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CHAPTER
INTRODUCTION
“Aside from the logical and mathematical sciences, there are three great branches of natural science which stand apart by reason of the variety of far reaching deductions drawn from a small number of primary postulates. They are mechanics, electromagnetics, and thermodynamics.
These sciences are monuments to the power of the human mind; and their inten- sive study is amply repaid by the aesthetic and intellectual satisfaction derived from a recognition of order and simplicity which have been discovered among the most complex of natural phenomena... Yet the greatest development of applied thermodynamics is still to come. It has been predicted that the era into which we are passing will be known as the chemical age; but the fullest employment of chemical science in meeting the various needs of society can be made only through the constant use of the methods of thermodynamics.”
Lewis and Randall (1923)
Lewis and Randall eloquently summarized the broad significance of thermodynamics as long ago as 1923. They went on to describe a number of the miraculous scientific developments of the time and the relevant roles of thermodynamics. Historically, thermodynamics has guided the develop- ment of steam engines, refrigerators, nuclear power plants, and rocket nozzles, to name just a few. The principles remain important today in the refinement of alternative refrigerants, heat pumps and improved turbines, and also in technological advances including computer chips, superconductors, advanced materials, and bioengineered drugs. These latter day “miracles” on first thought might appear to have little to do with power generation and refrigeration cycles. However, as Lewis and Randall point out, the implications of the postulates of thermodynamics are far reaching and will continue to be important in the development of even newer technologies. Much of modern thermo- dynamics focuses on characterization of the properties of mixtures, as their constituents partition into stable phases, inhomogeneous domains, and/or react. The capacity of thermodynamics to bring “quantitative precision in place of the old, vague ideas” 1 is as germane today as it was then.
- Lewis, G.N., Randall, M. Thermodynamics and the Free Energy of Chemical Substances, McGraw-Hill, NY, 1923.
4 Unit I First and Second Laws
Before overwhelming you with the details that comprise thermodynamics, we outline a few “primary postulates” as clearly as possible and put them into the context of what we will refer to as classical equilibrium thermodynamics. In their simplest human terms, our primary premises can be expressed as:
- You can’t get something for nothing.
- Generation of disorder results in lost work. 1
Occasionally, it may seem that we are discussing principles that are much more sophisticated. But the fact is that all of our discussions can be reduced to these fundamental principles. When you find yourself in the midst of a difficult problem, it may be helpful to remember that. We will see that coupling these two observations with some slightly sophisticated reasoning (mathematics included) leads to many clear and reliable insights about a wide range of subjects from energy crises to high-tech materials, from environmental remediation to biosynthesis. The bad news is that the level of sophistication required is not likely to be instantly assimilated by the average student. The good news is that many students have passed this way before, and the proper course is about as well-marked as one might hope.
There is less than universal agreement on what comprises “thermodynamics.” If we simply take the word apart, “thermo” sounds like “thermal” which ought to have something to do with heat, temperature, or energy, and “dynamics” ought to have something to do with movement. And if we could just leave the identification of thermodynamics as the study of “energy movements,” it would be sufficient for the purposes of this text. Unfortunately, such a definition would not clarify what makes thermodynamics distinct from, say, transport phenomena or kinetics, so we should spend some time clarifying the definition of thermodynamics in this way before moving on to the definitions of temperature, heat, energy, etc.
The definition of thermodynamics as the study of energy movements has evolved considerably to include classical thermodynamics, quantum thermodynamics, statistical thermodynamics, and irreversible thermodynamics as well as equilibrium and non-equilibrium thermodynamics. Classi- cal thermodynamics has the general connotation of referring to the implications of constraints related to multivariable calculus as developed by J.W. Gibbs. We spend a significant effort applying these insights in developing generalized equations for the thermodynamic properties of pure sub- stances. Statistical thermodynamics focuses on the idea that knowing the precise states of 10 23 atoms is not practical and prescribes ways of computing the average properties of interest based on very limited measurements. We touch on this principle in our introduction to entropy and in our ultrasimplified kinetic theory of ideal gases, but we will generally refrain from detailed formulation of all the statistical averages. Irreversible thermodynamics and non-equilibrium thermodynamics emphasize the ways atoms and energy move over periods of time. At this point, it becomes clear that such a broad characterization of thermodynamics would overlap with transport phenomena and kinetics in a way that would begin to be confusing at the introductory level. Nevertheless, these fields of study represent legitimate subtopics within the general domain of thermodynamics.
