Water, Water, Everywhere: Phase Diagrams of Ordinary Water

Research: Science and Education

Water, Water, Everywhere Phase Diagrams of Ordinary Water Substance L. Glasser Nanochemistry Research Institute, Department of Applied Chemistry, Curtin University of Technology, GPO Box U1987, Perth WA 6845, Australia; [email protected]

Water is everywhere about us: within, upon, and around the Earth (1, 2). It is essential to life and indeed, in space exploration, is regarded as the very signature for life. Its ubiquity is illustrated in the English language by the numerous words for water (much as the Inuit are reputed to have an extensive vocabulary for snow): bodies of water are described as oceans, seas, lakes, dams, reservoirs, aquifers, wells, ponds, puddles, and dewdrops; flowing water comes in rivers, streams, rivulets, billabongs, waterfalls, cascades, creeks, brooks, rain, drips, and drops; solid water appears as ice, icicles, icebergs, glaciers, hail, sleet, snow, and frost. Water enables the major processes of this planet’s energy transfers: the Sun’s energy powers the weather, cycling water between the oceans, vapor, and precipitation; engineers use water as the energy transfer medium in hydroelectric systems and steam generation; in chemistry, water is the (almost) universal solvent. For such reasons, the properties of ordinary water substance (that is, water as solid ice, liquid, or vapor and gas) are much studied (3–5): over 20,000 values in the literature have been collated in reporting the behavior of water (5). The purpose of this article is the presentation of the full phase diagram of water, in the form of a graphical representation of the three-dimensional (3D) pVT diagram using authentic

p

5

c

c + lq

p

g

4

lq +g

lq

c

lq +g

c+

cr

V

g

g

g

p

c + lq

T

5

lq cr

2

1 0

34

g

3

cr

+

0

01 2

0

cr

g

c

5

2

c

lq

4

V

3

T1

c+

g

3

2

V

4

5

1 0

T

Figure 1. An illustrative orthographic (isometric) three-dimensional pVT diagram, with exploded views of the pT, pV, and VT projections (adapted from ref 7, Figure 50, p 206).

414

data—such a diagram does not seem to have been published before. This presentation follows our recent publication of the 3D pVT diagram of carbon dioxide, where details of the theory and interpretation of such a diagram may be found (6). A generalized and diagrammatic pVT diagram, with “exploded” views of pV, pT, and VT projections (7) is shown in Figure 1. This diagram may be useful for broad discussions. However, the authentic diagram (Figure 2) emphasizes different features of the diagram. A principal difference between the sketched (Figure 1) and authentic (Figure 2) phase diagrams is that the ranges in values of the p and V variables are so great that it is essential to use logarithmic scales to encompass the three phase regions: solid, liquid, and vapor or gas. Figure 3 is a pT projection of the fluid region of the water phase diagram (4), while Figure 4 depicts the pt (where t is temperature in units of ⬚C) relations among the solid phases

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Figure 2. A perspective three-dimensional pVT diagram for water, with projections onto the pT, pV, and VT planes. To accommodate the full range of data, the logarithms of the pressure and molar volume axes are used. The horizontal lines (constant pT ) are tielines connecting phases in mechanical and thermal equilibrium across the phase gaps. The dot at the end of the liquid–vapor line in the pT projection represents its termination at the critical point. (Note: The carbon dioxide diagram, in ref 6, is also perspective, but is incorrectly described as “orthographic”.) Volumes in the diagram (corresponding to areas in the projections) are labelled solid, liquid, vapor, and, above the critical point, gas. I, (III and V not labeled), VI, and VII refer to the respective solid phases that can equilibrate with liquid water.

Vol. 81 No. 3 March 2004

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of water, even beyond the equilibria with fluid water (8). The space curves depicted in Figure 2 are:

104

ice VII

A. Solid sublimation, in equilibrium with vapor

e V ice III

B. Vapor condensation, in equilibrium with solid 102

C. Liquid saturation, in equilibrium with vapor

liquid

ice I

E. Solid–liquid melt equilibria. On the scale of the diagram the two separate curves, for solid and liquid in equilibrium, nearly overlay one another.

