The Outline. Fermi degenerate gas, repetition and specific heat calculations. ideal gas with Gibbs sum (grand sum) Gibbs free energy, definition

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1 The Outline ermi degenerate ga, repetition and pecific heat calculation ideal ga with Gibb um (grand um) Gibb free energy, definition Gibb free energy and chemical potential chemical reaction, concentration of reactant in equilibrium ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/

2 The ummary, lecture 4 Gibb um i a powerful tool to tudy ytem with variable number of particle: S exp exp!" <>, S exp AS Occupancy of a ingle orbital within the reervoir, for Boon, ermion, and in low occupancy =claical limit:: f f BE D f e claic exp AS exp In ermi ga at zero temperature all tate of energy le then ε_ are occupied. The ermi energy i given by the condition, that the number of occupied tate=umber of particle, and each orbital below ermi energy i occupied by exactly one particle. The entropy =ln The energy of the highet occupied orbital h 3 3 h Total energy in the ground tate U 3 5 m V ln m 3 n 3 ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/

3 & f(ε) 4 e ermi Dirac Boe Eintein and ermi Dirac ditribution function in the claical limit. # $ # % Claical In the limit of mall ORBITAL occupancy > large (ε µ)/τ > Boe Eintein and ermi Dirac ditribution approach an unique function, called claical ditribution Boe Eintein exp f BE f D exp exp f e claic (ε µ ) in unit of [τ] Claical ditribution i ued to tudy the propertie of the ideal ga > f(ε) 6 ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ Claical ditribution i reached for ytem with big( epilon mu)/tau). It i important that thi value i BIG for all poible energie (epilon) allowed for our ytem. If we gauge energie in uch a way that the lowet energy i=, than it i important that mu/tau i big. Thi mean that chemical potential mut be negative, and that the temperature mut be "low" compared to the value of chemical potential, which typically grow with temperature. A we ee next, thi mean for ideal ga approximation that the actual concentration of particle i much maller than quantum concentration > thu mall average occupancy of orbital. Thi i elf conitent with our aumption about ideal ga, for mall occupancy we can forget about ubtletie of boon v fermion and ue claical ditribution.

4 f ' ( ) exp ' *! ".5 τ=5 ermi occupancy for temperature τ=. f +,.5 for +, - (ε µ ) ame unit a [τ] or τ > all orbital with energy < µ are fully occupied, and all orbital with energy > µ have occupancy. Remember that µ typically depend on temperature. At zero temperature all orbital up to the ermi EERGY ε_ are occupied: -., / -, + / - ( OBS. olid tate phyicit often call µ ermi level but µ depend on temperature) h 3 3 h 3 3 n m V m ermi level for 3 dim ytem of particle orbital from free particle Schroedinger equation +pin ermi level for dimenional 4 particle(fermion) ytem ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ We ee that at temperature approaching zero the ermi Dirac ditribution function i a tep function. What doe it mean? All orbital of energy < µ(τ=) are occupied and hold exactly one particle each, while all orbital of energy above that are free. Thi i the reult we expect. We call thi highet filled orbital at zero temperature the ermi level. Thi i equal by definition to the chemical potential at zero temperature. In general the chemical potential will depend on the temperature however. The orbital which ha the energy equal to the chemical potential i "half occupied", it i the ermi ditribution function i equal to.5 for it.

5 @ The denity of tate and the ground tate energy By our definition: we mut thu have or our ga in 3 dimenion we have n 3 : ; <3 n= 3 ; : m 3> A h D d 3 (.. ) 3 d D 4 (.. ) d4 d ( ) ( k ) k dk = dk dk and V 3 < 3 m h We can now calculate any variable for the ytem by integrating over the energy, intead of umming up tate. Let calculate the ground tate energy: ε? D : ; 3 V A π d d: ; : h ( n ) x + ny nz > m 3> V h n, n, n = + x y z ml ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ where (ε) i the number of tate available for energie up to ε < D d 5 m 5 h = h n m ny nx n_ε.5" na L nz 3 V 3 5 Once we calculated the energy denity of tate we can integrate it now to the ermi level to obtain the total number of tate (particle), or, a we did before to calculate the ermi level a a function of. In the limit where we can approximate the um by an integral umming up over tate and integrating over energy denity i equivalent. The energy denity of tate we found here i good for a non relativitic free particle in a box, and it i of wide ue in many application, for example when conidering elementary particle reaction to figure out how many tate will be available for final tate particle with a given energy. We ue not the energy denity to calculate the energy of the ermi ga in the ground tate ( at zero temperature), it i the mean energy over all occupied orbital.

