Chapter 17: One- Component Fluids

Transcription

1 Chapter 17: One- Component Fluids Chapter 17: One-Component Fluids Introduction he Fundamental Surface he fundamental equation Behavior at a stability limit he behavior of cp and α at an instability Phase Equilibria he P- and -v phase diagrams of a one-component fluid Phase changes on increasing temperature Phase changes on increasing pressure he Clausius-Clapeyron Equation he Liquid-apor Critical Point he critical point in the P-v Plane he phase diagram near the critical point hermodynamic properties near the critical point he fundamental surface near the critical point he an der Waals Fluid he thermodynamics of the an der Waals fluid he phase diagram of a an der Waals fluid he Law of Corresponding States INRODUCION he simplest class of materials are one-component fluids. his class includes the ideal gases, whose properties we have already studied in some detail, imperfect gases and liquids, and one-component solids that are in equilibrium with a hydrostatic pressure. he general class of one-component fluids includes quantum fluids such as systems of identical electrons and bosons, although we shall not study their properties here. Despite the simplicity of the one-component fluids their thermodynamic properties are extremely difficult to calculate unless they are ideal gases, or "nearly ideal" gases whose behavior is governed by a small permutation of the fundamental equation of the ideal gas. he solved problems in the behavior of one-component fluids that are most pertinent to materials science treat the slope and curvature of the fundamental surface, its behavior near an instability, the nature of two- and three-phase equilibrium, the characteristics of phase transformations, and the geometry of the fundamental surface near the liquid-vapor critical point. We shall discuss these subjects in the context of classical thermodynamics. We shall end with a discussion of the "almost ideal" gas, which obeys the an der Waals equation of state. he latter is a very useful first approximation to the fundamental equation of a liquid or vapor phase since it naturally yields a vapor-liquid condensation on cooling and produces a critical point near which the fundamental surface of the fluid has the correct qualitative behavior. Page 374

2 17.2 HE FUNDAMENAL SURFACE he fundamental equation he fundamental equation of a one-component fluid is most conveniently written in one of three forms: the molar energy density, e = e(s,v) 17.1 where e is the energy per mole, s the entropy per mole and v the molar volume, the Gibbs- Duhem function, P = P(,µ) 17.2 which is also the negative of the volume density of the work function,, and the Gibbs free energy per mole, g = g(p,) = µ 17.3 Since the molar density of the Gibbs free energy is equal to the chemical potential, the Gibbs free energy function for a one-component fluid is simply an alternate form of the Gibbs-Duhem function, equation e v s Fig. 17.1: he energy surface of a one-component fluid. he shape of the fundamental surface of a stable fluid was discussed in Chapter 16. he form of the energy surface is shown in Fig Since de = ds - Pdv 17.4 Page 375

3 the energy increases monotonically with the entropy at constant volume and decreases monotonically with the volume at constant entropy. he matrix of thermodynamic properties, e ij, has the elements e ss = s = v c 17.5 v e vv = - P v = 1 s v s 17.6 e sv = e vs = v = 1 s vå 17.7 s All eigenvalues of the matrix are positive when the fluid is stable. Hence the fundamental surface is concave for all directions of motion, as shown in the figure. µ P Fig. 17.2: he potential surface of a one-component fluid. he molar Gibbs free energy, which is equivalent to the Gibbs-Duhem function, has the differential form and the property matrix dµ = - sd + vdp 17.8 µ = - s = - p c P 17.9 µ PP = v P = - v Page 376

4 µ P = µ P = v = vå P he condition of stability requires that the chemical potential be a convex function. he form of the fundamental surface is shown in Fig Behavior at a stability limit he least eigenvalue of the property matrix of the fluid vanishes at the stability limit. he eigenvalues of the property matrix of a one-component fluid are easily computed, and are = 1 [ ] 2 (e ss + e vv ) ± (e ss -e vv ) 2 + 4e 2 sv since e ss and e vv are positive definite when the fluid is stable the minimum eigenvalue, 1, is associated with the minus sign in he minimal eigenvector is a solution of the simultaneous equations (e ss - 1 ) 1 s + e sv 1 v = 0 e sv 1 s + (e vv - 1 ) 1 v = where 1 k is the component of the minimal eigenvalue in the direction of the coordinate k. For an infinitesimal change of state along the direction of the minimal eigenvector the molar entropy and volume change in the ratio 1 s 1 v s v = = 1 - e vv e sv at the stability limit 1 = 0. According to equation this requires e ss e vv = (e sv ) Either e ss or e vv may vanish at the limit of stability. he former case is particularly common in the instabilities that are associated with mutations such as ferromagnetic and ferroelectric transitions. In this case the specific heat has a singularity at the transition point, and the direction of the instability is along the entropic, or thermal axis. It may also happen that all three of the thermodynamic properties are non-zero at the stability limit. In this case the variation of state that leads to instability is a simultaneous variation of the entropy and volume in the ratio Page 377

