Lecture 13. hermodynamic Potential (Ch. 5) So far we have been uing the total internal energy U and ometime the enthalpy H to characterize variou macrocopic ytem. hee function are called the thermodynamic potential: all the thermodynamic propertie of the ytem can be found by taking partial derivative of the P. For each P a et of o-called natural variable exit: d U d S P d + μ d d H d S + dp + μ d oday we ll introduce the other two thermodynamic potential: thehelmhotzfree energy F and Gibb free energy G. Depending on the type of a proce one of thee four thermodynamic potential provide the mot convenient decription (and i tabulated). All four function have unit of energy. Potential U (S) H (SP) F () G (P) ariable S S P P When conidering different type of procee we will be intereted in two main iue: what determine the tability of a ytem and how the ytem evolve toward an equilibrium; how much work can be extracted from a ytem.
Diffuive Equilibrium and Chemical Potential For completene let recall what we ve learned about the chemical potential. d U U A A S A For ub-ytem in diffuive equilibrium: P μ d S P d + μ d d S 1 d U + d d U S S U A A A A he meaning of the partial derivative (S/) U : let fix A and (the membrane poition i fixed) but aume that the membrane become permeable for ga molecule (exchange of both U and between the ubytem the molecule in A and are the ame ). 0 S A A U A A 0 S A A S S U μ S μ S U U S - the chemical potential A In equilibrium A μ A μ P A P U A Sign - : out of equilibrium the ytem with the larger S/ will get more particle. In other word particle will flow from from a high μ/ to a low μ/.
Chemical Potential of an Ideal ga S μ U U μ ha unit of energy: it an amount of energy we need to (uually) remove from the ytem after adding one particle in order to keep it total energy fixed. S Monatomic ideal ga: S( U ) k 4π m ln U 3h ln 3/ 5/ + 5 μ S k π m ln k h ln U 3/ k h 3 P ( ) ( ) 3/ 5/ πm k At normal and P μ for an ideal ga i negative (e.g. for He μ ~ - 5 10-0 J ~ - 0.3 e). Sign - : by adding particle to thi ytem we increae it entropy. o keep ds 0 we need to ubtract ome energy thu ΔU i negative. he chemical potential increae with with it preure. hu the molecule will flow from region of high denity to region of lower denity or from region of high preure to thoe of low preure. μ 0 μ when P increae ote that μ in thi cae i negative becaue S increae with n. hi i not alway the cae. For example for a ytem of fermion at 0 the entropy i zero (all the lowet tate are occupied) but adding one fermion to the ytem cot ome energy (the Fermi energy). hu μ 0 E > ( ) 0 F
he Quantum Concentration At 300K P10 5 Pa n << n Q. When n n Q the quantum tatitic come into play. 3/ k h m n Q π - the o-called quantum concentration (one particle per cube of ide equal to the thermal de roglie wavelength). When n << n Q (In the limit of low denitie) the ga i in the claical regime and μ<0. When n n Q μ 0 3/ 3 1 h mk n mk h p h d Q d λ λ Q n n k mk h n k k h m k ln ln ln 3/ 3/ π π μ where n/ i the concentration of particle
Iolated Sytem independent variable S and Advantage of U : it i conerved for an iolated ytem (it alo ha a imple phyical meaning the um of all the kin. and pot. energie of all the particle). In particular for an iolated ytem δq0 and du δw. Earlier by conidering the total differential of S a a function of variable U and we arrived at the thermodynamic identity for quaitatic procee : du ( S ) ds Pd + μd he combination of parameter on the right ide i equal to the exact differential of U. hi implie that the natural variable of U are S Conidering S and a independent variable: U U U du ( S ) ds d S + + S S d Since thee two equation for du mut yield the ame reult for any ds and d the correponding coefficient mut be the ame: U S U S P U S μ Again thi how that among everal macrocopic variable that characterize the ytem (P μ etc.) only three are independent the other variable can be found by taking partial derivative of the P with repect to it natural variable.
