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W. M. White Geochemistry Appendix II Summary of Important Equations W. M. White Geochemistry Appendix II Summary of Important Equations Equations of State: Ideal GasLaw: PV = NRT Coefficient of Thermal Expansion: ∂ α≡α ≡ 1 V V ∂T Compressibility: ∂V β≡β≡± 1 V ∂P Van der Waals Equation: RT a P= - 2 V ± b V The Laws of Thermdynamics: First Law: ∆U = Q + W 1 written in differential form: dU = dQ + dW 2 work done on the system and heat added to the system are positive. The first law states the equivalence of heat and work and the conservation of energy. Second Law: dQrev = TdS 3 Two ways of stating the second law are Every system left to itself will, on average, change to a condition of maximum probability and Heat cannot be extracted from a body and turned entirely into work. Third Law: lim S= 0 T → 0 3 This follows from the facts that S = R ln Ω and Ω =1 at T = 0 for a perfectly crystalline pure substance. Primary Variables of Thermodynamics The leading thermodynamic properties of a fluid are determined by the relations which exist between the volume, pressure, termperature, energy and entropy of a given mass of fluid in a state of thermodynamic equilibrium - J. W. Gibbs The primary variables of thermodynamics are P, V, T, U, and S. Other thermodynamic functions can be stated in terms of these variables. For various combination of these variables there are 1 W. M. White Geochemistry Appendix II Equation Summary characteristics functions. The characteristic function for S and V is one of the primary variables: U. Thus dU = TdS + PdV 5 Other Important Thermodynamic Functions What then is the use of thermodynamic equations? They are useful precisely because some quantities are easier to measure than others. — M. L. McGlashan Enthalpy: H ≡ U + PV 6 In differential form in terms of its characteristic variables: dH = TdS + VdP 7 Helmholtz Free Energy: A ≡ U - TS 8 and: dA = -PdV -SdT 9 Gibbs Free Energy: G ≡ H - TS 10 The Gibbs Free Energy change of a reaction at constant temperature and pressure is: ∆ G r = ∆Hr – T∆Sr 10a and: dG = VdP - SdT 11 Your choice of which of these functions to use should depend on what the independent variables in your system are. In geochemistry, P and T are the most common independent variables, so the Gibbs Free Energy is often the function of choice. Exact Differentials and the Maxwell Relations Any expression that may be written: M(x,y)dx+N(x,y)dy 12 is an exact differential if there exists a function z = f(x,y) such that f(x,y) = M(x,y)dx+N(x,y)dy 13 The total differential of the function z(x,y) is written: ∂z ∂z dz = dx + dy = Mdx + Ndy 14 ∂x y ∂y x If dz is an exact differential, then ∂M ∂N = 15 ∂y ∂x which is equivalent to: ∂M ∂N = 16 ∂y y ∂x x 2 W. M. White Geochemistry Appendix II Equation Summary All thermodynamic variables of state are exact differentials. Thus the practical application of the properties of exact differentials can be illustrated as follows. Equation 11 (dG = VdP - SdT) has the form dz = M(x,y)dx+N(x,y)dy since V and S are functions of temperature and pressure. Equation 11 may also be written as ∂G ∂G dG = dP + dT 17 ∂P T ∂T P and comparing equations 11 and 16, we conclude that ∂G ∂G =V and = –S 18, 19 ∂P T ∂T P Applying the rule embodied in Equation 15, we can conclude that: ∂V ∂S = – 20 ∂T P ∂P T Playing similar games with Equations 5 through 9, we can develop a series of relationships: ∂ T ∂P from dE ∂ =± ∂ 21 V S S V ∂ T ∂V from dH ∂ = ∂ 22 P V S P ∂ P ∂S from dA ∂ = ∂ 23 T V V T Equations 20 - 23 are known as the Maxwell Relations. Derivatives of Entropy ∂ S α pressure: ∂ =± V 24 P T ∂ S CV ∂S CP temperature: ∂ = and ∂ = 25, 26 T V T T P T ∂S α volume ∂ = β 27 V T Derivatives of Enthalpy ∂ H α pressure ∂ = V(1 ± T) 28 P T ∂ H temperature ∂ =CP 29 T P Derivatives of Energy ∂ U ∂U and α temperature: ∂ =CV ∂ =CP±P V 30, 31 T V T P 3 W. M. White Geochemistry Appendix II Equation Summary ∂U Tα volume: ∂ = β ±P 32 V T Difference between CP and CV TVα2 CP ±CV= β 33 The Gibbs Phase Rule: The Gibbs Phase Rule is a rule for determining the degrees of freedom of a system. f = c - p + 2 34 f is the number of degrees of freedom, c is the number of components, and p is the number of phases. The minimum number of components needed to describe a system is: c = N - R where N is the number of species, and R is the number of reactions possible between these species. The Clapeyron