Pchem 365: hermodynamcs -SUMMARY- Uwe Burghaus, Fargo, 5 9 Mnmum requrements for underneath of your pllow. However, wrte your own summary! You need to know the story behnd the equatons : Pressure : olume n: partcle number trp : temperature of trple pont of water R: Gas constant α: phase of a system? : standard value of a therm. functon ( bar, and typcally at room temperature X: mole fracton P c : pressure at crtcal pont Be aware of typos. IDEAL GAS (no nteractons no volume of the speces (More realstc equatons such as the van der Waals equaton nclude the nteracton of the partcles and there volume. Boyl s law P for n & constant for n & P constant Gay-Lussac & Charles laws R S p for n & constant Avogadro s law n for P & constant Equal volumes of dfferent gases have equal numbers of partcles f and p are constant. Or, p,, const (.e. are the same for two dfferent gases than the number of partcles n s the same. General deal gas equaton P nr Dalton s law Ptot P + P + P+ 3... A reversble process s one where a system s always nfntesmally close to equlbrum, and an nfntesmal change n condtons can reverse the process to restore both system and surroundngs to ther ntal states. Extensve propertes are equal to the sum of ther values (e.g. volume, mass, heat, c p. Intensve propertes do not depend on the amount of matter (e.g. temperature, densty.
HERMODYNAMIC FUNCIONS mechancal work pressure work heat capactes Enthalpy adabatc expanson (Joule-homson, Lnde apparatus for lquefyng ar, fast expanson of a gas, open suddenly a gas cylnder wthout a regulator don t do that at home z r r w F( dr L r r z z w dw Pd tot L L c dq dq ( ; c ( d d P P const const H U c ( ; c ( c c nr (deal gas P P const const R S P H: U + P H q for P constant U q for constant compare p const (Boyl' s law wth p const (Posson eq / γ γ ( ; P P ; γ R c C P C γ state functons (e.g. U, Cp, Cv z L db b b b; db z non state functons L dq q b s ndependent of the way from ntal ( to fnal ( state z (e.g. q, w
Entropy ds dq ds H ; ( & P const (What s dfference between ds and C p? Gbbs equaton dg dp Sd + µ α dn α k α Important defntons: Enthalpy Helmholz free energy Gbbs energy H U + P A U S G H S 3
Carnot cycle Cycles: x sothermal and x adabatc work output q η + heat nput q Pressure P olume --> sothermal --> 3 adabatc 3 --> 4 sothermal 4 --> adabatc heat work q q w -q -R ln( / q 3 w 3 -c ( - q q 34 w 34 -q -R ln( 4 / 3 q 4 w 4 -c ( - Note that the equaton w 3 -c for an adabatc step s very useful for analyzng these cycles (see above and try to remember that. (Why?: U c for an deal gas that s bascally the defnton of an deal gas-, st law U w + q, adabatc step q. 4
MAERIAL EQUILIBRIUM dsσ dssystem + dssurroundng reversble process > rreversble A:U-S (Helmholts free energy R S U W G:H-S (Gbbs free energy cf., maxmum work for const process (Arbetsfunkton thermodynamc work functon da at equlbrum for & constant cf., maxmum work for P const process dg at equlbrum for &P constant chemcal potental G ( α µ α α n const P const all mole numbers except n const α j phase equlbrum k µ α α dn α α b µ µ µ of speces s the same n every phase that contans In approachng the equlbrum: flows from phase wth larger µ to phase wth lower µ. µ µ + R ln(p /P (deal gas & P const or @ & const reacton equlbrum ν A+ ν A+... ν mam + ν m+ Am+ +... νµ 5
