2.625 - Electochemical Systems Fall 2013 Lectue 2 - Themodynamics Oveview D.Yang Shao-Hon Reading: Chapte 1 & 2 of Newman, Chapte 1 & 2 of Bad & Faulkne, Chaptes 9 & 10 of Physical Chemisty I. Lectue Topics: A. Review 1st and 2nd laws of themodynamics Fist law: du = δq δw Second law: δ IS 0 B. Chemical Equilibium ΔG = ΔG + nrt ln K a = 0; whee K a = µ i = µ + RT ln a i ; µ i ν i n = 0 i C. Electochemical Equilibium ΔG = ΔG + nrt ln K a + nz F E = 0 ν II a i pod i I ν a i eact i Nenst Equation: E = E + RT ln K a z F II. Review 1st and 2nd laws of themodynamics A. 1st Law of Themodynamics Fo a closed system of constant mass, i. intenal enegy is constant: δ IU = 0 ii. extenal enegy changes though wok and heat: δ EU = δq δw, whee δq is heat tansfeed into the system, and δw is wok done by the system. Note: in most (fluid) systems, δw is PdV; fo electochemical systems, we will expand this definition to include electochemical wok. iii. the total enegy change of the system is the sum the intenal and extenal enegy changes: du = δ IU + δ EU = δq δw δ IU and δ EU ae intensive popeties wheeas δq and δw ae extensive popeties A. Intensive popety: A physical popety which does not depend on the size (volume, mass, numbe, etc.) of the system B. Extensive popety: A physical popety which does depend on the size (volume, mass, numbe, etc.) of the system 1
Lectue 2 Themodynamics Oveview 2.625, Fall 13 System δ IU Envionment δ EU Figue 1: The canonical themodynamic system, with intenal enegy change δ IU and extenal enegy change δ EU. B. 2nd Law of Themodynamics i. change in intenal entopy is equal to geneated entopy, which is neve less than zeo: δ IS = δ gs 0 ii. extenal entopy change is the change in heat scaled by the tempeatue of the system: δq δ ES = T iii. the total entopy change is, similaly, the sum of the intenal and extenal entopy changes: δq ds = δ IS + δ ES = δ IS + T Multiplying though by T, we have: T ds = T δ IS + δq And since δ IS is always geate than o equal to zeo, T ds δq. These expessions can be mapped into state space. The change in entopy while U and V ae held constant, o H and P ae held constant, ae shown below. C. Combining the 1st and 2nd Laws, δq = du + P dv (whee δw is given as P dv ) Subbing in T ds δ IS fo δq and eaanging, we have T δ IS = du + P dv T ds 0 (1) Fo evesible pocesses, δ IS = 0, so T ds = du + P dv. This is Gibbs fundamental equation. Futhe, a closed system appoaches equilibium by maximizing entopy, S. 2
Lectue 2 Themodynamics Oveview 2.625, Fall 13 S At constant U, V equilibium States Figue 2: Equilibium is achieved when entopy is maximized fo a system whee U and V ae constant. D. Altenatively, we can define equilibium by using Gibbs fee enegy. i. Fist, we define enthalpy, H: H = U + P V ii. Then define Gibbs fee enegy, G: G = H T S iii. G = U + P V T S, and dg = du + d(p V ) d(t S) By consideing the combined fist and second law equation, we can now wite: δ IS = dg V dp + SdT 0. Thus, a system appoaches equilibium by minimizing G such that dg 0 : 3
Lectue 2 Themodynamics Oveview 2.625, Fall 13 G At constant P, T equilibium States Figue 3: With the pope choice of vaiables held constant, the Helmholtz and Gibbs functions show simila pofiles. Equilibium is achieved when Helmholtz o Gibbs functions ae minimized. Note: fo a evesible system, dg = V dp SdT III. Gibbs fee enegy of multicomponent systems The geneal fom of the Gibbs fee enegy fo a multicomponent system o phase is G(T, P, n 1, n 2,...n i ), whee n i is the numbe of moles of species i in the system o phase. ( ) ( ) ( ) G G G dg(p, T, n i ) = dt + dp + dn i (2) T P n i P,n T,n i T,P,n j We can define the patial diffeentials in the above equation as follows: G S = T P,n G V = P T,n g G µ i = n i T,P,nj=n i Then we can wite the diffeential fom of the Gibbs fee enegy fo a multicomponent system as: dg(p, T, n i ) = SdT + V dp + µ i dn i (3) i A. Compaing this to the expession fo dg above, we see that the intenal entopy is expessed hee as the sum of the chemical potentials: T δ IS = i µ idn i This entopy is ggeneated when a new component i is intoduced into the system, with a chemical G potential, µ i = n i T,P,nj=n i Looking back, ecall that at constant T and P, T δ I S = dg = i µ idn i 0 4