These considerations should give some idea of the narrowness and the breadth that are possible within the general study of thermodynamics. This text will try to find a happy medium. One general unifying principle about the perspective offered by thermodynamics is that there are certain proper- ties that are invariant with respect to time. For example, the process of diffusion may indicate some changes in the system with time, but the diffusion coefficient can be considered to be a property
- The term “lost work” refers to the loss of capability to perform work, and is discussed in more detail in Sections 2.4, 3.3, and 3.4.
6 Unit I First and Second Laws
elements of hydrogen and oxygen (as in the heat of reaction), depending on the situation. In studies of activity and chemical reactions, we will need to be somewhat careful in specifying the exact nature of the reference state. Since we will focus on changes in kinetic energy, potential energy, and energies of reaction, we will not need to specify reference states any more fundamental than the elements, thus do not consider subatomic particles.
Potential Energy
Kinetic energy is commonly covered in detail during introductory physics courses so we will assume it is already familiar. Students may be somewhat less comfortable with the concept of potential energy, however, especially as it relates to molecular interactions. Therefore, we spend some time with it here.
Macroscopic Potential Energy
Potential energy is associated with the “work” of moving a system some distance through a force field. On the macroscopic scale, we are well aware of the effect of gravity. This is a very common force field. As an example, the earth and the moon are two spherical bodies which are attracted by a force which varies as r −^2. The potential energy represents the work of moving the two bodies closer together or farther apart, which is simply the integral of the force over distance. So the potential function varies as r −^1. These kinds of operations should be familiar from a course in introductory physics. They are the same at the microscopic level except that the forces vary with position according to different laws.
Intermolecular Potential Energy
In consideration of potential energy on a molecular scale, it is important to realize that uncharged atoms do exert forces on each other. For a rigorous description, the origin of the intermolecular potential must be traced back to the solution of Schrödinger’s quantum mechanics equation for the motions of electrons around nuclei. We cannot afford that level of rigor in this course. In fact, that level of rigor is not currently available for some of the complex molecules to be considered in this course. On the other hand, there are several generalities which arise from this rigorous approach which can be appreciated from a somewhat intuitive perspective. For instance, atoms are comprised of dense nuclei of positive charge with electron densities of negative charge built around the nucleus in shells. The outermost shell is referred to as the valence shell. Technically, we could say that this insight represents a generalization based on solutions of Schrödinger’s equation (which was in turn based on compiled experimental data detailing the differences between subatomic parti- cles and the macroscopic particles which Newton’s laws accurately describe). This much should be familiar from an elementary course in chemistry or physics. Another generalization which may be less familiar is that electron density often tends to concentrate in lobes in the valence shell for ele- ments like C, N, O, F, S, Cl. These lobes may be occupied by protons that are tetrahedrally coordi- nated as in CH 4 , or they may be occupied by unbonded electron pairs that fill out the tetrahedral coordination as in NH 3 or H 2 O. These elements (H, C, N, O, F, S, Cl) and some noble gases like He, Ne, and Ar provide virtually all of the building blocks for the molecules to be considered in this text. The two generalizations discussed here are sufficient to describe what we need to know about intermolecular forces at the introductory level.
By considering the implications of atomic structure and atomic collisions, it is possible to develop subclassifications of intermolecular forces. These are:
- Electrostatic forces between charged particles (ions) and between permanent dipoles, quadru- poles, and higher multipoles.
