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Chapter 8 Thermodynamic Potentials 1 Now that we have the 1st and 2nd Laws of thermodynamics, you might think that we can now apply these laws to many different situations using the concepts of internal energy and entropy. Although this is possible, it is very helpful to introduce three other state variables with the dimensions of energy. We have already made our acquaintance with one of them, the enthalpy. In this chapter we introduce two other concepts which have wide use. 2 In this chapter we introduce thermodynamic potentials. There is a fundamental set of four, all of which are energies and hence conserved. They are: U the internal energy H enthalpy ( old fashioned name is “heat content”) Important when discussing heat capacities and latent heats. F Helmholtz Function. Often A is used as the symbol. Important in statistical mechanics G Gibbs Function. Important in phase transitions and chemical reactions. The thermodynamic potentials are also called generating functions. These potentials, similar to potential energy in mechanics, are introduced for convenience in the analysis of certain processes. 3 • Legendre Transformation This is a very useful mathematical transformation. For example it is used in classical mechanics to go from the Lagrangian to the Hamiltonian. One often introduces the notion of canonically conjugate pairs, one variable being extensive and the other intensive. For example (V, -P), (S, T). In an analysis of certain processes U is often not a convenient function to work with. We have already introduced another useful function (H) and now we will introduce two new state functions: Helmholtz F Gibbs G 4 The thermodynamic potentials U, H, F and G are state functions. If we know these functions then all the thermodynamic properties of a system can be calculated by differentiation alone. Consider dU(S,V) = TdS - PdV The extensive variables S,V are taken as independent. We say that U is a generating function having as its natural variables S and V. Canonically conjugate pairs are (-P, V) and (T, S). The first pair are mechanical variables and the second pair are thermal variables. 5 We have dU(S,V) = TdS - PdV But we can write dU US V dS UV S dV Comparing gives US V T and UV S P These two equations are called the equations of state. Hence if we know U(S,V) we can obtain T and P by differentiation. There are four possible pairings of a thermal variable and a mechanical variable: (S, V) (S, P) (T, V) (T, P) We will obtain thermodynamic potentials for each of these pairs. This will be done using a Legendre Transformation. 6 Consider Z = Z(x, y) dZ = Xdx + Ydy _ _ _ _(1) with (x, X) and (y, Y) as canonical pairs. Suppose we wish to replace (x, y) by (X, Y). We use a transformation to form a new function M(X, Y) = Z - xX - yY dM = dZ - Xdx - Ydy - xdX - ydY dM = - xdX - ydY _ _ _ _(2) Z Z dZ dx dy x y y x M M dX dY dM X Y Y X The reciprocity relations are: Zx y X MX Y x Z y x Y MY X y 7 To replace only one variable, say y by Y: N(x,Y) = Z - yY _ _ _ _(3) and using (1) dN = Xdx - ydY _ _ _ _(4) N N dN dx dY x Y Y x with reciprocity relations Nx Y X NY x y With this introduction we proceed to the thermodynamic potentials. 8 Definition of the Thermodynamic Potentials We had, for the first law of thermodynamics, for a hydrostatic system: dU(S, V) = TdS - PdV _ _ _ _(5) We will replace U(S, V) by a new state function H (S, P) (natural variables). In this case we are replacing only one variable V by its conjugate -P. H=U-(-P)V → H=U+PV dH=dU+PdV+VdP dH=TdS-PdV+PdV+VdP → dH=TdS+VdP H H dS dP dH S P P S 9 Reciprocity relations: HS P T HP S V If H is written in terms of S and P differentiation permits us to calculate T and V. 10 The Helmholtz Function Now we wish to replace U(S, V) by F(T, V) We wish to replace S by T. The appropriate Legendre transformation is then F(V,T)=U-ST dF=dU-SdT-TdS=TdS-PdV-SdT-TdS dF(V,T) = -PdV – SdT F F dT dV dF T V V T F V T P F T V S Again, if F is a known function of V and T (natural variables), then we can obtain P and S. 11 The Gibbs Function Now we wish to replace U(S, V) by G(T, P). Two variables are changed so we use G(T, P) = U – ST + VP giving dG(T, P) = -SdT + VdP GT P S GP T V Let us summarize the thermodynamic potential functions. dU(S,V) = -PdV + TdS _ _ _ _(a) dH(S,P) = VdP + TdS _ _ _ _(b) dF(T,V) = -PdV – SdT _ _ _ _(c) dG(T,P) = VdP – SdT _ _ _ _(d) 12 Helmholtz Function In statistical mechanics the quantity that is usually calculated is the Helmholtz Function. Other thermodynamic properties are then calculated from this. Before pursuing a general discussion of this function, let us consider a example to show how U and F differ by means of a simple example. We consider a closed cylinder in which there is a piston which separates two gases which as not initially at the same pressure. The situation is shown on the next slide. 13 Bath at T V1 V2 Cylinder Piston (adiabatic) We let the piston move until equilibrium is achieved. Now we consider an infinitesimal reversible change about this equilibrium position. (a) Suppose that the wall of the cylinder is adiabatic. đ Q dU PdV đ Q 0 dU PdV dU dU 1 dU 2 P1dV1 P2 dV2 Since the pressures are now the same and obviously dV2 dV1 we have dU=0. At equilibrium the internal energy has its minimum value. 14 (b) Suppose that the wall of the cylinder is diathermal. dF SdT PdV dT 0 dF PdV dF dF1 dF2 P1dV1 P2dV2 Since the pressures are now the same and dV2 dV1 we have dF=0. In this situation, it is F, not U that is a minimum at equilibrium. (U is not minimized in this case because the total entropy of the system and the reservoir must be maximized and the Helmholtz Function takes this into account.) In continuing our discussion of the Helmholtz Potential, we consider isothermal processes. {As an aside we have (slide 11) F V T P } 15 Recall that if we have a system in contact with a reservoir (surroundings at constant temperature) then S (system) + S (reservoir) 0. If an amount of heat Q is transferred from the reservoir to the system, then (at T) S(system) - Q 0 giving T Q T (S) (system) The equality sign holds only for a reversible process. We have F= U – ST (This is just the Legendre Transformation.) At constant temperature: F = U - TS or F = Q - W - TS 16 If no new entropy is created in a process (reversible process) then Q = TS and F = -W or ( F f Fi ) W (constant T) The change in the Helmholtz function is then equal to the work done. If new entropy is created in a process (irreversible) then Q <TS and F < -W or ( F f Fi ) W (constant T) The work is all the work on or by the system, including any done by the system’s surroundings. Conclusion: If a system undergoes a reversible process at constant T and V (no work) then the Helmholtz Function does not change. If the system undergoes an irreversible process under constant T and V, then the Helmholtz Function decreases. 17 Conclusion: The change in F in an isothermal reversible process is the work done on or by the system. In an isothermal process, the maximum amount of work that can be done by a system is the decrease in the Helmholtz function. W Fi F f (constant T) One can also say that the decrease in F gives the maximum amount of energy that can be fixed in an isothermal process and made available for work. (The symbol A is often used for the Helmholtz function. Arbeit is German for work). For this reason F is often called the Helmholtz free energy. So F is intimately associated with work in an isothermal process. (See slide 22 for an example.) 18 If we consider constant T and constant V and no other types of work, W = 0 and F 0 and so (Ff - Fi) < 0 in an irreversible process. Natural processes at constant T and V undergoing change will move to lower values of F. Equilibrium occurs at minimum F. The Helmholtz potential is a function of T and V, its natural variables. In statistical mechanics these are the important variables, hence the importance of the Helmholtz potential in statistical mechanics. The example starting on the next slide is to show that, if the Helmholtz Function is known, many other thermodynamic properties can be calculated. 