Lectue 2 Themodynamics Oveview 2.625, Fall 13 B. Themodynamic quantities ae typically tabulated on a mola basis, meaning that the quantity is povided in units pe mole of eactant/poduct. Examples ae: g : mola Gibbs fee enegy s : mola entopy h : mola enthalpy C. We define mole faction: X i = ini n i G = n i g i ; whee g i = X i g i ; and n = n i D. We define the patial mola Gibbs fee enegy with espect to composition as: g g g G ḡ i = = = µ i X i T,P,Xj =X i n i T,P,n j =n i E. Fo an ideal gas, we define the patial pessue as: P i = X i P F. We can define a moe convenient fom of the chemical potential, whee we wite the chemical potential of a species i in tems of its chemical potential in the efeence state, µ i and its activity, a i : µ i = µ i + RT ln a i Defining the chemical potential of a species in tems of its activity is slightly moe convenient because the activity can be descibed by simple models, which will be discussed next. G. Activity models Ideal (Raoultian) Solution: a i = X i An ideal solution assumes that the activity of species i is equal to the mole faction of species i. Ideal Gas: a i = P i Fo an ideal gas, the activity of species i is equal to the patial pessue of species i. This is an extension of the ideal solution model. i. Figue?? demonstates how activities can be calculated fom the patial pessues of gases in a mixtue. ii. The enthalpy of mixing can be found by calculating the enthalpy befoe and afte mixing: h init = i X i h i (T, P i ) h final = i X ih i (T, P i ) = i X ih i Δh mix = h final h init = 0 iii. The entopy of mixing can be found in a simila way: s init = i X is i = i X ( i s i R ln ) P i P s final = s i = i X i s R ln Pi i X ( ) i i P Δs mix = s final s init = i X ir ln X i Note that since ΔS mix is geate than zeo, mixing is ievesible. This should line up with you intuitive undestanding of gas behavio. iv. Next, the Gibbs fee enegy ( of a mixtue: ) g init = X i g i = X i g + RT ln Pi i i i P g final = i X iḡ i = i X ( ) Pi i g i + RT ln P Δg mix = g final g init = i X i RT ln X i 0 When Δg mix 0, mixing will occu spontaneously and wok will be geneated. 5
Lectue 2 Themodynamics Oveview 2.625, Fall 13 Befoe Mixing: P 1 = P 2 = P 3 = P i = P P 1 P 2 System P 3 P 4 n 1 n 2 n 3 n 4 Afte Mixing: P = p 1 + p 2 + p 3 + p i p 1, p 2, p 3,... p i n 1, n 2, n 3,... n i Figue 4: Initially, a box contains seveal species of a gas, each having the same pessue P i, the same as the oveall box pessue P. Afte the patitions ae emoved, the gases mix and P is the sum of each gas patial pessue P i. IV. Chemical Equilibium A. Condition fo chemical equilibium µ i dn i = 0, o ΔG = 0 at constant T and P. Whee ΔG = µ i ν i n, and ν i is the stoichiometic coefficient of species i. 6
Lectue 2 Themodynamics Oveview 2.625, Fall 13 µ i dn i At constant T, P equilibium States Figue 5: Equilibium is achieved at constant T and P when the sum of chemical potentials of components in the system is minimized. B. Example 1 i. Conside the fomation of wate: H 2(g) + 2 O 2(g) H 2 O ii. ν H2(g) = 1, ν O2(g) = 0.5, and ν H2O = 1 C. Deiving the condition fo chemical equilibium: i. ΔG = µ i ν i n = n ν i (µ + RT ln a i i) ii. ΔG = n µ i ν i + nrt ν i ln a i iii. Let ΔG = n µ i ν i iv. ΔG = ΔG + nrt ln v. Remembe that K a = II ν a i pod i I ν i eact a i II ν i pod a i I ν a i eact i vi. At equilibium: ΔG = ΔG + nrt ln K a = 0 vii. Thus, at we can define the chemical equilibium condition: D. Two inteesting elations ΔG = nrt ln K a i. The natual h logaithm of the h equilibium constant, K a, can be witten as follows: ΔG ΔH K a = exp = A exp nrt ΔH ln K a = lna nrt nrt 7
Lectue 2 Themodynamics Oveview 2.625, Fall 13 ln K a H 2 + 0.5O 2 H 2 O CO 2 CO + 0.5O 2 (high T) 1/T (low T) Figue 6: Example plot of ln K a vs. 1 1 T ln K a vaies appoximately linealy with T.. ii. Assuming that ΔH is not a function of tempeatue (this is a good assumption fo a wide ange of T), the following elationship can be developed: Since ΔG (T, P ) = ΔH T ΔS, ΔG = ΔS, assuming that ΔH = f(t ) T 1 Example: H 2(g) + O 2(g) H 2 O 2 T (K) Δh (kj/mol) Δg (kj/mol) E 300-286 -237 1.23 500-242.4-219 1.13 1000-245 -192 0.99 8
Lectue 2 Themodynamics Oveview 2.625, Fall 13 ΔG 0 T -ΔS Figue 7: Visual epesentation of how the change in Gibbs fee enegy with tempeatue is appoximately equal to the change in entopy (i.e. the slope of ΔG as a function of tempeatue is equal to ΔS ). E. Electochemical Equilibium i. Relative to a chemical system, equilibium fo an electochemical system has an additional wok tem. ii. δw = Edq fo chaged paticles with a chage of dq iii. Condition fo electochemical equilibium (at constant T and P): µ i ν i + z F E = 0 9
MIT OpenCouseWae http://ocw.mit.edu 2.625 / 10.625J Electochemical Enegy Convesion and Stoage: Fundamentals, Mateials, and Applications Fall 2013 Fo infomation about citing these mateials o ou Tems of Use, visit: http://ocw.mit.edu/tems.