Engineering model potentials per- mit representation of attractive and repul- sive forces in a tracta- ble form.
Section 1.1 The Molecular Nature of Energy 7
- Induction forces between a permanent dipole (or quadrupole) and an induced dipole.
- Forces of attraction (dispersion forces) and repulsion between nonpolar molecules.
- Specific (chemical) forces leading to association and complex formation, especially evi- dent in the case of hydrogen bonding.
Further, attractive forces will be quantified by negative numerical values, and repulsive forces will be characterized by positive numerical values.
Electrostatic Forces
The force between two point charges described by Coulomb’s Law is very similar to the law of gravitation and should be familiar from elementary courses in chemistry and physics,
where qi and q (^) j are the charges, and r is the separation of centers. Upon integration, , the potential energy is proportional to inverse distance,
If all molecules were perfectly spherical and rigid, the only way that these electrostatic interac- tions could come into play is through the presence of ions. But a molecule like NH 3 is not perfectly spherical. NH 3 has three protons on one side and a lobe of electron density in the unbonded valence shell electron pair. This permanent asymmetric distribution of charge density gives rise to a perma- nent dipole on the NH 3 molecule. This means that ammonia molecules lined up with the electrons facing each other will repel while molecules lined up with the electrons facing the protons will attract. Since electrostatic energy drops off as r −^1 , one might expect that the impact of these forces would be long range. Fortunately, with the close proximity of the positive charge to the negative charge in a molecule like NH 3 , the charges tend to cancel each other as the molecule spins and tum- bles about through a fluid. This spinning and tumbling makes it reasonable to consider a spherical average of the intermolecular energy as a function of distance that may be constructed by averaging over all orientational angles between the molecules at each distance. In a large collection of mole- cules randomly distributed to each other, this averaging approach gives rise to many cancellations, and the net impact is that
This surprisingly simple result is responsible for a large part of the attractive energy between polar molecules. This energy is attractive because the molecules tend to spend somewhat more time lined up attractively than repulsively, and the r −^6 power arises from the many cancellations that occur in a fluid.
Induction Forces
When a molecule with a permanent dipole approaches a molecule with no dipole, the positive charge of the dipolar molecule will tend to pull electron density away from the nonpolar molecule and “induce” a dipole moment into the nonpolar molecule. The magnitude of this effect depends on
u =∫ F rd
u
q (^) i q (^) j r
Section 1.1 The Molecular Nature of Energy 9
Examples of Model Potentials
Based on the forms of these electrostatic, induction, and dispersion forces, it should be easy to appreciate the form of the Lennard-Jones potential in Fig. 1.1. Other models of the potential func- tion are possible, such as the square-well potential or the Sutherland potential also shown in Fig. 1.1. These latter potential models represent simplified forms of the Lennard-Jones model that are accu- rate enough for many applications.
0
- 5
1
- 5
2
0 0. 5 1 1. 5 2 2. 5 3 r/ σ
u(r)
/^ ε
u(r) = 4 ε [(σ / r ) 12 – (σ / r ) 6 ] The Lennard-Jones potential
0
- 5
1
- 5
2
0 0. 5 1 1. 5 2 2. 5 3 r/ σ
u(r)
/^ ε
R σ
The square-well potential for R = 1.5.
0
- 5
1
- 5
2
0 0. 5 1 1. 5 2 2. 5 3 r/ σ
u(r)
ε /
Figure 1.1 Schematics of three engineering models for pair potentials on a dimensionless basis.
The Sutherland potential.
10 Unit I First and Second Laws
The key features of all of these potential models are the representation of the size of the mole- cule by the parameter σ and the attractive strength (i.e. “stickiness”) by the parameter ε. Note that we would need a more complicated potential model to represent the shape of the molecule. Typi- cally, molecules of different shapes are represented by binding together several potentials like those above with each potential site representing one molecular segment. For example, n -butane could be represented by four Lennard-Jones sites that have their relative centers located at distances corre- sponding to the bond-lengths in n -butane. The potential between two butane molecules would then be the sum of the potentials between each of the individual Lennard-Jones sites on the different molecules. In similar fashion, potential models for very complex molecules can be constructed.
