PHASES AND PHYSICAL STATES OF MATTER

S YSTEMS AND T HEIR P ROPERTIES

2.2 PHASES AND PHYSICAL STATES OF MATTER

a subscript or a formula in parentheses. Examples are Molar volumeD V

n DVm (2.1.2)

Molar volume of substancei D V

ni DVm;i (2.1.3)

Molar volume of H2O DVm(H2O) (2.1.4) In the past, especially in the United States, molar quantities were commonly denoted with an overbar (e.g.,Vi).

2.2 PHASES AND PHYSICAL STATES OF MATTER

Aphaseis a region of the system in which each intensive property (such as temperature and pressure) has at each instant either the same value throughout (auniformorhomogeneous phase), or else a value that varies continuously from one point to another. Whenever this book mentions a phase, it is auniformphase unless otherwise stated. Two different phases meet at an interface surface, where intensive properties have a discontinuity or change over a small distance.

Some intensive properties (e.g., refractive index and polarizability) can have directional characteristics. A uniform phase may be eitherisotropic, exhibiting the same values of these properties in all directions, oranisotropic, as in the case of some solids and liquid crystals.

A vacuum is a uniform phase of zero density.

Suppose we have to deal with anonuniformregion in which intensive properties vary continuously in space along one or more directions—for example, a tall column of gas in a gravitational field whose density decreases with increasing altitude. There are two ways we may treat such a nonuniform, continuous region: either as a single nonuniform phase, or else as an infinite number of uniform phases, each of infinitesimal size in one or more dimensions.

2.2.1 Physical states of matter

We are used to labeling phases by physical state, or state of aggregation. It is common to say that a phase is a solid if it is relatively rigid, a liquid if it is easily deformed and relatively incompressible, and agasif it is easily deformed and easily compressed. Since these descriptions of responses to external forces differ only in degree, they are inadequate to classify intermediate cases.

A more rigorous approach is to make a primary distinction between asolidand afluid, based on the phase’s response to an applied shear stress, and then use additional criteria to classify a fluid as aliquid,gas, orsupercritical fluid. Shear stressis a tangential force per unit area that is exerted on matter on one side of an interior plane by the matter on the other side. We can produce shear stress in a phase by applying tangential forces to parallel surfaces of the phase as shown in Fig. 2.1 on the next page.

A solid responds to shear stress by undergoing momentary relative motion of its parts, resulting indeformation—a change of shape. If the applied shear stress is constant and small (not large enough to cause creep or fracture), the solid quickly reaches a certain

CHAPTER 2 SYSTEMS AND THEIR PROPERTIES

2.2 PHASES ANDPHYSICALSTATES OFMATTER 31

Figure 2.1 Experimental procedure for producing shear stress in a phase (shaded).

Blocks at the upper and lower surfaces of the phase are pushed in opposite directions, dragging the adjacent portions of the phase with them.

degree of deformation that depends on the magnitude of the stress and maintains this deformation without further change as long as the shear stress continues to be applied.

On the microscopic level, deformation requires relative movement of adjacent layers of particles (atoms, molecules, or ions). The shape of an unstressed solid is determined by the attractive and repulsive forces between the particles; these forces make it difficult for adjacent layers to slide past one another, so that the solid resists deformation.

Afluidresponds to shear stress differently, by undergoing continuous relative motion (flow) of its parts. The flow continues as long as there is any shear stress, no matter how small, and stops only when the shear stress is removed.

Thus, a constant applied shear stress causes a fixed deformation in a solid and continuous flow in a fluid. We say that a phase under constant shear stress is a solid if, after the initial deformation, we are unable to detect a further change in shape during the period we observe the phase.

Usually this criterion allows us to unambiguously classify a phase as either a solid or a fluid. Over a sufficiently long time period, however, detectable flow is likely to occur inanymaterial under shear stress of anymagnitude. Thus, the distinction between solid and fluid actually depends on the time scale of observation. This fact is obvious when we observe the behavior of certain materials (such as Silly Putty, or a paste of water and cornstarch) that exhibit solid-like behavior over a short time period and fluid-like behavior over a longer period. Such materials, that resist deformation by a suddenly-applied shear stress but undergo flow over a longer time period, are calledviscoelastic solids.

2.2.2 Phase coexistence and phase transitions

This section considers some general characteristics of systems containing more than one phase.

Suppose we bring two uniform phases containing the same constituents into physical contact at an interface surface. If we find that the phases have no tendency to change over time while both have the same temperature and the same pressure, but differ in other intensive properties such as density and composition, we say that theycoexistin equilibrium with one another. The conditions for such phase coexistence are the subject of later sections in this book, but they tend to be quite restricted. For instance, the liquid and gas phases of pure H2O at a pressure of1bar can coexist at only one temperature,99:61ıC.

Aphase transitionof a pure substance is a change over time in which there is a continuous transfer of the substance from one phase to another. Eventually one phase can

CHAPTER 2 SYSTEMS AND THEIR PROPERTIES

2.2 PHASES ANDPHYSICALSTATES OFMATTER 32

b b

cp solid

liquid

gas

supercritical fluid

bb bb

B C

Figure 2.2 Pressure–temperature phase diagram of a pure substance (schematic).

