PVT BEHAVIOR OF PURE SUBSTANCES - Volumetric Properties of Pure Fluids

Volumetric Properties of Pure Fluids

3.2 PVT BEHAVIOR OF PURE SUBSTANCES

Figure 3.1 displays an example of the equilibrium conditions of P and T at which solid, liquid, and gas phases of a pure substance might exist. Lines 1–2 and 2–C represent the conditions at which solid and liquid phases exist in equilibrium with a vapor phase. These vapor pressure

3.2. PVT Behavior of Pure Substances 71 versus temperature lines describe states of solid/vapor (line 1–2) and liquid/vapor (line 2–C) equilibrium. As indicated in Ex. 3.1(a), such systems have but a single degree of freedom.

Similarly, solid/liquid equilibrium is represented by line 2–3. The three lines display conditions of P and T at which two phases coexist, and they divide the diagram into single-phase regions.

Line 1–2, the sublimation curve, separates the solid and gas regions; line 2–3, the fusion curve, separates the solid and liquid regions; line 2–C, the vaporization curve, separates the liquid and gas regions. Point C is known as the critical point; its coordinates P_c and T_c are the highest pressure and highest temperature at which a pure chemical species is observed to exist in vapor/

liquid equilibrium.

The positive slope of the fusion line (2–3) represents the behavior of the vast majority of substances. Water, a very common substance, has some very uncommon properties, and exhibits a fusion line with negative slope.

The three lines meet at the triple point, where the three phases coexist in equilibrium.

According to the phase rule the triple point is invariant (F = 0). If the system exists along any of the two-phase lines of Fig. 3.1, it is univariant (F = 1), whereas in the single-phase regions it is divariant (F = 2). Invariant, univariant, and divariant states appear as points, curves, and areas, respectively, on a PT diagram.

Changes of state can be represented by lines on the PT diagram: a constant-T change by a vertical line, and a constant-P change by a horizontal line. When such a line crosses a phase boundary, an abrupt change in properties of the fluid occurs at constant T and P; for example, vaporization, the transition from liquid to vapor.

Figure 3.1: PT diagram for a pure substance.

Fusion curve Liquid region Vaporization

curve

Triple point Sublimation

curve Solid region

Fluid region

Temperature

Pressure

A C

T_c P_c

2 Vapor

region Gas region

Water in an open flask is obviously a liquid in contact with air. If the flask is sealed and the air is pumped out, water vaporizes to replace the air, and H2O fills the flask. Though the pressure in the flask is much reduced, everything appears unchanged. The liquid water resides at the bottom of the flask because its density is much greater than that of water vapor (steam), and the two phases are in equilibrium at conditions represented by a point on curve 2–C of Fig. 3.1. Far from point C, the properties of liquid and vapor are very different. However, if the

temperature is raised so that the equilibrium state progresses upward along curve 2–C, the properties of the two phases become more and more nearly alike; at point C they become iden- tical, and the meniscus disappears. One consequence is that transitions from liquid to vapor may occur along paths that do not cross the vaporization curve 2–C, i.e., from A to B. The transition from liquid to gas is gradual and does not include the usual vaporization step.

The region existing at temperatures and pressures greater than T_c and P_c is marked off by dashed lines in Fig. 3.1; these do not represent phase boundaries, but rather are limits fixed by the meanings accorded the words liquid and gas. A phase is generally considered a liquid if vaporization results from pressure reduction at constant temperature. A phase is considered a gas if condensation results from temperature reduction at constant pressure. Since neither process can be initiated in the region beyond the dashed lines, it is called the fluid region.

The gas region is sometimes divided into two parts, as indicated by the dotted vertical line of Fig. 3.1. A gas to the left of this line, which can be condensed either by compression at constant temperature or by cooling at constant pressure, is called a vapor. A fluid existing at a temperature greater than T_c is said to be supercritical. An example is atmospheric air.

PV Diagram

Figure 3.1 does not provide any information about volume; it merely displays the boundaries between single-phase regions. On a PV diagram [Fig. 3.2(a)] these boundaries in turn become regions where two phases—solid/liquid, solid/vapor, and liquid/vapor—coexist in equilibrium. The curves that outline these two-phase regions represent single phases that are in equilibrium. Their relative amounts determine the molar (or specific) volumes within the two-phase regions. The triple point of Fig. 3.1 here becomes a triple line, where the three phases with different values of V coexist at a single temperature and pressure.