Some important differences from the carbon dioxide phase diagram (6), which add complexity and interest to the water diagram, should be noted. Firstly, the solid (ice) phases are quite well-known (in contrast to those of carbon dioxide) and so those solid phases that are in equilibrium with liquid water are included in the diagram, which extends to pressures of 104 MPa. Secondly, the density of ordinary ice (hexagonal ice Ih) is less than that of liquid water at and near the triple point so that the solid–liquid equilibrium line has, unusually, a negative slope in that region, following the Clapeyron equation, dp ∆Hmelt = dT Tmelt ∆Vmelt

Pressure p / MPa

D. Vapor saturation, in equilibrium with liquid

critical point

pm

100

pσ solid 10ⴚ2

vapor triple point 10ⴚ4

psubl

200

300

400

500

600

700

Temperature T / K

• Sublimation–condensation curve: solid and vapor densities (or molar volumes). • Melting–freezing curves: solid and liquid densities. • Saturation curve: liquid and vapor densities.

Because of the extraordinary and broad interest in the properties of water, there has been established an International Association for the Properties of Water and Steam (IAPWS) that coordinates research in this area. IAPWS issues interna•

Figure 3. Projection on the pT plane of the phase diagram of ordinary water substance. Reprinted with permission from Figure 1 of ref 4: (Figure reprinted in W. Wagner and A. Pruβ, J. Phys. Chem. Ref. Data 2002, 31 (2), 387–535. Copyright 1994, 2002, American Institute of Physics & the U.S. Secretary of Commerce.)

4000 3500 3000

VII

VIII

2500

p / GPa

since the enthalpy of melting, ∆Hmelt, and the absolute temperature of melting at the triple point, Ttr, are necessarily positive and the volume change on melting, ∆Vmelt, is negative. This property is of significance in preventing bodies of water from freezing from the bottom up, which would destroy life therein. Thirdly, the liquid range of water lies in the range of pressures and temperatures normally found on the surface of the Earth. Fourthly, the liquid range is much larger than for most materials (the last two points are significant for sustaining life under the wide range of conditions that pertain on Earth). An interesting controversy regarding the phase behavior of water was the much-touted proposal of a solid phase of water, polywater (now rejected as simply a concentrated silica solution), supposedly stable under atmospheric conditions (9). Such a phase, if stable, would likely seed crystallization of all the liquid water on Earth, with catastrophic consequences for life! Indeed, such a phase (“Ice 9”) and its consequences had already formed the subject of the science-fiction novel, Cat’s Cradle, by Kurt Vonnegut, Jr., (10) based on an overheard proposal to H. G. Wells by the scientist Irving Langmuir at General Electric Corp. The data required for the phase diagram are equilibrium temperatures and pressures, and, for the:

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ice VI

ic

2000

VI

1500

liquid 1000

V II

500 0 -150

III Ih

-100

-50

0

50

100

150

200

t / °C Figure 4. Projection on the pt (T兾K = 273.2 + t兾⬚C) plane of the phase diagram of the principal solid phases of water, with broken lines representing extrapolations of the experimental data. Ice IV is a metastable phase that exists in the region of Ice V. Ice IX exists in the region below ᎑100 ⬚C and pressures in the range 200–400 MPa. Ice X exists at pressures above 44 GPa. (Adapted from Figure 3, legend, and data, p 12–202 of ref 8.)


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tional standards for water and has authorized the IAPWS-95 formulation (5) of the equation of state of water. IAPWS-95 is expressed in the form of a “fundamental” equation of thermodynamics explicit in the Helmholtz free energy (one of the four forms of the fundamental equations that apply when only pressure–volume work is involved, linking internal energy, enthalpy, entropy, Gibbs, and Helmholtz free energies through Legendre transforms (11); for example, A = U − TS is one such Legendre transform, forming the fundamental equation: dA = ᎑SdT − pdV in its natural variables, T and V . The IAPWS-95 equation for the Helmholtz free energy, f (using the European conventional symbol, F, for Helmholtz free energy rather than A, with lowercase denoting a specific, per kilogram, quantity rather than a molar quantity), is used with the intensive independent variables absolute temperature, T, and density, ρ. The function is split into a part, f ⬚, which represents the properties of the ideal gas, and a part, f r, which denotes the residual fluid behavior: r

