W. M. Whte Geochemstry Equatons of State: Ideal GasLaw: Coeffcent of Thermal Expanson: Compressblty: Van der Waals Equaton: The Laws of Thermdynamcs: Frst Law: Appendx II Summary of Important Equatons PV = NRT α 1 V β 1 V V T V P P= RT V b - a V 2 U = Q + W 1 wrtten n dfferental form: du = dq + dw 2 work done on the system and heat added to the system are postve. The frst law states the equvalence of heat and work and the conservaton of energy. Second Law: dq rev = TdS 3 Two ways of statng the second law are Every system left to tself wll, on average, change to a condton of maxmum probablty and Heat cannot be extracted from a body and turned entrely nto work. Thrd Law: lm S = 0 T 0 3 Ths follows from the facts that S = R ln Ω and Ω =1 at T = 0 for a perfectly crystallne pure substance. Prmary Varables of Thermodynamcs The leadng thermodynamc propertes of a flud are determned by the relatons whch exst between the volume, pressure, termperature, energy and entropy of a gven mass of flud n a state of thermodynamc equlbrum - J. W. Gbbs The prmary varables of thermodynamcs are P, V, T, U, and S. Other thermodynamc functons can be stated n terms of these varables. For varous combnaton of these varables there are 1
W. M. Whte Geochemstry Appendx II Equaton Summary characterstcs functons. The characterstc functon for S and V s one of the prmary varables: U. Thus du = TdS + PdV 5 Other Important Thermodynamc Functons What then s the use of thermodynamc equatons? They are useful precsely because some quanttes are easer to measure than others. M. L. McGlashan Enthalpy: H U + PV 6 In dfferental form n terms of ts characterstc varables: dh = TdS + VdP 7 Helmholtz Free Energy: A U - TS 8 and: da = -PdV -SdT 9 Gbbs Free Energy: G H - TS 10 The Gbbs Free Energy change of a reacton at constant temperature and pressure s: G r = H r T S r and: dg = VdP - SdT 11 Your choce of whch of these functons to use should depend on what the ndependent varables n your system are. In geochemstry, P and T are the most common ndependent varables, so the Gbbs Free Energy s often the functon of choce. Exact Dfferentals and the Maxwell Relatons Any expresson that may be wrtten: M(x,y)dx+N(x,y)dy 12 s an exact dfferental f there exsts a functon z = f(x,y) such that f(x,y) = M(x,y)dx+N(x,y)dy 13 The total dfferental of the functon z(x,y) s wrtten: If dz s an exact dfferental, then whch s equvalent to: dz = z x y dx + z y x dy = Mdx + Ndy 14 M y = N x M y y = N x x 16 10a 15 2
W. M. Whte Geochemstry Appendx II Equaton Summary All thermodynamc varables of state are exact dfferentals. Thus the practcal applcaton of the propertes of exact dfferentals can be llustrated as follows. Equaton 11 (dg = VdP - SdT) has the form dz = M(x,y)dx+N(x,y)dy snce V and S are functons of temperature and pressure. Equaton 11 may also be wrtten as dg = G dp + G dt 17 P T T P and comparng equatons 11 and 16, we conclude that G P T =V and G T P = S 18, 19 Applyng the rule emboded n Equaton 15, we can conclude that: V = 20 T P P T Playng smlar games wth Equatons 5 through 9, we can develop a seres of relatonshps: from de from dh from da T = P 21 V S V T P V = P T V = Equatons 20-23 are known as the Maxwell Relatons. V P 22 V T 23 Dervatves of Entropy pressure: P T = αv 24 temperature: T V = C V T and T P = C P T 25, 26 volume V T = α β 27 Dervatves of Enthalpy pressure temperature H P T = V(1 αt) 28 H T P =C P 29 Dervatves of Energy temperature: U T V =C V and U T P =C P PαV 30, 31 3
W. M. Whte Geochemstry Appendx II Equaton Summary volume: U V T = Tα β P 32 Dfference between C P and C V C P C V = TVα2 β 33 The Gbbs Phase Rule: The Gbbs Phase Rule s a rule for determnng the degrees of freedom of a system. f = c - p + 2 34 f s the number of degrees of freedom, c s the number of components, and p s the number of phases. The mnmum number of components needed to descrbe a system s: c = N - R where N s the number of speces, and R s the number of reactons possble between these speces. The Clapeyron Equaton The slope of a phase boundary n P-T space s: dt dp Solutons Raoult s Law: apples to deal solutons: = V r S r 35 P = X P total 36 Henry s Law: apples to very dlute solutons, and state that the partal pressure of a component n soluton s proportonal to t mole fracton: Chemcal Potental Chemcal potental s defned as: P =hx for X << 1 37 µ = G n P,T,n 38 where n s the number of moles of the th component. In multcomponent systems, the full expresson for the Gbbs Free Energy s: dg = VdP - SdT + µ dn 39 4