5 e vv s v = - e sv he behavior of c P and α at an instability In those cases where e ss and e vv are well-behaved at the stability limit, the complementary properties, the isobaric specific heat, c P, isothermal compressibility,, and isobaric coefficient of expansion, å, are singular. We establish this through a slight modification of the general theorem on singularities of derivatives of the Gibbs-Duhem function that was established in Chapter 16. Using equation 17.9 and the properties of the Jacobian µ = - c P = - s P = - (s,p) (,P) = - (s,-p) (s,v) (,-P) (s,v) -1 = - e vv det(e ij ) Since e vv is finite at the instability while det(e ij ) 0, c P at the limit of stability. Using equation and the properties of the Jacobian µ PP = - v = v P e ss = - det(e ij ) which has the consequence that at the limit of stability. o find the behavior of the isobaric coefficient of expansion consider the ratio e vv e = - P sv = - s c P vå Since the right-hand side of remains finite while c P, å at the limit of stability. In those cases where e ss and e vv are well-behaved at the stability limit, the isentropic compressibility, s, is finite as well. Since the isentropic and isothermal compressibilities are related by the equation s = c v c P Page 378

6 the properties and c P must increase at the instability in such a way that their quotient remains finite. Let be a coordinate along the minimum eigenvector near the instability, and let its value at the instability be c. hen near c the isobaric heat capacity behaves like c P = c 1 + c 2 ( - c) where c 1 and c 2 are finite coefficients and > 0 is the critical exponent. It follows from equation that the isothermal compressibility near the instability can be written = ( - c) and from equation that the isobaric coefficient of expansion can be written Hence å = å 1 + å 2 ( - c) the properties c P,, and å have the same critical exponent at an instability of a one-component fluid PHASE EQUILIBRIA he P- and -v phase diagrams of a one-component fluid ^ S L P Fig. 17.3: Phase diagram of a one-component fluid with solid, liquid and vapor phases. A possible phase diagram of a one-component fluid that has solid, liquid and vapor phases is shown in Fig his phase diagram is the projection of the lower envelope of the fundamental surfaces µ(,p) onto the,p plane. he liquid-vapor equilibrium line terminates at the critical point marked in the figure. Assuming that the solid phase is crys- Page 379

7 talline, there is a discontinuity in symmetry on passing from solid to liquid or vapor. Hence there is no critical point on the S- and L- equilibrium lines. Fig shows a possible form for the -v phase diagram of the same fluid. his diagram is the projection of the equilibrium surface generated by the Helmholtz free energy density, f(,v), on the,v plane. he two-phase equilibrium lines on the -P diagram are split into two-phase regions that are generated by common tangents to the fundamental surface in the direction of v. he three-phase point in the P- plot is replaced by a line in the -v diagram that connects the S- and -phase fields and touches the L-phase field. he L+ two-phase region has the form of a cap that terminates at the critical temperature, c, where the specific volumes of the L and phases become equal. S S + L L L + S + v Fig. 17.4: Possible form for the -v phase diagram of a one-component fluid. Several general principles apply to the two-phase equilibrium lines in the phase diagram Phase changes on increasing temperature First consider the phase changes that occur on increasing the temperature of a fluid at constant pressure. he behavior of the chemical potential, or molar Gibbs free energy, is shown in Fig Since the curve of µ against slopes down. Since µ = - s < µ 2 = - c P < the curve is convex. Since the Gibbs free energy is minimum at equilibrium the equilibrium surface is the lower envelope of the µ- plot. Page 380