Figure 3.2(a), like Fig. 3.1, represents the behavior of the vast majority of substances wherein the transition from liquid to solid (freezing) is accompanied by a decrease in specific volume (increase in density), and the solid phase sinks in the liquid. Here again water displays unusual behavior in that freezing results in an increase in specific volume (decrease in density), and on Fig. 3.2(a) the lines labeled solid and liquid are interchanged for water. Ice therefore floats on liquid water. Were it not so, the conditions on the earth’s surface would be vastly different.

Figure 3.2(b) is an expanded view of the liquid, liquid/vapor, and vapor regions of the PV diagram, with four isotherms (paths of constant T) superimposed. Isotherms on Fig. 3.1 are vertical lines, and at temperatures greater than T_c do not cross a phase boundary. On Fig. 3.2(b) the isotherm labeled T > T_c is therefore smooth.

The lines labeled T₁ and T₂ are for subcritical temperatures and consist of three segments. The horizontal segment of each isotherm represents all possible mixtures of liquid and vapor in equilibrium, ranging from 100% liquid at the left end to 100% vapor at the right end.

The locus of these end points is the dome-shaped curve labeled BCD, the left half of which (from B to C) represents single-phase liquids at their vaporization (boiling) temperatures and the right half (from C to D) single-phase vapors at their condensation temperatures. Liquids and vapors represented by BCD are said to be saturated, and coexisting phases are connected by the horizontal segment of the isotherm at the saturation pressure specific to the isotherm.

Also called the vapor pressure, it is given by a point on Fig. 3.1 where an isotherm (vertical line) crosses the vaporization curve.

3.2. PVT Behavior of Pure Substances 73

The two-phase liquid/vapor region lies under dome BCD; the subcooled-liquid region lies to the left of the saturated-liquid curve BC, and the superheated-vapor region lies to the right of the saturated-vapor curve CD. Subcooled liquid exists at temperatures below, and superheated vapor, at temperatures above the boiling point for the given pressure. Isotherms in the subcooled-liquid region are very steep because liquid volumes change little with large changes in pressure.

The horizontal segments of the isotherms in the two-phase region become progressively shorter at higher temperatures, being ultimately reduced to a point at C. Thus, the critical isotherm, labeled T_c, exhibits a horizontal inflection at the critical point C at the top of the dome, where the liquid and vapor phases become indistinguishable.

Critical Behavior

Insight into the nature of the critical point is gained from a description of the changes that occur when a pure substance is heated in a sealed upright tube of constant volume. The dotted vertical lines of Fig. 3.2(b) indicate such processes. They can also be traced on the PT diagram of Fig. 3.3, where the solid line is the vaporization curve (Fig. 3.1), and the dashed lines are constant-volume paths in the single-phase regions. If the tube is filled with either liquid or vapor, the heating process produces changes that lie along the dashed lines of Fig. 3.3, for example, by the change from E to F (subcooled-liquid) and by the change from G to H (superheated-vapor). The corresponding vertical lines on Fig. 3.2(b) are not shown, but they lie to the left and right of BCD respectively.

If the tube is only partially filled with liquid (the remainder being vapor in equilibrium with the liquid), heating at first causes changes described by the vapor-pressure curve (solid Figure 3.2: PV diagrams for a pure substance. (a) Showing solid, liquid, and gas regions. (b) Showing liquid, liquid/vapor, and vapor regions with isotherms.

(a) (b)

V_c V

V_c V P_c

P_c P Solid/liquid

Fluid

Liquid

Vapor Liquid/vapor

Solid/vapor

Solid

P_c T_c

T_c C

Gas

Liquid Vapor

Liquid/vapor C Q N

J K

T T_c

T₁ T_c T₂ T_c D

T_c

line of Fig. 3.3). For the process indicated by line JQ on Fig. 3.2(b), the meniscus is initially near the top of the tube (point J), and the liquid expands sufficiently upon heating to fill the tube (point Q). On Fig. 3.3 the process traces a path from (J, K) to Q, and with further heating departs from the vapor-pressure curve along the line of constant molar volume V₂^l .

The process indicated by line KN on Fig. 3.2(b) starts with a meniscus level closer to the bottom of the tube (point K), and heating vaporizes liquid, causing the meniscus to recede to the bottom (point N). On Fig. 3.3 the process traces a path from (J, K) to N. With further heating the path continues along the line of constant molar volume V ₂^v .

For a unique filling of the tube, with a particular intermediate meniscus level, the heating process follows a vertical line on Fig. 3.2(b) that passes through the critical point C. Physically, heating does not produce much change in the level of the meniscus. As the critical point is approached, the meniscus becomes indistinct, then hazy, and finally disappears.