Iolated Sytem independent variable S and (cont.) Work i the tranfer of energy to a ytem by a change in the external parameter uch a volume magnetic and electric field gravitational potential etc. We can repreent δw a the um of two term a mechanical work on changing the volume of a ytem (an expanion work) - Pd and all other kind of work δw other (electrical work work on creating the urface area etc.): If the ytem comprie only olid and liquid we can uually aume d 0 and the difference between δw and δw other vanihe. For gae the difference may be very ignificant. initially the ytem i not necearily in equilibrium he energy balance for an iolated ytem : du ds Pd + δ Wother 0 (for fixed ) δ W other Pd ds δ W Pd + δ W other If we conider a quai-tatic proce (the ytem evolve from one equilibrium tate to the other) than ince for an iolated ytem δqds0 δ W other Pd
U A A S A U S Suppoe that the ytem i characterized by a parameter x which i free to vary (e.g. the ytem might conit of ice and water and x i the relative concentration of ice). y pontaneou procee the ytem will approach the table equilibrium (x x eq ) where S attain it abolute maximum. Equilibrium in Iolated Sytem For a thermally iolated ytem δq 0. If the volume i fixed then no work get done (δw 0) and the internal energy i conerved: While thi contraint i alway in place the ytem might be out of equilibrium (e.g. we move a piton that eparate two ub-ytem ee Figure). If the ytem i initially out of equilibrium then ome pontaneou procee will drive the ytem toward equilibrium. In a tate of table equilibrium no further pontaneou procee (other than ever-preent random fluctuation) can take place. he equilibrium tate correpond to the maximum multiplicity and maximum entropy. All microtate in equilibrium are equally acceible (the ytem i in one of thee microtate with equal probability). hi implie that in any of thee pontaneou procee the entropy tend to increae and the change of entropy atifie the condition ds 0 ( S ) eq max S U cont x eq x
Enthalpy (independent variable S and P) he volume i not the mot convenient independent variable. In the lab it i uually much eaier to control P than it i to control. o change the natural variable we can ue the following trick: ( S ) U ( S ) P U + ( U + P ) du + Pd dp dh d + du ds Pd dh ds + dp dh ( S P ) ds + dp + μd H (the enthalpy) i alo a thermodynamic potential with it natural variable S P and. - the internal energy of a ytem plu the work needed to make room for it at Pcont. he total differential of H in term of it independent variable : dh H H H S P ( S P ) ds + dp + d P S S P Comparion yield the relation: H S P H P S H S P μ In general if we conider procee with other work: dh ds + dp + δ W other
Procee at P cont δw other 0 For what kind of procee i H the mot convenient thermodynamic potential? At thi point we have to conider a ytem which i not iolated: it i in a thermal contact with a thermal reervoir. dh ds + dp + δ W δ Q + dp + δ other W other Let conider the P cont procee with purely expanion work (δw other 0) ( dh ) δ Q P δ W 0 For uch procee the change of enthalpy i equal to the thermal energy ( heat ) received by a ytem. other C P Q P H P For the procee with P cont and δ W other 0 the enthalpy play the ame part a the internal energy for the procee with cont and δw other 0. Example: the evaporation of liquid from an open veel i uch a proce becaue no effective work i done. he heat of vaporization i the enthalpy difference between the vapor phae and the liquid phae.
Sytem in Contact with a hermal Reervoir When we conider ytem in contact with a large thermal reervoir (a thermal bath there are two complication: (a) the energy in the ytem i no longer fixed (it may flow between the ytem and reervoir) and (b) in order to invetigate the tability of an equilibrium we need to conider the entropy of the combined ytem ( the ytem of interet+the reervoir) according to the nd Law thi total entropy hould be maximized. What hould be the ytem behavior in order to maximize the total entropy? For the ytem in contact with a eat bath we need to invent a better more ueful approach. he entropy along with and determine the ytem energy U U (S). Among the three variable the entropy i the mot difficult to control (the entropy-meter do not exit!). For an iolated ytem we have to work with the entropy it cannot be replaced with ome other function. And we did not want to do thi o far after all our approach to thermodynamic wa baed on thi concept. However for ytem in thermal contact with a reervoir we can replace the entropy with another more-convenient-to-work-with function. hi of coure doe not mean that we can get rid of entropy. We will be able to work with a different energy-like thermodynamic potential for which entropy i not one of the natural variable.