We can gain considerable insight about the thermodynamics of fluids by intuitively reasoning about the relatively simple effects of size and stickiness. For example, a large molecule like buck- minsterfullerene would have a larger value for σ than would methane. Water and methane are about the same size, but their difference in boiling temperature indicates a large difference in their sticki- ness. As you read through this chapter, it should become apparent that water has a higher boiling temperature because it sticks to itself more strongly than does methane. With these simple insights, you should be able to understand the molecular basis for a large number of macroscopic phenomena.
1.2 THE MOLECULAR NATURE OF ENTROPY
To be fair to both of the central concepts of the course, we must mention entropy at this point, in parallel with the mention of energy. Unfortunately, there is no simple analogy that can be drawn like that of the potential energy between the earth and moon. The study of entropy is fairly specific to the study of thermodynamics. The proper development of the subject must await Chapter 3.
What we can say at this point is that entropy has been conceived to account for losses in the prospect of performing useful lost work. The energy can take on many forms and be completely accounted for without contemplating how much energy has been “wasted” by converting work into
Example 1.1 Intermolecular potentials for mixtures
Our discussion of intermolecular potentials has focused on describing single molecules, but it is actually more interesting to compare and contrast the potential models for different molecules that are mixed together. We can use the square-well potential as the basis for this exercise and focus simply on the size (σ (^) ij ) and stickiness (ε (^) ij ) of each potential model, where the subscript ij indicates an interaction of molecule i with molecule j. For example, ε 11 would be the stickiness of molecule 1 to itself, and ε 12 would be its stickiness to a molecule of type 2. The size parameter for interaction between different molecules is reasonably well represented by σ 12 = (σ 11 + σ 22 )/ 2. The estimation of the stickiness parameter for interaction between different molecules requires some intuitive reasoning. For mixtures of hydrocarbons, it is conventional to estimate the sticki- ness by a geometric mean. To illustrate, sketch on the same pair of axes the potential models for methane and benzene, assuming that the stickiness parameter is given by ε 12 = (ε 11 ε 22 )1/^.
Solution: Methane(1) has fewer atoms in it than benzene(2), so we can assume it is smaller. Let’s depict this by saying σ 22 = 2 σ 11. This means that σ 12 = 1.5σ 11. Similarly, methane’s boiling temperature is lower so its stickiness should be smaller in magnitude. Let’s depict this by ε 22 = 4 ε 11 and this means ε 12 = 2 ε 11. Thus we obtain Fig. 1.2.
12 Unit I First and Second Laws
Work
Work is a familiar term from physics. We know that work is a force acting through a distance. There are several ways forces may interact with the system which all fit under this category. We will dis- cuss the details of how we calculate work and determine its impact on the system in the next chapter.
Density
Density is a measure of the mass per unit volume and may be expressed on a molar basis or a mass basis. In some situations, it is expressed as number of particles per unit volume.
Pressure — An Ultrasimplified Kinetic Theory
Pressure is the force exerted per unit area. We will be concerned primarily with the pressure exerted by the molecules of fluids upon the walls of their containers. For our purposes, the kinetic theory of pressure should provide a sufficient description.
Suppose we have two hard spherical molecules in a container that are bouncing back and forth in the x -direction only and not contacting each other.