19 Example: Some expressions involving F(T,V). We previously obtained the expressions: F V T P F U TS F T V TS F U S H U PV G H TS (U PV ) ( F U ) F PV F G F V V T U F TS H U PV F U F T T V F F V H F T T V V T 20 U CV T V and we had F U F T T V 2 F F F CV T 2 T V T V T V F C V T 2 T V 2 Notice that the thermodynamic quantities are all expressed as functions of F F , F ,V , T , T V V T If one can theoretically calculate F(V,T) then any other thermodynamic function can be obtained as a function of V and T.21 EXAMPLE: A cylinder contains a piston on each side of which is one kilomole of an ideal gas. The walls of the cylinder are diathermal and the system is in contact with a heat reservoir at T=273K. The initial volumes of the gases are 12 liters (left) and 2 liters (right). The piston is now moved reversibly so that the final volume on both sides is 7 liters. What is the change in the Helmholtz potential? f 12 dF=-PdV-SdT=-PdV 2 F PdV i 1 kmole Since we have 1 kmole of each gas PV=RT and T is constant. 7 For the left side: Wl RT dV RT ln 7 1 kmole 273K 12 7 7 dV Wr RT RT ln V 2 2 12 7 7 1 kmole 1 kmole For the right side V 7 7 49 J F RT ln 8.314 103 (273 K ) ln K 12 2 24 (Some work is done and F decreases.) F 1.62MJ 22 EXAMPLE (Problem 8.5): The Helmholtz function for a certain gas is: n 2a F( V , T ) nRT ln( V nb ) J(T) V Obtain an expression for the pressure of the gas. We have the reciprocity relation 1 n2 a P 2 nRT V V nb 1 n2 a P 2 nRT V V nb F P V T n2 a nRT P 2 V nb V n2 a P 2 (V nb) nRT V Placing in a form using the specific volume: a P 2 ( v b) RT v Van der Waals gas (The reciprocity relationship has led to the equation of state.) 23 EXAMPLE: A van der Waals gas undergoes an isothermal expansion from some initial specific volume to a final specific volume. Calculate the change in the specific Helmholtz function. We have dF PdV SdT PdV df Pd v a P 2 ( v b) RT v RT a f 2 dv vb v v 1 v2 v2 b 1 1 a f RT ln v1 b v 2 v1 (Again the Helmholtz thermodynamic potential is closely associated with the work done in an isothermal process.) 24 EXAMPLE: The molar value for F for a gas is given by T 1 1 vb s 0 (T T0 ) f0 f c V (T T0 ) c V T ln a RT ln T0 v v 0 v0 b (a) Derive the equation of state. (b) Derive the energy equation. (a) From slide 12 (or from fact sheet) df=-Pdv-sdT f f f ( v , T) dT dv T V v T f (reciprocity relations) P ......(1) v T From (1) P a RT P a RT 2 2 v v b v vb so f s ......( 2) T v a P 2 ( v b) RT v 25 (b) From fact sheet F=U-TS Using (2) f U=F+TS u f T T V T vb s 0 u f Tc V c V ln c V R ln T0 v0 b T 1 1 vb s 0 (T T0 ) f 0 u c V (T T0 ) c V T ln a RT ln T0 v v 0 v0 b T vb s 0 T c V T ln RT ln T0 v0 b 1 1 u c V (T T0 ) a s 0T0 f 0 v v0 u cVT a u0 v 26 • Gibbs Function (very important in chemistry) G = H – TS To create a system one need not supply the entire enthalpy, as some energy (TS) can flow in as heat from the surroundings. Only the difference (H-TS) must be supplied by work. (See example on slide 34.) Consider a system in surroundings that constitute a T and P reservoir. This is the usual environment for chemical reactions and some phase changes. (T and P natural variables for the Gibbs Function.) Since H = U + PV G = U - TS + PV 27 Considering a process which is both isothermal and isobaric G = U - TS + PV and since U = Q - W, then G = Q - W - TS + PV In general, besides any mechanical work, we can have some other work, such as electrical work. W = PV + W(other) G = Q - TS -W(other) Suppose W(other) = 0 As before Q TS and so G 0 for constant T, P. Again, in an evolving process, equilibrium will be established when G reaches its minimum value. 28 Suppose W (other) 0 If we have a reversible process Q = TS and G = -W(other), so W(other) (Gi Gf ) Gf - Gi = -W (other) or The change in the Gibbs function gives the maximum energy that can be freed in an isothermal, isobaric process and made available for non-mechanical work. For this reason one often speaks of the Gibbs free energy. This can be confusing as we have the Helmholtz free energy also. It is best to avoid using free in both cases. N.B. The state variables T and P need not be fixed throughout the process, but must have the same initial and final values. 