Helmholtz Free Energy (independ. variable and ) ( ) Pd Sd ds Sd Pd ds S U d S U F he natural variable for F are : ( ) d F d F d F df + + Comparion yield the relation: μ F P F S F Let do the trick (Legendre tranformation) again now to exclude S : ( ) ( ) S S U S U Helmholtz free energy can be rewritten a: S U F P + he firt term the energy preure i dominant in mot olid the econd term the entropy preure i dominant in gae. (For an ideal ga U doe not depend on and only the econd term urvive). P F F i the total energy needed to create the ytem minu the heat we can get for free from the environment at temperature. If we annihilate the ytem we can t recover all it U a work becaue we have to dipoe it entropy at a nonzero by dumping ome heat into the environment. ( ) d Pd Sd df μ +
S R+ he Minimum Free Energy Principle ( cont) he total energy of the combined ytem ( the ytem of interet+the reervoir) i U U R +U thi energy i to be hared between the reervoir and the ytem (we aume that and for all the ytem are fixed). Sharing i controlled by the maximum entropy principle: S ( U U ) S ( U U ) + S ( U ) max Since U ~ U R >> U S S U R+ R R U R ( U U ) S ( U ) + ( U ) + S ( U ) S ( U ) S S ( U ) R+ R reervoir +ytem R lo in S R due to tranferring U to the ytem ytem parameter only R F gain in S due to tranferring U to the ytem F table equilibrium U ytem U ds R+ du ( U U ) ds [ du ds ] df hu we can enforce the maximum entropy principle by imply minimizing the Helmholtz free energy of the ytem without having to know anything about the reervoir except that it maintain a fixed! Under thee condition (fixed and ) the maximum entropy principle of an iolated ytem i tranformed into a minimum Helmholtz free energy principle for a ytem in thermal contact with the thermal bath. 1
Procee at cont In general if we conider procee with other work: For the procee at cont (in thermal equilibrium with a large reervoir): df Sd Pd + δ ( df ) ( Pd + δ W other ) W other he total work performed on a ytem at cont in a reverible proce i equal to the change in the Helmholtz free energy of the ytem. In other word for the cont procee the Helmholtz free energy give all the reverible work. Problem: Conider a cylinder eparated into two part by an adiabatic piton. Compartment a and b each contain one mole of a monatomic ideal ga and their initial volume are ai 10l and bi 1l repectively. he cylinder whoe wall allow heat tranfer only i immered in a large bath at 0 0 C. he piton i now moving reveribly o that the final volume are af 6l and bi 5l. How much work i delivered by (or to) the ytem? U A A S A U S For one mole of monatomic ideal ga: df Pd he proce i iothermal : ( ) ( ) he work delivered by the ytem: F U S δ W ai δ W a + δ W bi b af dfa + 3 3 R R ln R ln + f ( m) 0 0 af bf 3 δ W R ln + R ln.6 10 J ai bf bi df b
Gibb Free Energy (independent variable and P) Let do the trick of Legendre tranformation again now to exclude both S and : ( S ) U ( P) S P U + G U S + P - the thermodynamic potential G i called the Gibb free energy. Let rewrite du in term of independent variable and P : ( P ) + dp d( U S + P ) Sd dp du ds Pd d( S) Sd d + dg ( P ) Sd + dp + μd Conidering P and a independent variable: dg G G G P ( P ) d + dp + d P P Comparion yield the relation: G P S G P G P μ
Gibb Free Energy and Chemical Potential Combining U S P + μ with G U S + P G μ - thi give u a new interpretation of the chemical potential: at leat for the ytem with only one type of particle the chemical potential i jut the Gibb free energy per particle. he chemical potential μ G P If we add one particle to a ytem holding and P fixed the Gibb free energy of the ytem will increae by μ. y adding more particle we do not change the value of μ ince we do not change the denity: μ μ(). ote that U H and F whoe differential alo have the term μd depend on non-linearly becaue in the procee with the independent variable (S) (SP) and () μ μ() might vary with.