We wish to quantify the forces acting on each wall. Since the particles are only colliding with the walls at A 1 and A 2 in our idealized model, these are the only walls we need consider. Let us assume that particles bounce off the wall with the same speed which they had before striking the wall, but in the opposite direction (a perfectly elastic collision where the wall is perfectly rigid and absorbs no momentum). Thus, the kinetic energy of the particles will be fixed. If u is the initial velocity of the particle (recall that u is a vector quantity and u is a scalar) before it strikes a wall, the change in velocity due to striking the wall is − 2 u. The change in velocity of the particle indicates the presence of interacting forces between the wall and the particle. If we quantify the force neces- sary to change the velocity of the particle, we will also quantify the forces of the particle on the wall by Newton’s third principle. To quantify the force, we may apply Newton’s second principle stated in terms of momentum:
The time rate of change of the momentum of a particle is equal to the resultant force acting on the particle and is in the direction of the resultant force.
The application of this formula directly is somewhat problematic since the change in direction is instantaneous, and it might seem that the time scale is important. This can be avoided by determin- ing the time-averaged force, 1 F avg exerted on the wall during time ∆ t:
- See an introductory physics text for further discussion of time-averaged force.
A 1 A 2
Section 1.3 Brief Summary of Several Thermodynamic Quantities 13
where ∆ p is the total change in momentum during time ∆ t. The momentum change for each colli- sion is − 2 m u where m is the mass per particle. Each particle will collide with the wall every t (^) 1 sec- onds, where t (^) 1 = 2 L /u , where L is the distance between A 1 and A 2. The average force is then
where u is the velocity before the collision with the wall. Pressure is the force per unit area, and the area of a wall is L^2 , thus
where the subscripts denote the particles.
If the particle motions are generalized to motion in arbitrary directions, collisions with addi- tional walls in the analysis does not complicate the problem dramatically because each component of the velocity may be evaluated independently. To illustrate, consider a particle bouncing around the centers of four walls in a horizontal plane. From the top view, the trajectory would appear as below:
For the same velocity as the first case, the force of each collision would be reduced because the par- ticle strikes merely a glancing blow. The x -component of the force can be related to the magnitude of the velocity by noting that ux = uy , such that u = ( ux^2 + uy^2 )1/2^ = ux 2 1/2. The time between collisions with wall A 1 would be 4 L /( u 2 1/2 ). The formula for the average force in two dimensions then becomes:
and the pressure due to two particles that don’t collide with each other in two dimensions becomes:
P is propor- tional to the number of particles in a vol- ume and to the kinetic energy of the particles.
x
y
A 1 A 2
Section 1.4 Basic Concepts 15
Heat – Sinks and Reservoirs
Heat is energy in transit between the source from which the energy is coming and a destination toward which the energy is going. When developing thermodynamic concepts, we frequently will assume that our system transfers heat to/from a sink or reservoir. A heat reservoir is an infinitely large source or destination of heat transfer. The reservoir is assumed to be so large that the heat transfer does not affect the temperature of the reservoir. A sink is a special name sometimes used for a reservoir which can accept heat without a change in temperature. The assumption of constant temperature makes it easier to concentrate on the thermodynamic behavior of the system while making a reasonable assumption about the part of the universe assigned to be the reservoir.
The mechanics of heat transfer are also easy to picture conceptually from the molecular kinet- ics perspective. In heat conduction, faster moving molecules collide with slower ones, exchanging kinetic energy and equilibrating the temperatures. In this manner, we can imagine heat being trans- ferred from the hot surface to the center of a pizza in an oven until the center of the pizza is cooked. In heat convection, packets of hot mass are circulated and mixed, accelerating the equilibration pro- cess. Heat convection is important in getting the heat from the oven flame to the surface of the pizza. Heat radiation, the remaining mode of heat transfer, occurs by an entirely different mecha- nism having to do with waves of electromagnetic energy emitted from a hot body that are absorbed by a cooler body. Radiative heat transfer is typically discussed in detail during courses devoted to heat transfer.
1.4 BASIC CONCEPTS
The System
A system is that portion of the universe which we have chosen to study.
A closed system is one in which no mass crosses the system boundaries.
An open system is one in which mass crosses the system boundaries. The system may gain or lose mass or simply have some mass pass through it.