6 E E = E Ground tate energy repulive effect We ue denity of tate to calculate variable characterizing the ytem: U; C B <>; C : f : ; D f : ; D f,g,: D: d: ; D : f,g,: D: d: kinetic energy D: d: Thi equation can be ued to get µ(τ) <X>; D any other variable X : f,g,: D: d: ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ We have calculated the ground tate energy the kinetic energy of the ermi ga at zero temperature. We ee it i non zero, we can alo ee that it depend on the volume, thu the preure of the ermi ga ( derivative of energy over volume at contant entropy) i non zero even at the zero temperature. Thi i in contrat with ideal ga in claical approximation, where the energy and preure are proportional to the temperature, and vanih at zero temperature. We can now calculate variou variable characterizing the ytem umming up over the tate, taking into account the occupancy of a given tate (ermi ditribution "f"), and the um over tate i repreented by the integral over the energy denity of tate. We would like to calculate the heat capacity of the ermi ga.

7 S U H R I S L U T R Heat Capacity of the electron ga or the ideal claical ga we have U = 3/ τ and heat capacity (δu / δ τ)_v =3/. or electron in metal much lower value are found, and calculation for degenerate ga give good agreement with experiment. Let calculate the increae of the total energy when the ga i heated from τ=. U = U(τ) U() U ( ε) ( ε) dε ε ( ε ) = εf U T ε H V we will make ue of : dε J I K f M,, I D I d I J I K D I d I f W,X, D d Y T V D d P Q temperature independent term ε) P Q UZ D V f W,X, d Y... ε) f( ε) energy ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ O we ee that : We ee that contribution to the integral hould be larget from energie cloe to To calculate the heat capacity of the ermi ga we need to calculate it energy a a function of temperature. At the moment we ut calculated the ground tate energy (zero temperature), and it i eaier to calculate the difference between the energy at ome temperature and the energy at zero temperature. Anyway, the energy at zero temperature doe not depend on temperature ; ) by definition, o it would fall out of differentiation to calculate heat capacity at a contant volume. Here we make ue of the fact that the ga a a whole ha a contant number of particle, which doe not depend on temperature o the occupancie mut um up the the ame at different temperature. ow we hould integrate the temperature derivative of the ermi occupancy with the denity of tate. We are not able to do it... it i too complicated, and beide we would have to calculate firt the chemical potential a a function of temperature firt. So we need to make ome implifying aumption. We note that in U only the tate with energy cloe to chemical potential will matter.

8 ^ Heat Capacity of the Electron Ga We evaluate heat capacity auming that we can approximate ε by ε_ in P f ' ( ) ' * exp "!.5 τ=5 τ= τ=. C el[ D \ ] \ _ \ d f We make another aumption, that µ doe not depend trongly on temperature for τ << τ_ and again we can approximate it by ε_. We get after bit boring calculation (next page) C ela 3b D c d e d` b d\ d d (ε µ ) ame unit a [τ] f g3 n 3 f h m h D h f i m 3i h h V g 3i V 3 g ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ f 3 h f 3 k We note, that if for a given energy, we take a derivative of the ermi ditribution a a function of temperature, it will be very big for energie cloe to the ermi level. It will be mall for high and low energie, where the ermi ditribution function i cloe to zero or one, whatever i the temperature. A the denity of tate change rather "lowly" with energy, it i the D(ermi level) which matter. Second we aume that, for mall temperature we are intereted in the chemical potential doe not depend on temperature and it i equal to the ermi level. With thi aumption we can "eaily" calculate the heat capacity.