Example: Pr.5.9. Sketch a qualitatively accurate graph of G v. for a pure ubtance a it change from olid to liquid to ga at fixed preure. G G P S - the lope of the graph G( ) at fixed P hould be S. hu the lope i alway negative and become teeper a and S increae. When a ubtance undergoe a phae tranformation it entropy increae abruptly o the lope of G( ) i dicontinuou at the tranition. S olid liquid ga G P S ΔG SΔ - thee equation allow computing Gibb free energie at non-tandard (if G i tabulated at a tandard ) olid liquid ga
S R+ G he Minimum Free Energy Principle (P cont) he total energy of the combined ytem (the ytem of interet+the reervoir) i U U R +U thi energy i to be hared between the reervoir and the ytem (we aume that P and for all the ytem are fixed). Sharing i controlled by the maximum entropy principle: ds table equilibrium R+ reervoir +ytem U ytem U du S P d ( U U ) S ( U U ) + S ( U ) max R+ R R ( U U ) ds [ du ds + Pd ] 1 dg hu we can enforce the maximum entropy principle by imply minimizing the Gibb free energy of the ytem without having to know anything about the reervoir except that it maintain a fixed! Under thee condition (fixed P and ) the maximum entropy principle of an iolated ytem i tranformed into a minimum Gibb free energy principle for a ytem in the thermal contact + mechanical equilibrium with the reervoir. ( ) 0 dg hu if a ytem whoe parameter P and are fixed i in thermal contact with a heat reervoir the table equilibrium i characterized by the condition: G/ i the net entropy cot that the reervoir pay for allowing the ytem to have volume and energy U which i why minimizing it maximize the total entropy of the whole combined ytem. P G min
Procee at P cont and cont Let conider the procee at P cont and cont in general including the procee with other work: hen dg δ W Pd + δ W other d( U S + P ) P ( δq Pd + δ Wother ) ( δ Q) P + ( δ Wother ) ds ( δ Wother ) P P P ds + Pd he other work performed on a ytem at cont and P cont in a reverible proce i equal to the change in the Gibb free energy of the ytem. In other word the Gibb free energy give all the reverible work except the P work. If the mechanical work i the only kind of work performed by a ytem the Gibb free energy i conerved: dg 0. Gibb Free Energy and the Spontaneity of Chemical Reaction he Gibb free energy i particularly ueful when we conider the chemical reaction at contant P and but the volume change a the reaction proceed. ΔG aociated with a chemical reaction i a ueful indicator of weather the reaction will proceed pontaneouly. Since the change in G i equal to the maximum ueful work which can be accomplihed by the reaction then a negative ΔG indicate that the reaction can happen pontaneouly. On the other hand if ΔG i poitive we need to upply the minimum other work δ W other ΔG to make the reaction go.