System boundaries are established at the beginning of a problem, and simplification of balance equations depends on whether the system is open or closed. Therefore, the system boundaries should be clearly identified. If the system boundaries are changed, the simplification of the mass and energy balance equations should be performed again, because different balance terms are likely to be necessary. These guidelines will become more apparent in Chapter 2.
The Mass Balance
Presumably, students in this course are familiar with mass balances from an introductory course in material and energy balances. The relevant relation is simply:
A reservoir is an infinitely large source or destination for heat transfer.
The placement of system boundaries is a key step in prob- lem solving.
The mass balance.
16 Unit I First and Second Laws
where are the absolute values of mass flow rates entering and leaving, respectively.
We may also write
where mass differentials dm in^ and dm out^ are always positive. When all the flows of mass are ana- lyzed in detail for several subsystems coupled together, this simple equation may not seem to fully portray the complexity of the application. The general study of thermodynamics is similar in that regard. A number of simple relations like this one are coupled together in a way that requires some training to understand. In the absence of chemical reactions , we may also write a mole balance by replacing mass with moles in the balance.
Intensive Properties
Intensive properties are those properties which are independent of the size of the system. For exam- ple, in a system at equilibrium without internal rigid/insulating walls, the temperature and pressure are uniform throughout the system and are therefore intensive properties. Likewise, mass or mole- specific properties are independent of the size of the system. For example, the molar volume ([≡ ] length 3 /mole), mass density ([≡ ] mass/length 3 ), the specific internal energy ([≡ ] energy/mass) are intensive properties. Notationally in this text, intensive properties are not underlined.
Extensive Properties
Extensive properties depend on the size of the system. For example the volume ([≡ ] length 3 ) and energy ([≡ ] energy). Extensive properties are underlined, e.g. U = n U, where n is the number of moles and U is molar internal energy.
States and State Properties – The Phase Rule
Two state variables are necessary to specify the state of a single-phase pure fluid, i.e., two from the set P, V, T, U. Other state variables to be defined later in the text which also fit in this category are molar enthalpy, molar entropy, molar Helmholtz energy and molar Gibbs energy. State variables must be intensive properties. As an example, specifying P and T permits you to find the specific internal energy and specific volume of steam_._ Note, however, that you need to specify only one variable, the temperature or the pressure, if you want to find the properties of saturated vapor or liq- uid. This reduction in the needed specifications is referred to as a reduction in the “degrees of free- dom.” As another example in a ternary, two-phase system, the temperature and the mole fractions of two of the components of the lower phase are state variables (the third component is implicit in summing the mole fractions to unity), but the total number of moles of a certain component is not a state variable because it is extensive. In this example, the pressure and mole fractions of the upper phase may be calculated once the temperature and lower-phase mole fractions have been specified. The number of state variables needed to completely specify the state of a system is given by the Gibbs phase rule:
F = C − P + 2 1.
dm dm in
inlets
∑ dm^
out
outlets
= – ∑
The distinction between intensive and extensive properties is key in selecting and using variables for problem solving.
18 Unit I First and Second Laws
Eqn. 1.17 may seem to be confined to ideal monatomic gases since that was the origin of its derivation, but it is actually applicable to any monatomic classical system, including monatomic liquids and solids. This means that for a pure system of a monatomic ideal gas in thermal equilib- rium with a liquid, the average velocities of the molecules are independent of the phase in which they reside. The liquid molecular environment is still different from the gas molecular environment because liquid molecules are confined to move primarily within a much more crowded environment where the potential energies are more significant. Unless a molecule’s kinetic energy is sufficient to escape the potential energy, it simply collides with a higher frequency in its local environment. What happens if the temperature is raised such that the liquid molecules can escape the potential energies of its neighbors? We call that “boiling,” and the pressure must increase to keep the system in vapor-liquid phase equilibrium. The energy required to promote “boiling” is related to the heat of vaporization. Now you can begin to understand what temperature is and how it relates to other important thermodynamic properties.