8 µ å Fig. 17.5: Plot of µ against for a fluid that has two phases, å and. At the limit of zero temperature the equilibrium phase is the one that has the lowest value of the molar enthalpy, h. As the temperature increases the chemical potential decreases according to equation 17.24, which has the consequence that phases of successively higher entropy appear. If a phase ( ) transforms into its successor (å) at the å- equilibrium temperature the transformation is reversible. he heat that is liberated during the transformation is the latent heat of the transformation, Q å = (s - s å ) Equation can be used to measure the change in entropy in a transformation from the heat evolved. Since µ = h - s and µ å = µ at the equilibrium temperature, the latent heat is equal to the change in the enthalpy h - h å = (s - s å ) = Q å If the transformation from å to is a first-order transformation then the thermodynamic densities are discontinuous and the latent heat is finite. he latent heat is zero in a mutation. Since a transformation from a low-temperature to a high-temperature phase increases the entropy, the latent heat is negative. When heat is absorbed the transformation is said to be endothermic. he reverse transformation involves a decrease in the enthalpy; Q å = - Q å, and heat is released. A transformation that liberates heat is called exothermic. Equation can be used to estimate the thermodynamic driving force for a transformation from a metastable state. Let be preferred to å when the temperature falls below 0. If the latent heat, Q å, is reasonably insensitive to the transformation temperature (as it is when the difference in the specific heats of the two phases, Îc P, is nearly constant), then the change in potential for the a transformation from å to is ε() = µ () - µ å () = Îh å - Îs å Page 381

9 Phase changes on increasing pressure = Q å Similar results apply to transformations that happen when the pressure is increased at constant temperature. he equilibrium surface is the lower envelope of a plot of µ against P. Since µ P = v > the µ-p curves are monotonically increasing, so phases of successively lower molar volume appear with increasing pressure (Fig. 17.6). Since phases of lower volume usually (though not always) have lower entropy, phase transitions that occur on increasing pressure are usually exothermic. µ vapor liquid Fig. 17.6: Plot of µ vs. P for a typical fluid with a vapor-liquid transition on increasing pressure. he thermodynamic driving force for the transformation of a phase that is made metastable by increasing the pressure at constant temperature can be estimated in the following way. Let P 0 be the pressure at which phase å transforms to phase under equilibrium conditions. he chemical potential near P 0 at given is P µ(p) - µ(p 0 ) = P µ P dp = P P 0 P 0 vdp If the phase is a solid or liquid then its molar volume is relatively insensitive to its pressure. Hence µ(p) - µ(p 0 ) ~ v(p-p 0 ) Page 382

10 If the phase is a vapor, on the other hand, the ideal gas law provides an approximate relation between its molar volume and its pressure, and µ(p) - µ(p 0 ) ~ ln(p/p 0 ) Hence, for a transformation between condensed phases at pressure, P, ε(P) ~ (P - P 0 )Îv For a transformation from a vapor phase to a condensed phase, ε(P) ~ ln P P where we have assumed that the molar volume of the condensed phase is negligible compared to that of the vapor. he quantity (P/P 0 ) is called the supersaturation ratio for a vapor-liquid transition he Clausius-Clapeyron Equation he equilibrium between two one-component fluids is represented by a curve in the P, plane. he Clausius-Clapeyron equation gives the slope of the curve. Suppose that phases å and are in equilibrium at temperature and pressure P. heir chemical potentials must be equal at P, and must also be equal at every other point on the equilibrium line. Hence for a change in state that preserves the two-phase equilibrium dµ å - dµ = 0 = (v å - v )dp - (s å - s )d Hence the slope of the two-phase equilibrium line is dp d = så - s v å - v = Q å (v å - v ) Equation is the Clausius-Clapeyron equation. If å is the high temperature phase with respect to then the latent heat of the å transformation, Q å, is positive. If the volume of å is greater than that of, as is usually the case, then the slope of the equilibrium line is positive: the temperature of the transition increases with increasing pressure. his result holds for all solid-vapor and liquid-vapor transitions and asserts that the sublimation or boiling temperature increases with the pressure (a fact familiar to anyone who has tried to boil potatoes in the Sierra). It also holds for most solid-liquid and solid-solid equilibria; the transformation temperature usually increases with pressure. Page 383

