 2 Q W 0, U 0 Process is irreversible a Ideal gas U 1 2 fk B N, 13

14 * 1 2, because U 1 U 2 he cange in the entroy is thus S Nk B ln 2 1 b Other material he internal energy the number of article being constant du 0 we obtain from the exression du d + f the Joule coefficient U,N U,N 1 C d the fm α κ F a rocess taking lace in constant ressure temerature the Gibbs function is the suitable otential: where Befe mixing after mixing G U S fk B N S + Nk B [φ + ln ] Nµ,, φ µ0 k B ξ f + 1 ln 2 G b j G a j N j k B [φ j + ln ] N j k B [φ j + ln j ], Mixing entroy the change in the Gibbs function is G mix G a G b j N j k B ln j F * F * j N j k B ln x j Suose that initially A B A B he rocess 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 Way 1 Each constituent gas exs in turn into the 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 Because S we get f the mixing entroy G, P,{N j} S mix S a S b j N j k B ln x 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 contradiction can be removed by emloying quantum statitics of identical articles Chemical reaction Consider f examle the chemical reaction 2 H 2 S + 3 O 2 2 H 2 O + 2 SO 2 In generall the chemical reaction fmula is written as S j N j k B ln j 0 j ν j M j S mix Nk B x j ln x j S mix 0, since 0 x j 1 Way 2 j Here ν j I are the stoichiometric coefficients M j st f the molecular secies Examle j A B C D M j H 2 S O 2 H 2 O SO 2 ν j

15 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 we always have ξ 0 We suose that are constant in the reaction hen a suitable otential is the Gibbs function Its differential is We define G U S + j dg j r G r µ j dn j dξ j G ξ, j µ j N j ν j µ j ν j µ j r is thus the change in the Gibbs function er one reaction Since, is constant G has a minimum at an equilibrium he equilibrium condition is thus r G eq j ν j µ eq j 0 In a nonequilibrium dg/dt < 0, if r > 0 we must have dξ/dt < 0, ie the reaction roceeds to left vice versa We assume that the comonents obey the ideal gas equation of state hen where So µ j k B [φ j + ln + ln x j ], φ j r G k B j µ0 j k B η j f j ln ν j φ j + k B ln νj x ν j he equilibrium condition can now be written as where j x νj j j νj K, K e j ν jφ j j is the equilibrium constant of the reaction he equilibrium condition is called the law of mass action he reaction heat is the change of heat 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 ln K + k B j ν j ln x 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,N r G d k B ln K, d that r H k B 2 d ln K d his exression is known as the reaction heat 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, f each secies there are F 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 15

16 F F F + M H 1F + 2 variables, Y HF 1 equations he simultaneous equations can have a lution only if M Y F H + 2 his condition is known as the Gibbs hase rule F 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 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 second der derivatives are discontinuous Otherwise the transition is continuous 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 First der transitions are asciated with the heat of hase transitions but not the higher der transitions 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 that G 1 + d, + d, N G 2 + d, + d, N, 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 a ressure curve We consider the transition B K I E? K H L A? H E J E? F E J I B K I K > E J E L F H? K H L A L F H F H A I I K H A J H E F A F E J? K H L A liquid va Suosing that we are dealing with ideal gas we have since v Nk B liquid va, 1

17 F F F F + I 8 8 Because the vaization heat the hase transition heat H lv is roughly constant on the va ressure curve we have d d H lv Nk B 2 Integration gives us 0 e H lv /Nk B b Fusion 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 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, since S S, using Maxwell relations definitions of thermodynamic rense functions ds d + d d + C d he va ressure is small d 0, d d H vs vs H vs Nk B 2, where H vs H s H v F a mono atomic ideal gas C 5 2 k BN, that ln H0 vs Nk B +5 2 ln 1 k B N 0 Cs d d +constant Here H 0 vs is the sublimation heat at 0 temerature ressure Coexistence range E I J D A H * L A H D A J A H? Matter is mechanically stable only if d d < 0 hus the range of stability lies outside of the oints A B Overheated liquid undercooled va are metastable Accding to the Gibbs-Duhem relation we have on an itherm 2 dµ S N d + N d dµ N d hus, when the hases A B are in equilibrium, µ A µ B E I J D A H B A * d 0 N Maxwell s construction: he oints A B have to be chosen that the area I area II H s H 0 s + H v H 0 v C s d C v d lid hase va gas Let us suose that the va satisfies the ideal gas state equation hen vs Nk B s Nk B, 17