29 Reversible processes: Let us summarize some results for some particular reversible processes. Isochoric Isobaric Isothermal Isothermal, isobaric dU = đQ dH = đQ dF = -đW dG = -đW(other) 30 Finally let us consider a change in phase (sublimation, fusion and vaporization) that takes place isothermally and isobarically. No work (other) is involved and so, in a reversible process, G = 0 or G = constant. We use g to indicate the molar Gibbs function. Notation: g' = saturated solid g" = saturated liquid g"' = saturated vapor Equation for fusion curve is g' = g" Equation for vaporization curve is g" = g"' Equation for sublimation curve is g' = g"' At the triple point g' = g" = g"' [Note: saturated means that the state of the system is at one of the phase transformation (or coexistence) curves on a P-T diagram.] 31 The quantity G is tabulated for a variety of chemical reactions and other processes. G = H - TS In most work we do not need to choose a reference point as, usually, only changes in the quantities are of interest. {One source of tabulations of thermodynamic properties is the CRC Handbook of Chemistry and Physics.} 32 The relationship amongst the thermodynamic potentials can be summarized as follows: - TS U F H G + PV 33 1 EXAMPLE: Electrolysis of 1 mole of water. H2O H2 O2 2 1 mle 1 mle+0.5 mle Process takes place at P=1 atm. and T=298K From a table we find that the change in enthalpy required for this phase change is H 286 kJ Some data (for 1 mole): S H2O 70 J K S H2 131 J K SO2 205 J K The enthalpy must be supplied in some fashion. Must all of this energy be supplied by work or can some enter from the environment as heat? Considering some reversible process and since it is isothermal and isobaric: H G TS But G W (other) W (electrical ) H W(electrical ) TS (1) J 205 J S S f Si 131 70 163 2 K K 34 Since T=298K TS 49 kJ This increase in entropy is due to energy coming into the system by means of the heating process. (Note that the volume increases.) The electrical work that must be performed is, from equation (1): H TS W(electrical ) 286 kJ 49 kJ 237 kJ W(electrical ) The – sign indicates that work must be done on the system The gas must push the environment aside. We obtain the work done by the gas by assuming an ideal gas (neglect the volume of the water) 1 J (298 K ) 4 kJ PV nRT 1 mole mole 8.31 2 mole K So, of the total energy supplied (237+49)kJ=286kJ, (286-4)kJ=282kJ goes into an increase in the internal energy of the system. 35 {Or G=U-TS+PV U G PV TS } PV 4kJ 237kJ (electrical energy) U 282 kJ SYSTEM 49kJ (heat) 36 Fuel Cell. If one combines hydrogen and oxygen to produce water in a controlled way one can extract, in principle, 237 kJ of electrical work for each mole of hydrogen. This is the principle of the fuel cell. This is the reverse of the process that we have just examined. 49 kJ of energy is expelled as waste heat. A reasonable measure of the efficiency is W (electrical ) U PV ( gas) 237 0.83 282 4 In practice, the efficiency will be somewhat less, but still better than a heat engine. {Where do the hydrogen and oxygen gases come from?) 37 EXAMPLE: The specific Gibb’s potential of a gas is given by: P g RT ln BP with B B(T ) P0 (Natural Derive everything. Express answers in terms of P and T variables) (a) Equation of state g g dg sdT vdP dg dT dP T P P T giving g s .....(1) T P From (2) RT v B P (reciprocity g v .....( 2 ) relations) P T P( v B) RT (b) Specific entropy From (1) P dB s R ln P P0 dT P dB s R ln P P dT 0 38 (c) Helmholtz potential g f Pv RT v B P (d) enthalpy f g Pv P f RT ln BP Pv P0 P f RT ln BP RT PB P0 P f RTln 1 P0 P P dB g h Ts h g Ts RT ln BP RT ln PT P P dT 0 0 dB h BP PT dT (e) Internal energy f u Ts P P u RT ln 1 T R ln P0 P0 dB B h P T dT u f Ts P P dB dB RT ln RT RT ln PT P dT P P dT 0 0 u RT PT dB dT dB u T P R dT 39 (f) Specific heat at constant pressure dB h d 2 B dB cP cP P T 2 T dT dT dT P d2B cP PT dT 2 (g) Specific heat at constant volume u cV T V RT dB RT2 dB dB u TP R T R RT dT ( v B ) dT ( v B ) dT 2RT dB RT2 d 2B 1 dB 2 dB cV RT R 2 2 ( v B) dT ( v B) dT dT ( v B) dT ! 