11 However, there are materials whose volume decreases in a solid-liquid transition, and for these the freezing point decreases with the pressure. he most familiar example is water. he suppression of the freezing point of water with pressure has the important practical consequence that ice is slick when the temperature is not too far below the freezing point; the pressure imposed by a shoe, ice-skate or tire creates a thin film of liquid. here are also solid-solid transformations in which the low-temperature phase has the higher volume. A classic example is the martensitic transformation in metastable austenitic steel. he higher volume of the low-temperature phase has the consequence that the transformation is promoted by hydrostatic tension. A practical case in which a hydrostatic tension develops is in the material immediately ahead of the tip of a sharp crack perpendicular to an axis of tension. he tendency of a metastable austenitic steel to transform to martensite immediately ahead of a crack tip is responsible for a type of transformation toughening that raises the fracture toughness of the material. he volume change is also positive in the martensitic transformation of metastable zirconia (ZrO 2 ), which has made possible the design of "transformation toughened" ceramics. he Clausius-Clapeyron equation can also be used to find the temperature dependence of the vapor pressure of a condensed phase. If the vapor pressure is low the molar volume of the vapor is large compared to that of the condensed phase. Equation can then be written dp d = Q v where Q is the latent heat of fusion and v is the molar volume of the vapor. If the molar volume of the vapor is approximated from the ideal gas law equation becomes d[ln(p)] d = Q he latent heat of fusion is usually a weak function of the temperature. In this case equation can be solved to give the vapor pressure in the form P = A exp - Q Equation shows that the vapor pressure varies with the temperature according to an Arrhenius equation in which the activation energy is the latent heat of fusion, Q HE LIQUID-APOR CRIICAL POIN he critical point in the P-v Plane he critical point of a one-component liquid-vapor system is the classical example of a critical state. While liquid-vapor equilibrium is of secondary interest in materials sci- Page 384

12 ence, the liquid-vapor critical point is worth studying as a simple, familiar example of a situation in which two phases become indistinguishable. he critical point of a one-component fluid is a single point in the P, plane, as shown in Fig No interesting behavior is revealed in the P, phase diagram; the equilibrium line between the liquid and vapor phases simply terminates there. he,v diagram, which is shown in Fig. 17.4, and the P,v diagram are more informative. he latter is most often used to visualize the critical point. A P-v phase diagram that shows the liquidvapor region and the critical point is drawn in Fig For pressures P < P c the P,v diagram contains two phases, liquid (L) and vapor (). he two phases are separated by a two-phase region that is bounded by the curves v L (P) and v (P), which give the molar volumes of the liquid and vapor states at equilibrium at given P. he two-phase region is capped at P c since the molar volumes of the liquid and vapor are identical there. P L L + v Fig. 17.7: A section of the P-v phase diagram of a one-component fluid showing the geometry near the critical point at P c. An isothermal line () is also shown. since An isotherm in the P,v diagram has a negative slope in any single phase region, P 1 v = - v and the isothermal compressibility,, is positive for a stable phase. When > c the equilibrium isotherm is a monotonically decreasing curve. When = c the equilibrium isotherm passes through the critical point. he slope of the isotherm is negative everywhere except at the critical point itself. If we write the Fundamental Equation as the molar Helmholtz free energy, f, f = f(,v) then the condition that the critical state be stable, which was discussed in the previous chapter, has the consequence that at the critical point P v = - 2 f v 2 = Page 385

13 2 P v 2 = - 3 f v 3 = P v 3 = - 4 f v 4 < he isotherms that pass below the critical point are more complex. An example is shown in Fig he equilibrium isotherm is a line that decreases through the liquid phase until it meets the two-phase region at the volume v L (P,). Since the pressure of the liquid and vapor phases are the same when they are in equilibrium, the equilibrium isotherm continues as a horizontal line through the two-phase region (the dashed line in the figure) to the point v (P,) and then decreases monotonically through the vapor-phase region. However, there is no reason to assume that the isotherms for the liquid and the vapor simply end at the boundaries of the two-phase region. he isotherm for either phase should extend to metastable states within the two-phase region. he extension on the liquid side must initially slope down into the two-phase region as the volume increases toward v (P,) while the extension on the vapor side slopes up into the two-phase region as the volume decreases toward v L (,P). he two curves diverge from one another. hey are shown in Fig as the bold portions of the extensions of the isotherm into the twophase region. Moreover, there is no reason to assume that the two phases cannot change continuously into one another below the critical point since they do merge above the critical point and the physical connection between them (a simple change in v) is the same below the critical point as it is above. If it were possible to vary the molar volume continuously from the liquid to the vapor below the critical point, however, there would necessarily have to be a range of (v) for which the isotherm has a positive slope to connect the two branches of the curve. his range of molar volumes corresponds to unstable states of the fluid, and is shown by the light connecting line on the isotherm in Fig hese considerations suggest that below the critical point the isotherms for the liquid and vapor phases extend into the two-phase region to provide a range of metastable states. he limit of metastability is determined by the condition P v = he solutions to equation define a stability gap, which is called the spinodal, which lies within the two-phase region and closes at the critical point. If the compressibility is well-behaved the spinodal widens as the temperature decreases, just as the equilibrium twophase region does. he regions between the two-phase equilibrium lines and the spinodal lines for the liquid and vapor are the regions for which metastable states can exist. Page 386