On Fig. 3.3 the path first follows the vapor-pressure curve, proceeding from point (J, K) to the critical point C, where it enters the single-phase fluid region, and follows V_c, the line of constant molar volume equal to the critical volume of the fluid.³

PVT Surfaces

For a pure substance, existing as a single phase, the phase rule tells us that two state variables must be specified to determine the intensive state of the substance. Any two, from among P, V, and T, can be selected as the specified, or independent, variables, and the third can then be regarded as a function of those two. Thus, the relationship among P, V, and T for a pure substance can be represented as a surface in three dimensions. PT and PV diagrams like those illustrated in Figs. 3.1, 3.2, and 3.3 represent slices or projections of the three-dimensional PVT surface. Figure 3.4 presents a view of the PVT surface for carbon dioxide over a region including liquid, vapor, and supercritical fluid states. Isotherms are superimposed on this

3A video illustrating this behavior is available in the Connect eBook, if available, or contact your instructor for instructions on accessing this video.

Figure 3.3: PT diagram for a pure fluid showing the vapor-pressure curve and constant-volume lines in the single-phase regions.

V_c

V₂^v V₂^l

V₁^v C Q N G H F

(J, K) V₁^l

T Liquid

Vapor

3.2. PVT Behavior of Pure Substances 75 surface. The vapor/liquid equilibrium curve is shown in white, with the vapor and liquid portions of it connected by the vertical segments of the isotherms. Note that for ease of visualization, the molar volume is given on a logarithmic scale, because the vapor volume at low pressure is several orders of magnitude larger than the liquid volume.

Figure 3.4: PVT surface for carbon dioxide, with isotherms shown in black and the vapor/liquid equilibrium curve in white.

200 250

300

350 0

40 20 80 60

100 101

102 103 104 105

P (bar)

V (cm

3 /mol)

T (K)

Single-Phase Regions

For the regions of the diagram where a single phase exists, there is a unique relation connect- ing P, V, and T. Expressed analytically, as f(P, V, T) = 0, such a relation is known as a PVT equation of state. It relates pressure, molar or specific volume, and temperature for a pure homogeneous fluid at equilibrium. The simplest example, the equation for the ideal-gas state, PV = RT, has approximate validity for the low-pressure gas region and is discussed in detail in the following section.

An equation of state can be solved for any one of the three quantities P, V, or T, given values for other two. For example, if V is considered a function of T and P, then V = V(T, P), and

dV = ₍ _{___}∂ V

∂ T ₎

dT + ₍ _{___}∂ V

∂ P ₎

dP (3.2)

The partial derivatives in this equation have definite physical meanings and are related to two properties, commonly tabulated for liquids, and defined as follows:

∙ Volume expansivity:

β ≡ _{__}1

V ₍ _{___}∂ V

∂ T )

(3.3)

∙ Isothermal compressibility:

κ ≡ − _{__}1

V ₍ ∂ _{___}V

∂ P ₎

(3.4)

Combining Eqs. (3.2) through (3.4) yields:

___V = β dT − κ dP (3.5) The isotherms for the liquid phase on the left side of Fig. 3.2(b) are very steep and closely spaced. Thus both ( ∂ V / ∂ T₎ P and ( ∂ V / ∂ P₎ T are small. Hence, both β and κ are small. This char- acteristic behavior of liquids (outside the critical region) suggests an idealization, commonly employed in fluid mechanics and known as the incompressible fluid, for which both β and κ are zero. No real fluid is truly incompressible, but the idealization is useful, because it pro- vides a sufficiently realistic model of liquid behavior for many practical purposes. No equation of state exists for an incompressible fluid, because V is independent of T and P.

For liquids, β is almost always positive (liquid water between 0°C and 4°C is an excep- tion), and κ is necessarily positive. At conditions not close to the critical point, β and κ are weak functions of temperature and pressure. Thus for small changes in T and P little error is introduced if they are assumed constant. Integration of Eq. (3.5) then yields:

ln _{___}V 2

V 1 = β ( T 2 − T 1 ) − κ ( P 2 − P 1 ) (3.6) This is a less restrictive approximation than the assumption of an incompressible fluid.

Example 3.2

For liquid acetone at 20°C and 1 bar,

β= 1.487 × 10 ⁻³ ° C ⁻¹ κ= 62 × 10 ⁻⁶ bar ⁻¹ V = 1.287 cm ³ ⋅g ⁻¹ For acetone, find:

(a) The value of ⁽∂P / ∂T⁾ V at 20°C and 1 bar.

(b) The pressure after heating at constant V from 20°C and 1 bar to 30°C.

Dalam dokumen Introduction to Chemical Engineering Thermodynamics (Halaman 89-96)