2 dB P 2 ( v B ) 2 d 2 B RT2 dB c V 2P R 2 2 dT R( v B ) dT ( v B ) dT c V 2P 2 dB 1 2 d B P dB P ( v B) 2 R dT R dT R dT 2 cV 2 P 2 2 dB d B P dB R PT 2 dT R dT dT 2 2 40 (h) Isothermal compressibility 1 v RT v RT v B 2 v P T P P T P 1 RT v P2 RT P( RT PB ) (i) Volume expansivity 1 v v T P RT R dB v v B P T P P dT RP Pv dB dT 1 R dB v P dT dB dT RT PB RP 41 (j) Joule-Thomson coefficient T P h T P h dB h T dT P T T P h 1 P h h T T P h P T dB h PT PB dT h T P 2 h dB d B dB B PT 2 P P dT dT dT T P dB dT d2B PT dT 2 B T 42 • Maxwell’s Relations Let us apply the condition for an exact differential to the differential forms of the potentials. dU = TdS - PdV dH = TdS + VdP dF = - SdT - PdV dG = -SdT + VdP VT S PS V TP S VS P VS T TP V PS T VT P 43 The relationships among the partials are Maxwell’s relations. These relations are enormously useful. Each partial involves a state variable that can be integrated along any convenient reversible path between two states to obtain the difference in the value of a state variable between two equilibrium states. They can be used to obtain values, or differences between values, for quantities not easily measured from more easily measured quantities, such as P, V or T. They are often used to replace a partial, which cannot be evaluated, by another partial which can be determined (see slide 47). 44 The VFT-VUS diagram is a device for remembering the differential forms of the potentials. Notice the direction of the arrows. If you go in a direction opposite to an arrow, the quantity is negative. The potentials are at the sides. They are functions of the state variables at the corresponding corners. The corresponding conjugate coordinate to the variable is obtained by following the diagonal arrow. 45 V F U S T G H P Consider the Gibbs function, dG will consist of a dT and a dP. The coefficient of dT will be -S and the coefficient of dP will be V. dG=-SdT+VdP 46 EXAMPLE: Ideal gas undergoes an isothermal reversible process P0 P Reversible process so Q=TdS đ Consider S=S(T,P) S S dT dP dS T P P T S dP P T dT=0 (isothermal) so dS đ Q T S P T dP We cannot evaluate the derivative involving S so we get rid of S by S V using the Maxwell relation P T V đ Q T dP T P nRT đQ dP P If P P0 Ideal gas so T P V nR T PV=nRT P P P Q nRT P0 dP P P Q nRT ln P0 Q<0 and heat flows out of the system. 47 • Clausius-Clapeyron Equation This is an important relation describing how the P varies with T for a system consisting of two phases in equilibrium. From the VFT-VUS diagram the molar Gibbs function is given by: dg = -sdT + vdP Hence s g T P and v g P T (reciprocity relations) {Note that we are concerned with the variables P and T and these are the natural variables for the Gibb’s Potential.} 48 In a first order phase transition these variables change as follows: v s phase (i) phase (f) T phase (i) phase (f) T (In higher order phase transformations s and v are the same.) 49 At the T of the phase transition g is single valued but the slope is discontinuous. slope discontinuity g phase (f) phase (i) T 50 Two phases can exist in equilibrium at various combinations of (P, T). On a P, T diagram there will be a phase boundary (coexistence curve). Phase equilibrium line with g (i) = g (f) phase (i) P b • a phase (f) • T 51 We consider a reversible phase change at constant T and P. As was discussed earlier, (slide 31) the Gibbs function will remain constant: g(i) = g(f) This is true for all phases in equilibrium, i.e. on a phase equilibrium line. Consider the two neighboring points a and b, separated by dP and dT. The g’s will change as we go from a to b, but they must remain equal, so dg(i) = dg(f) -s(i) dT + v(i) dP = -s(f) dT + v(f) dP [Note: P and T are not independent since we are moving along a phase boundary.] 