14 he phase diagram near the critical point While the phase diagram as a whole depends on the fundamental equations of the various phases, its geometry in the immediate vicinity of the critical point can be found by expanding the thermodynamic functions about their values at the critical point. In particular, we can expand the derivative ( P/ v) about the critical point, where its value is known to be zero. Using the variables Î = - c Îv = v - v c the result is, to second order in the deviations Î and Îv, P v = P vv Î + P v Î P vvv(îv) 2 + P vv ÎvÎ P v(î) where the symbols P ij, P ijk are the values of the partial derivatives of the function P (,v) at the critical point. From equation P vv = Hence if we keep only the lead terms in Î and Îv the expansion is P v = P v Î P vvv(îv) (he terms in ÎÎv and (Î) 2 are always smaller than the term in Î when the variations are small and the coefficients do not vanish, but the term in (Îv) 2 may be comparable in magnitude since the relative magnitudes of Î and Îv are unspecified.) he integral of equation is the thermal equation of state near the critical point, and has the form where P l () is a function of only such that and the functions P 2 and P 3 are given by P(Îv,Î) = P 1 () + P 2 ÎvÎ + P 3 (Îv) P l ( c ) = P c P 2 = P v P 3 = l 6 P vvv Page 387

15 When Î > 0 the equation of state, 17.52, determines the isotherm of a fluid that is stable for every Îv. Since stability requires ( P/ v) < 0 for all positive values of Îv and Î, it follows from equation that P v = P 2 < P vvv = 6P 3 < When Î = 0 equation has a stability limit (an inflection) at the critical point, Îv = 0, since equation vanishes there. When Î < 0 the isotherm determined by equation has two extrema (stability limits) which are located at the points Îv 0 = ± - P 2 Î 3P Since ( P/ ) > 0 when Îv < Îv 0, the region bounded by the solutions to equation is a region of instability that lies between stable fluid states. he equilibrium volumes, v L and v can also be found for small values of Î. Since the chemical potential is the same in the two equilibrium phases, an integral of dµ from the liquid (L) to the vapor () equilibrium states vanishes. Hence L dµ = 0 = L vdp = L (v c + Îv) dp = L ÎvdP = L Îv P v dv = L [P 2 ÎvÎ + 3P 3 (Îv) 3 ]dv where we have used the equality of the pressure of the L and states. he last form of equation shows that the integrand is an odd function of Îv (the kernel of the integral is antisymmetric about Îv = 0). It follows that Îv L = - Îv Since P L = P, equation then has the consequence that which has the solution P 2 ÎÎv + P 3 (Îv ) 3 = Îv = - Îv L = - P 2 Î P Page 388

16 Equations and determine the stable and metastable phase diagrams near c. he widths of both the stable two-phase region and the spinodal are proportional to (Î) 1/2 near c ; the two equilibrium lines and the two spinodal lines all converge at the critical point. he latent heat of fusion at temperature near the critical point is Q = (s - s L ) ~ c s v (v - v L ) ~ 2 c s v (Îv ) It follows that the latent heat of fusion is proportional to (Î) l/2 near the critical point and vanishes there hermodynamic properties near the critical point We found in Section 17.2 that the thermodynamic properties c P, å and are all singular at a stability limit of a one-component fluid. Hence they must be singular at the critical state. Using the expansion employed above we can both establish their singularity and calculate the critical exponents. From equation the isothermal compressibility near the critical point is given by the relation 1 = - P v v = - P 2 Î - 3P 3 (Îv) Now let the critical state be approached along a line with v = v c. When the critical state is approached from below this value of the molar volume corresponds to a two-phase equilibrium state. However, the two phases have the same value of (Îv) 2, and, therefore, have the same isothermal compressibility. Using equation to evaluate (Îv) 2 gives the result Hence near c, 1 = - P 2 Î - 3P v 3 - P 2 Î P = 2P 2 Î ~ 1 2vP 2 Î which behaves like (Î) -1. Page 389