Equation The slope of a phase boundary in P-T space is: dT ∆ V = r 35 dP ∆Sr Solutions Raoult’s Law: applies to ideal solutions: P i = XiPtotal 36 Henry’s Law: applies to very dilute solutions, and state that the partial pressure of a component in solution is proportional to it mole fraction: Pi =hXi for Xi << 1 37 Chemical Potential Chemical potential is defined as: ∂ µ G 38 i= ∂ ni P,T,n th where ni is the number of moles of the i component. In multicomponent systems, the full expression for the Gibbs Free Energy is: µ dG = VdP - SdT +∑ idni 39 i 4 W. M. White Geochemistry Appendix II Equation Summary The Gibbs-Duhem Relation At equilibrium and at constant pressure and temperature: µ 40 ∑ nid i =0 i Thermodynamic Variables in Ideal Solutions µ µ0 i, ideal = i +RTlnXi 41 ∆ Videal mixing = 0 and therefore: Videal = ∑ Xivi = ∑ XiVi i i ∆ Hideal mixing = 0 and therefore: Hideal = ∑ Xihi = ∑ XiHi i i ∆S ideal mixing = -R∑ Xiln Xi i Sideal solution =X∑iSi-R∑ Xiln X 42 i i ∆ Gideal mixing =RT∑ Xiln Xi 43 i µo Gideal solution =X∑ii+RT∑ Xiln X 44 i i Thermodynamic Variables in Non-Ideal Solutions Fugacity: Fugacity can be thought of as the escaping tendency of a gas in non-ideal solutions. Because systems tend toward ideal at low pressure, it has the property: lim ¦i P→ 0 =1 45 Pi ¦ µ µo i and i = i +RTln o 46 ¦i Activity: Activity is defined as: f ≡ i ai o 47 fi hence: µ µo i = i +RTlnai 48 5 W. M. White Geochemistry Appendix II Equation Summary The activity in an ideal solution is: ai,ideal = Xi 49 The activity coefficient, λ, is defined as: ai = Xi λi 50 When Henry's Law law holds: λi = hi 51 The Debye-Hückel equation is used to calculate activity coefficients in aqueous solutions. It is: 2 -Azi I log γ = 52 10 i 1+Bå I where z is charge, I is ionic strength, å is the hydrated ionic radius (significantly larger than ionic radius), and A and B are solvent parameters. I is calculated as: 1 2 I= ∑mz 2 i i 53 i Excess Free Energy and activity coefficients: G =RT∑ Xlnλ excess i i 54 i Excess Free Energy and Margules Parameters of a Regular Solution: Gex =X1X2WG 55 Excess Free Energy and Margules Parameters of an Asymmetric Solution: G =W X +W X X X 56 excess G1 2 G2 1 1 2 Equilibrium Constant The equilibrium constant is defined as: ν i K=∏ ai 57 i It is related to the Gibbs Free Energy change of the reaction by: ±∆GÊ/RT K=e 58 It is related to enthalphy and entropy changes of the reaction by: ∆ o ∆ o Hr Sr ln K = ± + 59 RT R Pressure and temperature dependencies of the equilibrium constant are: ∆ o ∂ln K Vr ∂ =± 60 P T RT Oxidation and Reduction: The redox potential is related to the Gibbs Free Energy change of reaction as: ∆G = -nFE 61 6 W. M. White Geochemistry Appendix II Equation Summary where E is the redox potential, n is the number of electrons exchanged and F is the Faraday constant. The Nernst Equation is: ν E = E° – RT ln Πa i nF i 62 The pε is defined as: ε ± p = -log ae 63 and is related to hydrogen scale redox potential, EH, as: FEH pε = 64 2.303RT Kinetics Reaction Rates: For a reaction such as: aA + bB ® cC + dD A general form for the rate of a reaction is: 1 dA 1 dB 1 dC 1 dD n n n n = =± =± = k A AB B C c D D 65 a dt b dt c dt d dt where nA, etc. can be any number. For an elementary reaction, this reduces to: 1 dA 1 dB 1 dC 1 dD a b = =± =± = k A B 66 a dt b dt c dt d dt The temperature dependence of the rate constant is given by the Arrhenius Relation: EB k = A exp - 67 RT Rate constants of elementary reactions are related to the equilibrium constant as: k+ [B]eq app = =K 68 k- [A]eq ∂c Diffusion: Fick’s First Law is: J=±D 69 ∂x where J is the diffusion flux and D is the diffusion coefficient. Fick’s Second Law is: 2 ∂ c ∂ c =D 70 ∂ ∂ 2 t x x t 7 W. M. White Geochemistry Appendix II Equation Summary The temperature dependence of the diffusion coefficient is: E D=D exp - A 71 o RT ∂C i ∂Fi Diagentic Equation: =± +RΣ 72 ∂ ∂ i t x x t Trace Elements Equilibrium or Batch Partial Melting: Ci 1 o = 73 Ci Ds/ (1 ± F) + F Ci 1 1/D ± 1 Fractional Partial Melting: o = (1±F) 74 Ci D Ci 1 1 ±DR Zone Refining: o= ±( ±1)e 75 Ci D D l Ci 1 Equilibrium Crystallization: o = 76 Ci DX + (1 – X) l Ci D–1 Fractional Crystallization: o = 1 – X 77 Ci Isotope Geochemistry W–M 2 Binding Energy per Nucleon: E = c 78 b A dN Basic Equation of Radioactive Decay: =–λN 79 dt Isotope Growth (or Isochron) Equation: λ R = R0 + RP/D (e t – 1) 80 8.
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