9 o r o! o o o n r o " o " Calculation leading to the heat capacity reult ^ d f C el[ D \ ] \ _ \ d` f x ( ) exp x, x( ' * " l =D pnq C elm D n x d f dx d\ where f ',,! " ( ) ' * exp! ' * ' ", ( " dx, " ( * x d d f dx d d dx dx m D df dx exp x dx d' d d =D n p o exp x n p q d f dx x d f dx dx D d dx f x n x d f dx dx d d dx dx = o n n ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ x d f dx dx x f π / 6 π / 6 3 D

10 y v ermi ga in metal. The reult that the heat capacity fall down with temperature i in agreement with what we oberve in metal. In alcali metal (copper, ilver) there i one conduction electron per atom, thu their concentration can be eaily evaluated an i of the order of Thi correpond to h m 3 3 h m 3 n 3 5t cmu 3 ~ 5 ev ~ 5 K V o in room temperature the electron ga i indeed degenerate, and T<< T_. In metal we oberve C totv C elw C vibv x y y w A y 3 z C tot x where the "vib" contribution come from lattice vibration and we will calculate it when dicuing Boe ga (Planck radiation) Thi formula i in excellent agreement with experiment. y w A y ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/

11 Ideal ga with Gibb um rule We will tudy now again the propertie of the ideal ga in the claical limit, uing the Gibb um rule. We will aume again the claical ditribution function, an approximation valid for n<< n_q: f e claic We ue : number of particle { Z L 3 Z ~ d n x f { mƒ h d n y <> { exp } f to evaluate the chemical potential for a given We take energie for the free particle in the box, and we can evaluate the um uing either the denity of tate, or remind ourelve that we did it already it wa ingle particle partition function Z_: 3 V n Q Thu : exp ˆ V n Q n n Q d n z exp # $ & n x n y n z which give a familiar expreion for chemical potential ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ The way we proceed here, tarting from occupancie of orbital to calculate the chemical potential for particle then calculating free energy, then preure, entropy, energy of the ga i a tandard way to procede whenever free energy can be integrated over with COSTAT volume and temperature, and when we have the orbital occupancie pecified

12 Ž ˆ ˆ Can be evaluated uing : š ree energy, preure, energy, entropy V n Q,V, Š and ln Œ,V ln ln V n Q d ln ln V n Q ln n n Q ln ln V n Q p V, V pv U V, mƒ h 3 n Q ln x xln x x 3 ln n Q V, n 5 ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ ote that the entropy for the ideal ga i proportional to the number of particle. If we put together two identical batche of the ideal ga, of the ame temperature and concentration of particle the final entropy will be ut the um of the entropy of two batche.

13 ˆ ˆ Chemical potential, dicuion We can ee now that the chemical potential can be modified by many effect, for a lightly non ideal ga: if every tate ha ome additional degeneracy, for example connected to pin (+) { f { exp ˆ ln œ œ exp } V œ n Q ln n n Q Z n n Q œ If the ga ha energy level aociated with internal degree of freedom, rotation, vibration, then : monÿ int ž { = Z { int f { mon, int exp } int ln Z int œ ln ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ exp } = Z Z int n n Q

14 ƒ ƒ ƒ ƒ Heat capacity of the ideal ga With contant number of particle we have : du d pdv Heat capacity at contant volume Heat capacity at contant preure Uing: pv U 3 C p C V C V C p U ƒ V Heat we get C V 5 d(heat) d(work) ƒ ƒ V p 3 ƒ or C p C V p U andc p C V k B ƒ p 5 k B Specific heat at contant preure i larger, becaue the ga can do ome work. The reult i valid for a ga without internal degree of freedom p p V ƒ p p p ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ A the volume change with temperature at contant preure the ga at contant preure will do the work while heated up. Thu heat capacity at contant preure i higher than thi at contant volume.

15 ± ± ± ± ± The relation : Experimental tet of Sackur Tetrode relation C p ƒ ƒ p ƒ ª «to meaure the entropy, a C_p i known for variou tate C p ƒ dƒ ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ i often ued Experimentally, the following contribution have to be conidered (example for neon, mol) ) Entropy increae on heating from abolute to olid melting point S T melt * S ( C p T dt( 4.9 J K ) + in the olid to liquid tranformation 3) + on heating liquid from melting point to boiling point at atm (7. K) T boil S( T melt 4) + in liquid to ga tranformation S( U melting T melting C p T dt( 3.85 J K S( U vaporize T boiling ( 3.64 J K ( 64.6 J K Total entropy of mol ga at atm and 7. K hould be the um of it : = 96.4 J/K Calculated from Sackur Tetrode relation: J/K The heat capacity wa meaured down to around K, and extrapolated down from Debaye law ( vibration component dicued previouly). ote that we peak here about conventional entropy, S=kσ