W. M. Whte Geochemstry Appendx II The Gbbs-Duhem Relaton At equlbrum and at constant pressure and temperature: Equaton Summary n dµ =0 40 Thermodynamc Varables n Ideal Solutons µ, deal = µ 0 +RTlnX 41 V deal mxng = 0 and therefore: V deal = X v H deal mxng = 0 and therefore: H deal = X h S deal mxng = -R S deal soluton = G deal mxng =RT G deal soluton = X ln X X S -R X ln X o X µ +RT X ln X X ln X = X V = X H 42 43 44 Thermodynamc Varables n Non-Ideal Solutons Fugacty: Fugacty can be thought of as the escapng tendency of a gas n non-deal solutons. Because systems tend toward deal at low pressure, t has the property: lm ƒ P 0 =1 45 P and Actvty: Actvty s defned as: µ = µ o +RTln ƒ ƒ o 46 a f f o 47 hence: µ = µ o +RTlna 48 5
W. M. Whte Geochemstry Appendx II The actvty n an deal soluton s: The actvty coeffcent, λ, s defned as: Equaton Summary a,deal = X 49 a = X λ 50 When Henry's Law law holds: λ = h 51 The Debye-Hückel equaton s used to calculate actvty coeffcents n aqueous solutons. It s: 2 log 10 γ = -Az I 1+Bå I where z s charge, I s onc strength, å s the hydrated onc radus (sgnfcantly larger than onc radus), and A and B are solvent parameters. I s calculated as: I= 1 2 Excess Free Energy and actvty coeffcents: G excess =RT m z 2 X lnλ Excess Free Energy and Margules Parameters of a Regular Soluton: G ex =X 1 X 2 W G 55 Excess Free Energy and Margules Parameters of an Asymmetrc Soluton: G excess =W G1 X 2 +W G2 X 1 X 1 X 2 56 Equlbrum Constant The equlbrum constant s defned as: 52 53 54 ν K= a It s related to the Gbbs Free Energy change of the reacton by: 57 K=e G /RT 58 It s related to enthalphy and entropy changes of the reacton by: o o ln K = H r RT + S r R Pressure and temperature dependences of the equlbrum constant are: o ln K P = V r T RT 59 60 Oxdaton and Reducton: The redox potental s related to the Gbbs Free Energy change of reacton as: G = -nfe 61 6
W. M. Whte Geochemstry Appendx II Equaton Summary where E s the redox potental, n s the number of electrons exchanged and F s the Faraday constant. The Nernst Equaton s: E = E RT nf ln Πa ν 62 The pε s defned as: and s related to hydrogen scale redox potental, E H, as: pε = -log a e 63 pε = FE H 2.303RT 64 Knetcs Reacton Rates: For a reacton such as: aa + bb cc + dd A general form for the rate of a reacton s: 1 a da dt = 1 b db dt = 1 c dc dt = 1 d dd dt = k An A B n B C n c D n D 65 where n A, etc. can be any number. For an elementary reacton, ths reduces to: 1 a da dt = 1 b db dt = 1 c dc dt = 1 d dd dt = k Aa B b 66 The temperature dependence of the rate constant s gven by the Arrhenus Relaton: k = A exp - E B 67 RT Rate constants of elementary reactons are related to the equlbrum constant as: k + = [B] eq =K app 68 k - [A] eq Dffuson: Fck s Frst Law s: J= D c x 69 where J s the dffuson flux and D s the dffuson coeffcent. Fck s Second Law s: c =D t x 2 c x 2 t 70 7
W. M. Whte Geochemstry Appendx II Equaton Summary The temperature dependence of the dffuson coeffcent s: D=D o exp - E A RT 71 Dagentc Equaton: C t x = F + ΣR 72 x t Trace Elements Equlbrum or Batch Partal Meltng: C o = 1 C D s/ (1 F) + F 73 Fractonal Partal Meltng: C C o = 1 D ( 1 F) 1/D 1 74 Zone Refnng: C C o = 1 D (1 D 1)e DR 75 Equlbrum Crystallzaton: C l C o = 1 DX + (1 X) 76 Fractonal Crystallzaton: C l C o = 1 X D 1 77 Isotope Geochemstry Bndng Energy per Nucleon: E b = W M c 2 78 A Basc Equaton of Radoactve Decay: dn = λn 79 dt Isotope Growth (or Isochron) Equaton: R = R 0 + R P/D (e λt 1) 80 8