17 When the critical state is approached from the single-phase side, Îv = 0 and the isothermal compressibility is = - 1 vp 2 Î which is, again, proportional to (Î) -1. he isobaric specific heat near the critical point can be found from the relation c P = c v + vå Using the Jacobian transformation it can be shown that vå 2 = - P2 P v he partial derivative P = [ P/ ] v is well-behaved near the critical point, so the numerator of equation is finite there. But the denominator, P v, vanishes at the critical point and at every point on the boundary of the spinodal. It follows that the value of c P near the critical point is dominated by the second term on the right in 17.68, and is approximated by c P = - P 2 P 2 (Î) + 3P 3 (Îv) When the critical point is approached from below v c, Îv is determined by the two-phase equilibrium condition, 17.62, and c P ~ P 2 2P 2 (Î) Equation shows that c P varies as (Î) -1, in agreement with the result obtained above for the isothermal compressibility. he behavior of the isobaric coefficient of thermal expansion, å, near the critical point can be found by using equation Since c v is negligible near c, Page 390

18 å ~ c P v which shows that å is also singular near c, and varies like (Î) he fundamental surface near the critical point o find the shape of the fundamental surface µ = µ(,p) near the critical point it is simplest to start by computing the change in the chemical potential along an isotherm in the (P,v) plane. he chemical potential along an isotherm is determined by the equation µ(p,) = µ 0 (P 0,) + P 0 P v(p)dp where P 0 is a reference value of the pressure. For a liquid-vapor system it is best to choose P 0 in the dilute vapor state where the system behaves as an ideal gas. As the vapor is compressed along the isotherm the pressure increases monotonically. he chemical potential increases with the pressure. his behavior is maintained until the two-phase equilibrium curve is reached, at which point µ = µ L. If equilibrium is achieved the vapor transforms to liquid at this value of the pressure. he transformation is indicated by a discontinuous decrease in the slope of the chemical potential as a function of the pressure since v > v L. Beyond the transformation point the chemical potential of the liquid phase increases monotonically with the pressure with slope v L (P). he potential curves of both the liquid and the solid have metastable extensions beyond the equilibrium point. he overall shape of the curve is shown in Fig he curve for the vapor can be obtained by continuing the integral into the two-phase region. he potential of the vapor continues to increase with the pressure until the instability point is reached. Suppose that it is possible to extend the properties of the fluid into the region of instability using an equation of state like he instability limit is a maximum in the P-v curve. If we continue to compress the fluid while keeping it homogeneous, the pressure decreases. he chemical potential decreases with the pressure, so the µ-p curve reverses itself. he chemical potential continues to decrease until the state reaches the minimum of the P-v isotherm, which corresponds to the instability point for the liquid. At that point the chemical potential and pressure begin to increase again, and generate the metastable extension of the chemical potential curve for the liquid. Since the specific volume of the liquid is smaller than that of the vapor, the chemical potential of the metastable liquid is below that of the metastable vapor. he liquid µ-p curve that is generated by con- Page 391

19 tinuing the integral cuts through the µ-p curve of the vapor at the equilibrium point and then continues as the µ-p curve of the stable liquid. he behavior of the µ-p curve at the instability points is striking. Since the integrand, v(p), is continuous at the instability point the slope of the µ-p curve has the same magnitude on both sides of the instability point. But the sign of dp is reversed because of the extremum in the P-v isotherm. It follows that the curve has a sharp cusp at the instability point; the slope has the same magnitude but opposite direction. Hence the unstable portion of the µ-p curve is terminated at either side by sharp cusps, from which the metastable portions slope down to intersect at the equilibrium point. µ P Fig. 17.8: he shape of the µ-p curve below the critical point of a onecomponent fluid. he left-hand branch is the vapor phase, the right-hand branch the liquid phase. he µ-p curve is a constant-temperature section through the surface µ(p,). As temperature is varied toward c, the instability points draw closer together. he "wings" of the unstable portion of the curve constrict, until they merge together at the critical point (this behavior was illustrated in Chapter 15). Above the critical point the µ-p curve is well-behaved everywhere; the chemical potential increases monotonically with P. he cusps create a figure on the fundamental surface that resembles a spine; hence the term spinodal HE AN DER WAALS FLUID he thermodynamics of the an der Waals fluid he an der Waals fluid is a one-component fluid that obeys a simple fundamental equation, yet has the property that it separates into two phases when the pressure or temperature fall below the critical values P c and c. Since it is one of the simplest model fluids that has this property, it is worth detailed study. he fundamental equation for the an der Waals fluid can be found in a number of ways. Its derivation from the statistical mechanics of "almost ideal" gases is discussed by Landau and Lifshitz. An alternate, semi-classical derivation can be given as follows (which, I understand, is close to an der Waals' original reasoning). Page 392