Electrolyi of Water y providing energy from a battery water can be diociated into the diatomic molecule of hydrogen and oxygen. Electrolyi i a (low) proce that i both iothermal and iobaric (P cont). he tank i filled with an electrolyte e.g. dilute ulfuric acid (we need ome ion to provide a current path) platinum electrode do not react with the acid. Diociation: H When I i paed through the cell H + move to the - electrode: he ulfate ion move to the + electrode: he um of the above tep: he electrical work required to decompoe 1 mole of water: (neglect the Joule heating of electrolyte) + SO4 H + SO4 + - H + e H -- 1 - SO4 + HO H SO4 + O + e 1 H O H + O 1 ΔW other ΔG G H + G O G HO ( ) ( ) ( ) In the able (p. 404) the Gibb free energy ΔG repreent the change in G upon forming 1 mole of the material tarting with element in their mot table pure tate: ΔG ( ) 0 ΔG( O ) 0 ΔG( H O) kj/mole H 37 Δ W other ΔG 37 kj/mole - + I I H O
Electrolyi of Water (cont.) ΔG 86 kj - ΔH 49 kj - ΔS 37 kj H + O H O Convenience of G: let conider the ame reaction but treat it in term of ΔU Δ and ΔS: Δ W other ΔG ΔU + PΔ ΔS PΔ: we will neglect the initial volume of water in comparion with the final volume of ga. y diociating 1 mole of water we ll get 1.5 mole of ga. he work by ga: ΔW P nr ( 1.5 mol)( 8.3 J/mol K)( 300 K) 3.7 kj -ΔS: the entropy of a mole of ubtance (from the ame able p.404) S(H )130.7 J/K S(O )05.1 J/K S(H O)69.9 J/K ΔS ΔU:???? not in the able... ( 300 K)( 130.7 J/K + 0.5 05.1 J/K - 69.9 J/K) 49 kj Well we got ΔH in the able - ΔH(H ) 0 ΔH(O ) 0 ΔH(H O) - 85.8 kj (ΔH upon forming 1 mol of the material tarting with element in their mot table pure tate). 1
Electrolyi of Water (cont.) he proce mut provide the energy for the diociation plu the energy to expand the produced gae. oth of thoe are included in ΔH. Since the enthalpy H U+P the change in internal energy ΔU i then: ΔU ΔH - PΔ 86 kj - 4 kj 8 kj However it i not neceary to put in the whole amount in the form of electrical energy. Since the entropy increae in the proce of diociation the amount ΔS can be provided from the environment. Since the electrolyi reult in an increae in entropy the environment help the proce by contributing ΔS. ΔW he min. voltage required for electrolyi: * 37 kj I Δt Q ΔG e.37 10 1.93 10 J/mole C/mole 5 0 5 A (Pr. 5.4) 1.3 ( e) A I H + O H O Fuel cell Electrolyi 1 If < 0 the reaction will proceed from right to left provided gaeou hydrogen i available at the + electrode and gaeou oxygen at the - electrode. 0
Fuel Cell Hydrogen and oxygen can be combined in a fuel cell to produce electrical energy. FC differ from a battery in that the fuel (H and O ) i continuouly upplied. y running the proce of electrolyi in revere (controllable reaction between H and O ) one can extract 37 kj of electrical work for 1 mole of H conumed. he efficiency of an ideal fuel cell : (37 kj / 86 kj)x100% 83%! hi efficiency i far greater than the ideal efficiency of a heat engine that burn the hydrogen and ue the heat to power a generator. he entropy of the gae decreae by 49 kj/mol ince the number of water molecule i le than the number of H and O molecule combining. Since the total entropy cannot decreae in the reaction the exce entropy mut be expelled to the environment a heat.
G Fuel Cell at High Fuel cell operate at elevated temperature (from ~70 0 C to ~600 0 C). Our etimate ignored thi fact the value of ΔG in the able are given at room temperature. Pr. 5.11 which require an etimate of the maximum electric work done by the cell operating at 75 0 C how how one can etimate ΔG at different by uing partial derivative of G. G P At 75 0 C (348K): G G S ΔG SΔ - thee equation allow computing Gibb free energie at non-tandard and P: Δ Subtance ΔG(1bar 98K) kj/mol S(1bar 98K) J/K mol H 0 130 O 0 05 ( H ) 0 ( 130 J/K)( 50 K) 6.5 kj ( O ) 0 ( 05 J/K)( 50 K) 10.5 kj ( H O) 37 kj ( 70 J/K)( 50 K) 40.5 kj 1 G G( O) G( H ) G( O ) 40.5 kj + 6.5 kj + 5.1kJ -8.9 kj H O -37 70 H hu the maximum electrical work done by the cell i 9 kj about 3.5% le than the room-temperature value of 37 kj. Why the difference? he reacting gae have more entropy at higher temperature and we mut get rid of it by dumping wate heat into the environment.
Potential U (S) H (SP) F () G (P) ariable S S P P Concluion: du dh df dg ( S ) ds Pd + μd ( S P ) ds + dp + μd ( ) S d Pd + μd ( P ) S d + dp + μd