16 @ } Reverible iothermal expanion Conider X atom of He at initial volume of l= cm at 3 K. Let the ga expand lowly to twice that volume, at contant temperature in contact with reervoir. At low, REVERSIBLE expanion the ytem at AY intant i at it MOST probable configuration. preure after expanion, from pv=τ, final preure i half initial value. change of entropy ² V, ln V³ V independent µ ², ln V V, ln,.69 3 work done by ga : V V V V V dv ln V V ln The energy of the ga did not change ince U=3/ τ, o the work done had to be compenated by the heat flowing from reervoir du d pdv ln reflecting the work However = U τσ changed done by our ytem ¹ heat = τ ln ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ We conider here a reverible proce, paing cloe to equilibrium (mot probable configuration) throught the proce, and the thermodynamic identity hold all the time. It i very intructive to analyze a well udden expanion into vacuum. Here there i no mean of work to be done by the ytem, and there i no heat flow, thu no temperature change. The only thing which change i the entropy. You can notice that during uch a proce the thermodynamic identity du=τdσ pdv doe not hold the proce i irreverible, doe not pa thru thermal equilibrium

17 ˆ The ummary. In ermi ga at zero temperature all tate of energy le then ε_ are occupied. The ermi energy i given by the condition, that the number of occupied tate=umber of particle, and each orbital below ermi energy i occupied by exactly one particle. The entropy =ln The energy of the highet occupied orbital h 3 3 h 3 3 n m V m Total energy in the ground tate f e { claic š d ln n n Q U 3 5 f { p V, and the heat capacity exp } ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ C elº x Z V n Q exp ˆ pv m» h y y V n Q n n Q 3¼ n Q

18 È Gibb ree Energy We will conider now chemical reaction > we will be conidering ytem with exchanging particle, but at COSTAT PRESSURE. Chemical reaction are often tudied in ytem at atmopheric preure. ½ Ut ¾ We defined before Helmholtz free energy : i minimal for ytem at a contant temperature, volume, and contant number of particle G½ Ut ¾ À pv We will define Gibb free energy : and how that it take minimum for ytem at a contant preure, temperature and number of particle dgç dut d¾ t ¾ d À VdpÀ pdv dg S Ç dut ¾ d SÀ dv S pç in general : dgç É dt d¾ À Vdp duá Â dã Ä pdvå Æ d Temperaure reervoir τ=cont Preure reervoir, p=cont ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ Syt em Temperaure reervoir τ=cont Here we find analogou variable to Helmholtz free energy Gibb free energy. Helmholtz free energy take minimum for ytem at contant temperature and volume, while Gibb free energy ha a minimum for ytem at contant temperature and preure. It i particularly ueful when conidering chemical reaction

19 ˆ we can write : dgç Ê G Ê p,ë and identify : Gibb free energy, intenive and extenive variable dgç É dt d¾ À Vdp dà Ê G Ê ¾ p, Ê G Ê p,ë G d¾ À Ê p Ê Ë, G Ç É, Ê Ê ¾ dp a,g(,τ,p), G Ç t, Ê Ç V p, p Ê Ë, Let conider how G change when two identical ytem are put together : G½ Ut ¾ À pv p and τ are ITESIVE quantitie, they do not change. U,V, σ and thu G double when two identical ytem are put together. U,V, σ, Ν are EXTESIVE quantitie. A G i proportional to we mut be able to write : G G,¾, p Ç f ¾, p a Ê Ç É Ì f ¾, p Ç É ¾, p,v, Š Ê Ë, p intenive Œ,V G, p, Í p, ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/

20 ˆ and G, comparion Let compare now the Helmholtz free energy and Gibb free energy in relation to the chemical potential µ. We have :,V, Š Œ,V ln and G Ê Ê p,ë ln Ç É ¾, p ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ or ideal ga : uing p V= τ => p V n Q n Q The chemical potential at a contant temperature and volume DEPEDS on. Thu Î µ But the chemical potential at a contant preure and temperature doe OT depend on the number of particle. Thu we can write G=µ p Intead we can integrate over and get : and check conitency : G Ï + pv G ln = ln V n Q G ln n Q ln Ð pv ln Ð = V n Q V n Q V n Q