Point cp is the critical point, and point tp is the triple point. Each area is labeled with the physical state that is stable under the pressure-temperature conditions that fall within the area. A solid curve (coexistence curve) separating two areas is the lo- cus of pressure-temperature conditions that allow the phases of these areas to coexist at equilibrium. Path ABCD illustratescontinuity of states.

completely disappear, and the substance has been completely transferred to the other phase.

If both phases coexist in equilibrium with one another, and the temperature and pressure of both phases remain equal and constant during the phase transition, the change is anequilib- rium phase transition. For example, H2O at99:61ıC and1bar can undergo an equilibrium phase transition from liquid to gas (vaporization) or from gas to liquid (condensation). Dur- ing an equilibrium phase transition, there is a transfer of energy between the system and its surroundings by means of heat or work.

2.2.3 Fluids

It is usual to classify a fluid as either a liquidor agas. The distinction is important for a pure substance because the choice determines the treatment of the phase’s standard state (see Sec. 7.7). To complicate matters, a fluid at high pressure may be asupercritical fluid.

Sometimes a plasma (a highly ionized, electrically conducting medium) is considered a separate kind of fluid state; it is the state found in the earth’s ionosphere and in stars.

In general, and provided the pressure is not high enough for supercritical phenomena to exist—usually true of pressures below25bar except in the case of He or H2—we can make the distinction between liquid and gas simply on the basis of density. Aliquidhas a relatively high density that is insensitive to changes in temperature and pressure. Agas, on the other hand, has a relatively low density that is sensitive to temperature and pressure and that approaches zero as pressure is reduced at constant temperature.

This simple distinction between liquids and gases fails at high pressures, where liquid and gas phases may have similar densities at the same temperature. Figure 2.2 shows how we can classify stable fluid states of a pure substance in relation to a liquid–gas coexistence curve and a critical point. If raising the temperature of a fluid at constant pressure causes a phase transition to a second fluid phase, the original fluid was a liquid and the transition

CHAPTER 2 SYSTEMS AND THEIR PROPERTIES

2.2 PHASES ANDPHYSICALSTATES OFMATTER 33

occurs at the liquid–gas coexistence curve. This curve ends at acritical point, at which all intensive properties of the coexisting liquid and gas phases become identical. The fluid state of a pure substance at a temperature greater than the critical temperature and a pressure greater than the critical pressure is called asupercritical fluid.

The termvaporis sometimes used for a gas that can be condensed to a liquid by increasing the pressure at constant temperature. By this definition, the vapor state of a substance exists only at temperatures below the critical temperature.

The designation of a supercritical fluid state of a substance is used more for convenience than because of any unique properties compared to a liquid or gas. If we vary the temperature or pressure in such a way that the substance changes from what we call a liquid to what we call a supercritical fluid, we observe only a continuous density change of a single phase, and no phase transition with two coexisting phases. The same is true for a change from a supercritical fluid to a gas. Thus, by making the changes described by the path ABCD shown in Fig. 2.2, we can transform a pure substance from a liquid at a certain pressure to a gas at the same pressure without ever observing an interface between two coexisting phases! This curious phenomenon is calledcontinuity of states.

Chapter 6 will take up the discussion of further aspects of the physical states of pure substances.

If we are dealing with a fluidmixture(instead of a pure substance) at a high pressure, it may be difficult to classify the phase as either liquid or gas. The complexity of classification at high pressure is illustrated by thebarotropic effect, observed in some mixtures, in which a small change of temperature or pressure causes what was initially the more dense of two coexisting fluid phases to become the less dense phase. In a gravitational field, the two phases switch positions.

2.2.4 The equation of state of a fluid

Suppose we prepare a uniform fluid phase containing a known amountniof each constituent substance i, and adjust the temperature T and pressurep to definite known values. We expect this phase to have a definite, fixed volumeV. If we change any one of the properties T, p, or ni, there is usually a change inV. The value of V is dependent on the other properties and cannot be varied independently of them. Thus, for a given substance or mixture of substances in a uniform fluid phase, V is a unique function ofT,p, andfnig, where fnigstands for the set of amounts of all substances in the phase. We may be able to express this relation in an explicit equation: V D f .T; p;fnig/. This equation (or a rearranged form) that gives a relation amongV,T,p, andfnig, is theequation of stateof the fluid.

We may solve the equation of state, implicitly or explicitly, for any one of the quantities V,T,p, andni in terms of the other quantities. Thus, of the3Csquantities (wheresis the number of substances), only2Csare independent.

Theideal gas equation,p D nRT =V (Eq. 1.2.5 on page 23), is an equation of state.

It is found experimentally that the behavior of any gas in the limit of low pressure, as temperature is held constant, approaches this equation of state. This limiting behavior is also predicted by kinetic-molecular theory.