For completeness, we may also mention that kinetic energy is the only form of energy for an ideal gas, so the internal energy of a monatomic ideal gas is given by:
(ig) 1.
The proportionality constant between temperature and internal energy is known as the ideal gas heat capacity at constant volume, denoted C (^) V. Eqn 1.18 shows that C (^) V = 1.5 R for a monatomic ideal gas. If you refer to the tables of constant pressure heat capacities ( C (^) P ) on the end flap of the
log 10 [V(cm 3 /mol)]
P(MPa)
Temperature oC
Figure 1.3 Ideal gas behavior at five temperatures.
Section 1.4 Basic Concepts 19
text and note that C (^) P = C (^) V + R , you may be surprised by how accurate this ultrasimplified theory actually is for species like helium, neon, and argon at 298 K.
Note that Eqn. 1.18 shows that U ig^ = U ig ( T ). In other words, the internal energy depends only on the temperature for an ideal gas. The observation that U ig^ = U ig ( T ) is true for any ideal gas, not only for ultrasimplified, monatomic ideal gases. We make the most of this fact in Chapter 5, where we show how to compute changes in energy for any fluid at any temperature and density by system- atically correcting the relatively simple ideal gas result.
Real Fluids
The thermodynamic behavior of real fluids differs from the behavior of ideal gases in most cases. Real fluids condense, evaporate, freeze and melt. Characterization of the volume changes and energy changes of these processes is an important skill for the chemical engineer. Many real fluids do behave as if they are ideal gases at typical process conditions. Application of the ideal gas law simplifies many process calculations for common gases, e.g ., air at room temperature and moderate pressures. However you must always remember that the ideal gas law is an approximation (some- times an excellent approximation) that must be applied carefully to any fluid. P-V behavior of a real fluid (water) and an ideal gas can be compared in Figs. 1.3 and 1.4. The behaviors are presented along isotherms (lines of constant temperature) and the deviations from the ideal gas law for water are obvious. Water is one of the most common substances that we work with, and water vapor behaves nearly as an ideal gas at 100°C ( Psat^ = 0.1014 MPa), where experimentally the vapor vol- ume is 1.6718 m 3 /kg (30,092 cm 3 /mol) and by the ideal gas law we may calculate V = RT/P = 8. · 373.15 / 0.1014 = 30,595 cm 3 /mol. However, the state is the normal boiling point, and we are well aware that a liquid phase can co-exist at this state. This is because there is another density of water at these conditions that is also stable. 1
We will frequently find it convenient to work mathematically in terms of molar density or mass density, which is inversely related to molar volume or mass volume, ρ = 1/ V. Plotting the isotherms in terms of density yields a P - ρ diagram that qualitatively looks like the mirror image of the P-V diagram. Density is convenient to use because it always stays finite as P → 0, whereas V diverges. Examples of P - ρ diagrams are shown in Fig. 6.1 on page 195.
The conditions where two phases coexist are called saturation conditions. The terms satura- tion pressure and saturation temperature are used to refer to the state. The volume (or density) is called the saturated volume (or saturated density). Saturation conditions are shown in Fig. 1.4 as the “hump” on the diagram. The hump is called the phase envelope. Two phases coexist when the sys- tem conditions result in a state inside or on the envelope. The horizontal lines inside the curves are called tie lines that show the two volumes (saturated liquid and saturated vapor) that can coexist. The curve labeled “sat’d liquid” is also called the bubble line, since it represents conditions where boiling (bubbles) can occur in the liquid. The curve labeled “sat’d vapor” is also called a dew line, since it is the condition where droplets (dew) can occur in the vapor. Therefore, saturation is a term that can refer to either bubble or dew conditions. When the total volume of a system results in a sys- tem state on the saturated vapor line, only an infinitesimal quantity of liquid exists, and the state is indicated by the term saturated vapor. Likewise, when a system state is on the saturated liquid line, only an infinitesimal quantity of vapor exists, and the state is indicated by term saturated liquid. When the total volume of the system results in a system in between the saturation vapor and satura- tion liquid volumes, the system will have vapor and liquid phases coexisting, each phase occupying a finite fraction of the overall system. Note that each isotherm has a unique saturation pressure.