21 Õ Ú Ø Ú Ø G½ Ut ¾ À pv Equilibrium in reaction With and G, p, Í p, we can expre G for a combination of ytem of different particle with different chemical potential a : G and Í p, dgç É d t d¾ À Vdp or a ytem at contant temperature and preure, coniting of everal pecie of "particle", which can form "molecule", the "particle" ditribute themelve among variou molecule in uch a way to minimize G. The total number of "particle" tay contant. Example: H Ñ Cl HCl 5 H Cl Ñ H cont Ò H Cl Ñ Cl cont or one chemical reaction which take place we can write : or more general if reaction take place, the change of number of given pecie will be : HCl Ó H Ó Cl Ô A Ö Ô,Ö Ô Ó,Ö 3Ô Ó d Ø Ù dg Ú W d W ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ W chemical equilibrium condition Here we will try to ue the minimum of G to try to decribe equilibrium of reactant (reacting ubtance) in variou chemical reaction. ote that thi equilibrium will depend on chemical potential of variou reactant which need to carry the information about how energetically favorable a reaction i ( we will have to take interaction into account, cannot ue ut the chemical potential of the non interacting ideal gae). The reaction coefficient ν decribe the change of the number of pecific molecule in a given poible reaction. The total number of "atom" tay contant, the reaction coefficient take thi into account. ote that reaction coefficient aume that reaction are time reverible ( if you know omething about particle phyic you can think how to repreent K > antik tranformation, which i not time reverible)

22 ˆ ˆ Chemical potential, dicuion We can ee now that the chemical potential can be modified by many effect, for a lightly non ideal ga: if every tate ha ome additional degeneracy, for example connected to pin (+) If the ga ha energy level aociated with internal degree of freedom, rotation, vibration, then : { f { exp ˆ { = Z { ln int f exp } V œ n Q ln { mon, int exp } int ln Z int œ n n Q ln ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ Z n n Q œ ln œ exp } = Z Z int n n Q We will need the form of chemical potential which allow for the interaction, the one with the internal binding energy of a molecule

23 { ˆ Ú Ø Û Û } int W ln Z int œ We can write : int exp } Chemical equilibrium for ideal gae. ln exp } int Z int = τ ln Z n n Q Let try to ue the form of the chemical potential for the ideal ga, to try to find concentration of variou inter reactant a a function of temperature. A we will conider non mono atomic gae, and binding of particle, we have to ue the form of potential which allow for the internal energy. In practice, for each pecie there will be ut one energy tate ε, connected with the binding energy of the pecie relative to the energy of free, unbound component, for "free" component Z int, int ˆ ln Z int œ introduce c Š Z int n,ˆ Q ln Z int ln n n Q int ln n ln Z int n Q ln n ln c "C" depend on temperature, and binding propertie, but do not depend on the concentration of the reactant. In equilibrium: Ü ln n Ü ln c Š ln K ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ rom the equilibrium condition, expreed a a um of reaction coefficient and chemical potential of variou pecie vanihing, we get a condition binding the equilibrium concentration of reactant to "coefficient C" which depend on the temperature and internal propertie of a reaction binding energy, but not on the concentration. Thu if we know C, which in our model are a function of binding energie and quantum concentration of given reactant ( it depend on the ma and temperature) we can calculate the concentration of reactant We introduce here "internal " which, for our purpoe (ut one bound tate) i equal to the internal binding energy, but might be more complicated for a more complicated binding

24 Û Û Ü Ü Û Û á Ü Ü ß Û Ü ¾ ln Ý ln n Þ n ln c we get for K : Ç ln Ý Law of ma action, and example ln c Š ln K Þ c ½ ln K ¾ Ì Ý ln Z int n Q n Q Example: Conider diociation of molecular hydrogen: let introduce: can be written in a form: ß ln n Q exp Ü int Š K Þ n Ç K ¾ int à H HÑ H 5 H H 5 ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ Š ln K Õ, Õ n H ½ H etc, H H u Ç K ¾ and ln K ¾ Ç ln n Q H t ln n Q H t H We obtain here two reult: if we know K(T), how to calculate concentration of reactant how to calculate "theoretically" K(T) from our model > ideal ga + internal binding energy. ote that pecifying internal binding energy for a given reactant need ome care. We will calculate here a an example the equilibrium concentration of atomic hydrogen. Here the only the molecular hydrogen will have internal, > internal binding energy. ote the notation for concentration of a reactant [H] i a concentration of atomic hydrogen, etc.