f (ρ,T ) = f ° (ρ,T ) + f (ρ,T ) In dimensionless form, φ = f 兾(RT ), with δ = ρ兾ρc, the reduced density, and τ = Tc兾T the inverse reduced temperature, ρ c being the critical density, and T c the critical temperature: φ(δ, τ) = φ° ( δ, τ) + φr(δ, τ) These functions are obtained by nonlinear curve-fitting on experimental data and are exceedingly complex—φº has 13 fitted terms while φr has 56! All the thermodynamic properties of the pure substance are then obtainable from derivatives or combinations of derivatives of the dimensionless Helmholtz free energy. For example, the pressure is given by the derivative, p = ρ2

∂ f ∂ ρ

T

so that, in terms of the dimensionless Helmholtz free energy, φ:

p( δ, τ) = 1 + δ φr(δ, τ) ρRT Clearly, it is impractical to attempt individual evaluation of thermodynamic quantities from such a complex source. Instead, FORTRAN subroutines and a simple, compiled test program are available (12) by which required data may be generated. Certain auxiliary equations for the vapor–liquid phase equilibria are also reported (5, 13); these are not thermodynamically consistent with IAPWS-95, but the differences are extremely small and they may be used for convenience (indeed, these were used to generate the data for Figure 2, and are listed in the Appendix below). Additional correlation equations for the melting–pressure curves (solid–liquid phase boundary) and the sublimation–pressure curve (solid–vapor phase boundary) have been reported (4, 5). These are not part of IAPWS-95 but establish the boundaries between the solid phases and the fluid phases, to which IAPWS-95 ap-

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plies; these equations also appear in the Appendix. These latter equations do not provide information on the vapor density, which has been generated using the program EQTEST.EXE (12) provided by W. Wagner. The article on the IAPWS-95 formulation (5) provides extensive tables of thermodynamic properties calculated therefrom, while the NIST WebBook (14) can be used to dynamically generate data for the fluid phase boundaries, based on IAPWS-95. The WebBook also lists references to authoritative articles on the properties of ordinary water substance. Bowers (15) has published extensive tables of p–V–T and related properties of fluid water, carbon dioxide, and mixtures of water with carbon dioxide, derived from published equations of state. Acknowledgments The assistance of W. Wagner (Bochum) and E. W. Lemmon (NIST) in provision of program files and documentation is gladly acknowledged; this article could not have been prepared in its present form without their assistance. Literature Cited 1. Water: A Comprehensive Treatise; Franks, F., Ed.; Plenum: New York, 1972. 2. Franks, F. Water: A Matrix of Life, 2nd ed.; Royal Soc. Chem.: Cambridge, United Kingdom, 2000. 3. Dorsey, N. E. Properties of Ordinary Water-Substance; ACS Monograph Series, Reinhold: New York, 1940. 4. Wagner, W.; Saul, A.; Pruβ, A. J. Phys. Chem. Ref. Data 1994, 23, 515. 5. Wagner, W.; Pruβ, A. J. Phys. Chem. Ref. Data 2002, 31, 387. 6. Glasser, L. J. Chem. Educ. 2002, 79, 874. Reference was omitted to the following important equation of state (EoS) for fluid carbon dioxide: Span, R.; Wagner, W. J. Phys. Chem. Ref. Data 1996, 25, 1509. This EoS takes the form of a Helmholtz free energy function, as presented in the present paper, which itself is based on ref 5. 7. Karapetyants, M. Kh. Chemical Thermodynamics; translated by Leib, G.; MIR: Moscow, 1978. 8. Handbook of Chemistry and Physics, 82nd ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 2001. 9. Franks, F. Polywater; MIT Press: Cambridge, MA, 1981. 10. Vonnegut, K., Jr. Cat’s Cradle; Delacorte Press: New York, 1963. 11. Alberty, R. A. Chem. Rev. 1994, 94, 1457; Alberty, R. A. Pure Appl. Chem. 2001, 73, 1349. 12. Wagner, W.; Pruβ, A. Jahrbuch 97, VDI Gesellschaft Verfahrenstechnik und Chemieingenieurwesen; VDI-Verlag: Düsseldorf, 1977. The equation of state is used in a revised version programmed by the Lehrstuhl für Thermodynamik, Ruhr-Universität Bochum. 13. Wagner, W.; Pruβ, A. J. Phys. Chem. Ref. Data 1993, 22, 783. 14. NIST WebBook Home Page. http://webbook.nist.gov (accessed Dec 2003). 15. Bowers, T. S. In A Handbook of Physical Constants; Rock Physics and Phase Relations, Vol. 3; Amer. Geophys. Union: Washington, DC, 1995. Also available as http://www.agu.org/ reference/rock/8_bowers.pdf (accessed Dec 2003).