52 dP dT This yields s( f ) s( i ) v( f ) v( i ) Hence the slope of the phase boundary is determined by the entropies and volumes of the two phases . đQ dS S From T T in which ℓ is the appropriate latent heat, we therefore have dP dT T v (f) v ( i ) This is the famous Clausius-Clapeyron equation. It applies to the slope of any phase boundary line on a PT diagram. 53 Let us consider a sublimation or vaporization curve. The vapor will be approximated as an ideal gas. Since v(i) << v(f) we can make the excellent approximation v(f) - v(i) = v(f). From the ideal gas law v ( f ) RT ,P the Clausius-Clapeyron equation now becomes dP dT P2 dP P1 P ln P RT 2 R T2 dT 2 T1 T P2 P1 1 R T2 , giving 1 T1 54 • Example A pressure cooker At P = 1 atmosphere the BP of water is 373k. What is the BP at 2 atmospheres? The latent heat of vaporization is 40.68 kJ/mole. Using the above equation: ln 2 R 1 T2 1 373 k giving T2 = 393.8k = 120.8 0C. The experimental value is 120.6 0C (Latent heat has some temperature dependence, but this can usually be ignored if the range is not very great.) 55 EXAMPLE: Clausius-Clapeyron Equation. s always increases except Gas Liquid Solid 3 dP s( f ) s( i ) (f ) dT v v ( i ) specific volume usually increases He below 0.3K Consider a solid to liquid transition. s ( f ) s (i ) 0 dP (f ) (i ) v v 0 0 If curve has positive slope dT dP (f ) (i ) v v 0 0 If curve has negative slope dT One of few substances with a negative slope is water 56 S-L (water) S-L P S-L slopes exaggerated L S G L-V V S-V T 57 Diamonds and Graphite These are two phases of C, differing in crystal structure. At P=1 atm. the most stable phase is graphite, so diamond will convert spontaneously to graphite, but the process is very slow at room temperature. The process increases with T, so do not store your diamonds near the fireplace. Graphite is more stable because it has a smaller Gibb’s function. T=298K g(kJ) 5 1 bar =10 Pa diamond 2.9 kJ graphite 0 5 10 15 P (kbar) Point on phase equilibrium line 20 58 Above about 15kbar diamond is more stable than graphite at room T. (Natural diamonds are formed at great depths below the surface of the Earth, some 100-200 km deep). Suppose that T is raised to 398K. At what P do these two phases have the same Gibb’s Function? We start at (298K, 15kbar) At room temperature (diamond converting to graphite) 3 J m s 3.4 v 1.9 10 6 mole K mole From the Clausius-Clapeyron Equation (slope of phase boundary) J 3.4 (an approximation because entropy P s mole K 3 increases with T) T v 6 m 1.9 10 mole So for a T 100 K P 1.8kbar The point on the phase transformation line is then (398K, 16.8kbar). The first synthesis of diamond from graphite was accomplished at 59 1800 K, 60 kbar EXAMPLE: Let us use the Clausius-Clapeyron equation to calculate the decrease in temperature from the triple point to the NMP. H2 0 P TP 0.01C 273.16K 612Pa magnified 5 fusion curve {NMP 0.00C 273.15K 1.013 10 Pa } NMP Apply the Clausius-Clapeyron equation to fusion curve TP T 3 m v 10 3 kg dP dT H 2O T( v v ) dT dP T( v v ) kg 10 m3 3 ice 3 m v 1.09 10 3 kg kg 916 3 m 1 v J 3.34 10 kg 5 60 3 m 273.16K 10 3 1.09 10 3 kg dT (1.01 105 612)Pa 5 J 3.34 10 kg dT=-0.0074K Presence of dissolved air in ice-water mixture further reduces the temperature by 0.0023K. Hence dT=-0.0097K 61 EXAMPLE: Prove that, on a phase diagram, the slope of the sublimation curve at the triple point is greater than that of the vaporization curve at the same point. We consider a substance, such as water, that expands upon freezing. v v ! v v v v H 2O Fusion curve From VFT-VUS dH=TdS+VdP At P=constant dH=TdS= đ Q At a phase transformation Liquid P đ Q dS T (i ) Solid Vapor dP s( f ) s (f ) (i ) dT v v T v(f ) v(i ) h( f ) h(i ) T h h h h h h (h h) (h h) (h h) 62 SV LV SL at the triple point dP dP dP T( v v ) T( v v ) T( v v ) dT SV dT LV dT SL Remember that, in this case, v v dP dP ( v v) dP ( v v) dT dT ( v v ) dT ( v v ) SV LV SL >1 (-) (-) dP dP dT SV dT LV 63