20 A gas becomes ideal in the limit of infinite dilution, that is, when its molecules are so far apart that they collide only occasionally and otherwise do not interfere with one another. When the gas is compressed just enough to begin to lose its ideality, two effects should become apparent. First, since the atoms have finite volume, the actual free volume available per atom is not the total volume,, but the modified volume, -Nb, where b is the atomic volume. Second, since the atoms are not an infinite distance apart, they sense one another. Considering only the average interaction of atoms with their neighbors, the net change in the energy per atom should be proportional to the atomic density, n = N/. It follows that the internal energy is given, approximately, by the caloric equation of state, E = 3 2 N - N2 a where the first term is the caloric equation of state and we have assumed an average interaction energy given by the constant, a, in writing the correction term. he fundamental equation of the an der Waals gas is most conveniently given by the Helmholtz free energy. Since the free energy of an ideal gas is F i (,,N) = - 3N 2 ln() - N ln() + Nç where ç is a constant. Adding the correction terms for the an der Waals fluid gives the fundamental equation of the an der Waals fluid: F(,,N) = F i - N ln 1 - Nb - N2 a = - 3N 2 ln() - N ln(-nb) - N2 a + Nç Differentiation with respect to gives the thermal equation of state P = N - Nb - N2 a which is known as the an der Waals equation of state. o complete the thermodynamics of the an der Waals fluid, the entropy is S = S i + N ln 1 - Nb where S i is the entropy of an ideal gas. Note that the parameter (a) does not affect the value of the entropy. he chemical potential is Page 393

21 µ = µ i - ln 1 - Nb + N (b - 2a) where µ i is the chemical potential of an ideal gas, and we have neglected a term of order b 2. he thermodynamic properties are the differentials of these relations. It is left as an exercise to find them. Note in particular that the isometric specific heat is equal to that of the ideal gas. he isobaric specific heat is not; it depends on the volume, and is singular at the critical point he phase diagram of a an der Waals fluid he utility of the an der Waals fluid as a model material lies in the fact that it has a very simple fundamental equation, but nonetheless exhibits a critical point and divides into two phases when (P,) lie below P c, c. It hence incorporates almost all of the thermodynamic behavior we have considered for the one-component fluid. It is left as an exercise to find the phase diagram. he critical point and the spinodal lines can be found analytically from the an der Waals equation of state. A numerical integration is required to find the liquid-vapor equilibrium lines. However, the equilibrium lines can be drawn near the critical point by applying the general expansion procedure presented in Section 17.4, and can also be found near = 0. he qualitative form of the phase diagram can be sketched by extrapolation. he P-v phase diagram appears as shown in Fig he P- and -v phase diagrams can be inferred from it. he critical state is determined by the thermal equation of state of the an der Waals fluid by the relations and has the coordinates P v = 2 P v 2 = c = 8a 27b P c = a 27b v c = 3b he spinodal lines are given by the solutions to the equation [ P/ v] = 0. Page 394

22 he Law of Corresponding States In terms of the reduced coordinates, * = / c, P* = P/P c, v* = v/v c, the an der Waals equation of state takes the dimensionless form P* + 3 v* 2 (3v* - 1) = 8* his equation contains no material constants and therefore governs every fluid that obeys the an der Waals equation of state. Equation is the basis of the Law of Corresponding States in the thermodynamics of liquid-vapor systems, which asserts that all such systems behave the same when their states are referred to reduced coordinates. Like all "Laws" that are extrapolated from very simple models, the Law of Corresponding States is not entirely accurate. However, it is a useful approximation for many purposes and is widely used in Chemistry. Page 395