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Liquid–Ice-III–Ice-V:

Appendix Regarding Ordinary Water Substance (IAPWS-95) (5)

T3-III-V = 256.164 K; p3-III-V = 3501 bar;

Notes on Subscripts, Suffixes, and Constants

VIII(s) = 15.7 cm3 mol᎑1; VV(s) = 14.5 cm3 mol᎑1;

m is molar; c is the critical point value; 3 is the triple-point value; I–VII are ice structures; (s) is solid; (l) is liquid; (g) = gas; 1 bar ≡ 1 × 105 Pa ≡ 0.1 MPa Molar mass: 18.015268 g mol᎑1

Liquid–Ice-V–Ice-VI: T3-V-VI = 273.31 K; p3-V-VI = 6324 bar; VV(s) = 14.5 cm3 mol᎑1; VVI(s) = 13.8 cm3 mol᎑1;

Critical point: Tc = 647.096 K; pc = 220.64 bar,; Vm,c = 55.95 cm3 mol᎑1 Table 1 synopsizes the structures and parameters of different phases of ice (8).

Cell Parameters/ Å

V3(l) = 15.15 cm3 mol᎑1

Liquid–Ice-VI–Ice-VII: T3-VI-VII = 355 K; p3-VI-VII = 22,160 bar; VVI(s) = 13.8 cm3 mol᎑1; VVII(s) = 11.5 cm3 mol᎑1;

Table 1. Stucture and Parameters of Different Phases of Ice (8) Phase Crystal System

V3(l) = 15.90 cm3 mol᎑1

V3(l) = 13.36 cm3 mol᎑1

Za

nb

ρ/ (g cm᎑3)

Ih

Hexagonal

a = 4.513; c = 7.352

4

4

0.93

Ic

Cubic

a = 6.35

8

4

0.94

II

Rhombohedral

a = 7.78; α = 113.1°

12

4

1.18

III

Tetragonal

a = 6.73; c = 6.83

12

4

1.15

IV

Rhombohedral

a = 7.60; α = 70.1°

16

4

1.27

V

Monoclinic

a = 9.22; b = 7.54; c = 10.35; ß = 109.2°

28

4

1.24

10

4

1.31

Values of V3(l) are obtained using EQTEST.EXE (12) based on IAPWS-95, whose range of validity is 251.2 K ≤ T ≤ 1273 K and p = 10,000 bar. The last value is, therefore, an extrapolation outside the cited pressure range. Saturated Vapor Pressure/Temperature

ln

pσ pc

=

3 Tc a1ϑ + a2 ϑ 2 + a3ϑ3 T 7 15 + a 4 ϑ 2 + a 5ϑ 4 + a 6 ϑ 2

where ␽ = (1 − T兾Tc); Tc = 647.096 K; pc = 220.64 bar a1 = ᎑7.85951783; a2 = 1.84408259; a3 = ᎑11.7866497;

VI

Tetragonal

a = 6.27; c = 5.79

VII

Cubic

a = 3.41

2

8

1.56

VIII

Tetragonal

a = 4.80; c = 6.99

8

8

1.56

IX

Tetragonal

a = 6.73; c = 6.83

12

4

1.16

X

Cubic

a = 2.83

2

8

2.51

a

a4 = 22.6807411; a5 = ᎑15.9618719; a6 = 1.80122502.