25 Ì Example, chemical equilibrium H H u Ç K ¾ Ì H H Ç H K ¾ raction of diociated H grow for the mall concentration of molecular H_ and for mall K. What i K? ln K ¾ Ç ln n Q H t ln n Q H t H ¾ K ¾ Ç n Q H n Q H ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ exp t â H ¾ ε i the energy of the H_ molecule meaured toward energy of "free" molecule, thu it i typically negative. At mall temperature ε= ev. The more tightly bound H_, the bigger K, and the maller proportion of diociated H. H diociation i the "entropy diociation", the gain of entropy with the diociation compenate lo of binding energy. Entropy i a part of Gibb energy, from the minimum of which we obtain thi reult. ote that a ytem at contant preure and temperature i not iolated, o we cannot obtain it propertie from maximum entropy principle ut for thi ytem alone. In the galactic pace hydrogen i very dilute > the fraction of diociated H i uppoed to be ubtantial

26 Example, ph and ionization of water In liquid water, ionization proceed to a light extent: H ã 5 H + Ñ OH 5 H Ñ H + Ñ OH 5 H + H OH Ç K ¾ H + OH Ç u 4 mol H + Ç OH Ç u 7 Õ, Õ, Õ 3 becaue concentration of H and OH i mall, it doe not really affect the concentration of (H_ ) and we have approximately, at room temperature: liter mol liter o the product of concentration tay contant at given temperature. In pure water there mut be the ame number of H and OH ion o we have : We can increae concentration of H+ ion by adding acid to the water, then the concentration of OH will decreae to keep the product contant. Similarly adding bae to the water will increae OH concentration, and decreae H. We define ph of water a a logarithm of H concentration in mole/liter: ph½ t log H + water phç t logu 7 Ç 7 tronget acid have ph cloe to, blood plama ha lightly "baic" ph ~ 7.5 ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/

27 Kinetic model of ma action Suppoe A and B can "collide" to form a molecule AB. rom our model we can write: ABã 5 AÑ B 5 ABÑ AÑ B 5,, and Õ Õ Õ 3 A B Ç AB K ¾ where [A], [B], [AB] are concentration of A,B, AB repectively We can look at the formation rate of AB from the kinetic point of view: d AB dt change in AB concentration Ç C A B t D AB formation rate decay rate In equilibrium : d [AB]/ dt = and thu: C will decribe the probability of forming AB in a colliion of A and B, while D decribe decay probability. A B AB Ç Dä C ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ which i conitent from what we got from the law of ma action What if the reaction proceed via a catalyzer? : AÑ Eã 5 AE and AEÑ Bã 5 ABÑ E If the lifetime of AE i very hort and we can neglect number of A bound up in AE the final equilibrium concentration will be the ame a before but the reaction rate will differ

28 Reaction rate d AB Ç C A B t D AB Equality of the formation and decay rate in equilibrium i called a principle of detailed dt balance. change in AB formation decay rate concentration rate The law of ma action tell u what are the concentration of reactant at equilibrium, but tell u nothing about how fat the reaction i proceeding. The reaction might have to go via "activation barrier" > for example Coulomb barrier in nuclear reaction The energy difference between the initial and the final tate H determine A...B the equilibrium concentration ratio Potential energy activation energy A + B H= ε Ditance between molecule AB [A][B]/[AB], for example: K ¾ Ç n Q A n Q B n Q AB However, the height of the activation barrier determine how fat the reaction of formation and decay are taking place ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/ exp â AB ¾

29 Ý The Summary The Gibb free energy defined a : equilibrium at contant temperature and preure G½ Ut ¾ À pv ha a minimum in thermal Ê G Ê p,ë Ç É, Ê G Ê ¾ G, p, Í p, G Ç t, Ê Ç V p, p Ê Ë, The law of ma action tate that ratio of concentration of the reactant in chemical equilibrium are a function of the temperature alone Þ Ç K ¾ n ile: /home/lipniack/tatitic_phyic/lecture5/lecture5.dd Date: 4&9/4/