If the fluid has only one constituent (i.e., is a pure substance rather than a mixture), then at a fixedT andp the volume is proportional to the amount. In this case, the equation of

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2.2 PHASES ANDPHYSICALSTATES OFMATTER 34

state may be expressed as a relation amongT,p, and the molar volumeVm D V =n. The equation of state for a pure ideal gas may be writtenp DRT =Vm.

The Redlich–Kwong equationis a two-parameter equation of state frequently used to describe, to good accuracy, the behavior of a pure gas at a pressure where the ideal gas equation fails:

p D RT Vm b

Vm.VmCb/T1=2 (2.2.1)

In this equation,aandbare constants that are independent of temperature and depend on the substance.

The next section describes features ofvirialequations, an important class of equations of state for real (nonideal) gases.

2.2.5 Virial equations of state for pure gases

In later chapters of this book there will be occasion to apply thermodynamic derivations to virial equations of state of a pure gas or gas mixture. These formulas accurately describe the gas at low and moderate pressures using empirically determined, temperature-dependent parameters. The equations may be derived from statistical mechanics, so they have a theo- retical as well as empirical foundation. There are two forms of virial equations for a pure gas: one a series in powers of1=Vm:

pVmDRT

1C B Vm C C

Vm2 C

(2.2.2) and the other a series in powers ofp:

pVm DRT 1CBppCCpp2C

(2.2.3) The parameters B,C, : : :are called the second, third, : : :virial coefficients, and the parameters Bp, Cp, : : :are a set of pressure virial coefficients. Their values depend on the substance and are functions of temperature. (Thefirst virial coefficient in both power series is 1, because pVm must approach RT as 1=Vm orp approach zero at constant T.) Coefficients beyond the third virial coefficient are small and rarely evaluated.

The values of the virial coefficients for a gas at a given temperature can be determined from the dependence ofp onVm at this temperature. The value of the second virial coef- ficientBdepends on pairwise interactions between the atoms or molecules of the gas, and in some cases can be calculated to good accuracy from statistical mechanics theory and a realistic intermolecular potential function.

To find the relation between the virial coefficients of Eq. 2.2.2 and the parametersBp, Cp,: : :in Eq. 2.2.3, we solve Eq. 2.2.2 forpin terms ofVm

pDRT 1

Vm C B Vm2 C

(2.2.4) and substitute in the right side of Eq. 2.2.3:

pVm DRT

1CBpRT 1

Vm C B Vm2 C

CCp.RT /2 1

Vm C B Vm2 C

(2.2.5)

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2.2 PHASES ANDPHYSICALSTATES OFMATTER 35

Then we equate coefficients of equal powers of1=Vm in Eqs. 2.2.2 and 2.2.5 (since both equations must yield the same value ofpVmfor any value of1=Vm):

B DRTBp (2.2.6)

C DBpRTBCCp.RT /2D.RT /2.Bp2CCp/ (2.2.7) In the last equation, we have substituted forBfrom Eq. 2.2.6.

At pressures up to at least one bar, the terms beyondBpp in the pressure power series of Eq. 2.2.3 are negligible; thenpVmmay be approximated byRT .1CBpp/, giving, with the help of Eq. 2.2.6, the simple approximate equation of state3

Vm RT

p CB (2.2.8)

(pure gas,p1bar) Thecompression factor(or compressibility factor)Zof a gas is defined by

Z defD pV

nRT D pVm

RT (2.2.9)

(gas) When a gas at a particular temperature and pressure satisfies the ideal gas equation, the value ofZis1. The virial equations rewritten usingZare

ZD1C B Vm C C

Vm2 C (2.2.10)

ZD1CBppCCpp2C (2.2.11)

These equations show that the second virial coefficientBis the initial slope of the curve of a plot ofZversus1=Vmat constantT, andBpis the initial slope ofZversuspat constant T.

The way in whichZ varies with p at different temperatures is shown for the case of carbon dioxide in Fig. 2.3(a) on the next page.

A temperature at which the initial slope is zero is called theBoyle temperature, which for CO2is710K. BothBandBp must be zero at the Boyle temperature. At lower temper- aturesBandBpare negative, and at higher temperatures they are positive—see Fig. 2.3(b).

This kind of temperature dependence is typical for other gases. Experimentally, and also according to statistical mechanical theory,BandBp for a gas can be zero only at a single Boyle temperature.

The fact that at any temperature other than the Boyle temperature B is nonzero is significant since it means that in the limit aspapproaches zero at constantT and the gas approaches ideal-gas behavior, thedifferencebetween the actual molar volumeVm and the ideal-gas molar volumeRT =pdoes not approach zero. Instead,Vm RT =p approaches the nonzero valueB(see Eq. 2.2.8). However, theratioof the actual and ideal molar volumes,Vm=.RT =p/, approaches unity in this limit.

Virial equations of gasmixtureswill be discussed in Sec. 9.3.4.

3Guggenheim (Ref. [71]) calls a gas with this equation of state aslightly imperfect gas.

CHAPTER 2 SYSTEMS AND THEIR PROPERTIES

THE SYSTEM, SURROUNDINGS, AND BOUNDARY

SOME BASIC PROPERTIES AND THEIR MEASUREMENT