Saturated Liquid Density 5 1 2 ρ′ = 1 + b1ϑ 3 + b2ϑ 3 + b3ϑ 3 ρc 43 110 16 + b4 ϑ 3 + b5ϑ 3 + b6 ϑ 3

where

Z is the number of formula units per unit cell.

b

n is the oxygen coordination number.

b1 = 1.99274064; b2 = 1.09965342; b3 = ᎑0.510839303; b4 = ᎑1.75493479; b5 = ᎑45.5170352; b6 = ᎑6.74694450 × 105.

Triple Points

Liquid–Vapor–Ice-I:

Saturated Vapor Density

T3 = 273.16 K; p3 = 6.117 × 10᎑3 bar; V3(s) = 29.09 cm3 mol᎑1; V3(l) = 18.019 cm3 mol᎑1;

ln

V3(g) = 371.098 cm3 mol᎑1

ρ′′ ρc

= c1ϑ

+ c 2ϑ

mol᎑1;

VIII(s) = 15.7

cm3

4

6

18

6

+ c 3ϑ

8

+ c 5ϑ

6

37

6

+ c 6ϑ

71

6

where

T3-I-III = 251.165 K; p3-I-III = 2099 bar; VI(s) = 19.4

6

+ c 4ϑ

Liquid–Ice-I–Ice-III: cm3

2

mol᎑1;

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c1 = ᎑2.03150240; c2 = ᎑2.68302940; c3 = ᎑5.38626492; c4 = ᎑17.2991605; c5 = ᎑44.7586581; c6 = ᎑63.9201063.

V3(l) = 16.52 cm3 mol᎑1

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Ice-VII (355 K to 715 K):

Melting/Pressure Curves

Ice-I (273.16 K to 251.165 K):

ln

pm,I = 1 − 0.626 × 10 6 1 − θ −3 pn + 0.197135 × 10 6 1 − θ 21.2

pm ,VII = 1.73683(1 − θ −1) pn − 0.0544606 (1 − θ 5 ) + 0.806106 x10 −7 (1 − θ 22 )

where θ = T兾Tn; Tn = 355 K; pn = 22,160 bar

where θ = T兾Tn; Tn = 273.16 K; pn = 0.00611657 bar

Sublimation/Pressure Curve

Ice-III (251.165 K to 256.164 K):

ln

pm,III = 1 − 0.295252(1 − θ 60 ) pn

psubl pn

= −13.928169 (1 − θ −1.5) + 34.7078238(1 − θ 1.25 )

where

where

θ = T兾Tn; Tn = 273.16 K; pn = 0.00611657 bar

θ = T兾Tn; Tn = 251.165 K; pn = 2099 bar

Table 2 lists the values for sublimation vapor volumes at specific temperatures.

Ice-V (256.164 K to 273.31 K):

Table 2. Sublimation Vapor Volumes

pm,V = 1 − 1.18721(1 − θ 8 ) pn

where θ = T兾Tn; Tn = 256.164 K; pn = 3501 bar

Ice-VI (273.31 K to 355 K): pm, VI = 1 − 1..074 76 (1 − θ 4.6) pn

where θ = T兾Tn; Tn = 273.3 K; pn = 6324 bar

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T/K

p/bar

Vv/(cm3 mol᎑1)

190

3.23 x 10᎑7

4.90 x 1010

200

1.62 x 10᎑6

1.03 x 1010

220

2.65 x 10᎑5

6.90 x 108

240

2.73 x 10᎑4

7.32 x 107

260

1.96 x 10᎑3

1.10 x 107

273.16

6.12 x 10᎑3

3.72 x 106

NOTE: The sublimation volumes are calculated using the ideal gas law. The van der Waals equation, using a = 5.353 x 106 bar cm6 mol᎑2, b = 30.5 cm3 mol᎑1 (8), gives essentially the same values. (The factor 106 was incorrectly omitted from the value for the van der Waals a of CO2 in ref 6).


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Water, Water, Everywhere: Phase Diagrams of Ordinary Water

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