PHASE DIAGRAMS Phase rule: f pc+ p c( p r a Isotherms of H O f: degree of freedom; constrants: number of p: phases; c: speces; r: reactons; a: other restrctons such as stochometry, conservaton of charge Example: H O sold phase regon (f ; lqud-gas equlbrum (f. > c densty (lqud densty (gas H m vap for c Clausus-Clapeyon dln P d H ; R ln( P Hm P R ( Metastable phase G α > G β leads to a slow phase change α β example: damond graphte Phase transttons (Ehrenfest classfcaton st µ order: s dscontnuous nd µ µ order: s dscontnuous and s dscontnuous 6
SANDARD HERMODYNAMIC FUNCIONS ν A+ ν A+... ν mam + ν m A Standard enthalpy of formaton + m+ ν A H ν Hf Def.: H f s the enthalpy change for sothermally ( const converson of pure elements n ther reference forms (typcally ther most stable forms to a standard-state substance. Standard state s typcally atm & 5C. e.g. C(graphte + H (deal gas+"o(deal gas" H CO(d. gas H H ( H CO H ( graphte + H ( H deal gas + H ( oxygen d. gas f f f f Hess s law: elements + O combuston product desred molecule + O We need f H for a product that cannot be syntheszed from ts elements. H s a state functon,.e., we can add chemcal equatons and the correspondng f H together. Krchhoff s law (problem we know H ( we need H ( H H C d p z Entropy S ν S,, m ν m,, Gbbs energy G G 7
th law Defnton of the temperature. lm ( P; (P P P LAWS OF HERMODYNAMICS P trp trp (Gas thermometer st law E q+ w; system at rest: U q+ w nd law A nd order perpetuum moble does not exst. It s mpossble to buld a cyclc machne whch converts heat nto work wth an effcency of %. he total entropy s ncreasng n any rreversble process S. 3 rd law Part A: G lm ( P ; lm S K K (Leads to the defnton of a standard entropy. Part B: Unattanablty prncple: he absolute zero temperature Kelvn cannot be reached. GAS MIXURES. P Chemcal potental µ µ + R ln( Equlbrum constant G Rln( Kp wth Kp: K p ( P / P ; ν products reactants P ( P / P j ν e G / R van t Hoff equaton dln( Kp H d R Kp ( ln( K ( H R ( p 8
SOLUIONS Partal molar volume otal volume j (,, ( n j P n j const n j j (obeyed or all extensve propertes of a soluton j Important equatons mxg P mx mxg ; mx S pure Ideal soluton def. µ µ (, P + Rln( x S R n ln( x ; G R n ln( x ; mx mx H ; U mx Raoult s law he rato of the partal vapor pressure of each component of the soluton to the correspondng vapor pressure of the pure lquds s equal to the mole fracton of that component n the lqud mxture. mx vapor pressure of A n the soluton vapor pressure of pure A mole fracton of A P X P lqud pure Ideally dlute soluton X solvent µ µ (, P + Rln( x solute solute solute solvent solvent +R ln( xsolvent µ µ Henry s law P solute Constsolute X lqud solute (Raoult s law s obeyed for the solvent 9
useful but not necessarly to memorze: Gbbs equatons hermal expanson coeffcent α dh ds + dp da Sd Pd dg Sd + dp ( & P const P ( m α κ Isothermal compressblty κ ( & (for an deal gas α P n const κ P ; Joule homson experment (adabatc expanson of a gas, H µ ( p a Rb H ( C p P R S > < for < "coolng" for > "heatng" (nverson temperature, for He well below room temperature Maxwell Relatons ( ( P ( ( S P S S ( ( P S ( ( P S S P P
Mxng entropy S n Rln( x ; mx n: mole fracton he entropy of an deal gas mxture s equal to the sum of the entropes each pure gas would have f t alone occuped the volume of the mxture (at the temperature of the mxture. Follows bascally from Daltons law (P tot P + P + P 3 +... REAL GASES an der Waals equaton ( P an + ( nb nr pm m Z( P, Compresson factor deal R m
deal gas real gas p nr p ZR Z( P, : compresson factor 3 ral equaton P R( + B / + C / + D / +... Maxwell constructon Law of correspondng states: If we defne reduced parameters (e.g. P r P/P c then r equals approxmately f(p r, r for all gases.