- This stability is determined by the Gibbs energy and we will defer proof until Chapter 8.
Real fluids have saturation conditions, bubble points, and dew points.
Section 1.4 Basic Concepts 21
Steam tables are divided into four tables. The first table presents saturation conditions indexed by temperature. This table is most convenient to use when the temperature is known. Each row lists the corresponding saturation values for pressure (vapor pressure), internal energy, volume and two other properties we will use later in the text: enthalpy and entropy. Special columns represent the energy, enthalpy, and entropy of vaporization. These properties are tabulated for convenience although they can be easily calculated by the difference between the saturated vapor value and the saturated liquid value. Notice that the vaporization values decrease as the saturation temperature and pressure increase. The vapor and liquid phases are becoming more similar as the saturation curve is followed to higher temperatures and pressures. At the critical point , the phases become identical. Notice in Fig. 1.4 that the two phases become identical at the highest temperature and pressure on the saturation curve, so this is the critical point. For a pure fluid, the critical temperature is the temperature at which vapor and liquid phases are identical on the saturation curve, and is given the notation T (^) c. The pressure at which this occurs is called the critical pressure, and is given the symbol Pc.
The second steam table organizes saturation properties indexed by pressure, so it is easiest to use when the pressure is known. Like the temperature table, vaporization values are presented. The table duplicates the saturated temperature table, i.e. plotting the saturated volumes from the two tables would result in the same curves. The third steam table is the largest portion of the steam tables, consisting of superheated steam values. Superheated steam is vapor above its saturation tempera- ture at the given pressure. The adjective superheated specifies that the vapor is above the saturation temperature at the system pressure. The adjective is usually used only where necessary for clarity. The difference between the system temperature and the saturation temperature, ( T − T sat^ ), is termed the degrees of superheat. The superheated steam tables are indexed by pressure and temperature. The saturation temperature is provided at the top of each pressure table so that the superheat may be quickly determined without referring to the saturation tables.
The fourth steam table has liquid-phase property values at temperatures below the critical tem- perature and above each corresponding vapor pressure. Liquid at these states is sometimes called subcooled liquid to indicate that the temperature is below the saturation temperature for the speci- fied pressure. Another common way to describe these states is to identify the system as compressed liquid, which indicates that the pressure is above the saturation pressure at the specified tempera- ture. The adjectives subcooled and compressed are usually only used where necessary for clarity. Notice by scanning the table that pressure has a small effect on the volume and internal energy of liquid water. By looking at the saturation conditions together with the general behavior of Fig. 1. in our minds, we can determine the state of aggregation (vapor, liquid or mixture) for a particular state.
The critical tem- perature and critical pressure are key char- acteristic properties of a fluid.
Superheat.
Subcooled, com- pressed.
22 Unit I First and Second Laws
Linear Interpolation
Since the information in the steam tables is tabular, we must interpolate to find values at states that are not listed. To interpolate, we assume the property we desire (e.g., volume, internal energy) var- ies linearly with the independent variables specified (e.g., pressure, temperature). The assumption of linearity is almost always an approximation, but is a close estimate if the interval of the calcula- tion is small. Suppose we seek the value of volume, V , at pressure, P , and temperature, T , but the steam tables have only values of volume at P 1 and P (^) 2 which straddle the desired pressure value as shown in Fig. 1.5. The two points represent values available in the tables and the solid line repre- sents the true behavior. The dotted line represents a linear fit to the tabulated points.
If we fit a linear segment to the tabulated points, the equation form is y = mx + b , where y is the dependent variable (volume in this case), x is the independent variable (pressure in this case), m is
the slope , and b is the intercept. We can interpolate to find V without directly
determining the intercept. Since the point we desire to calculate is also on the line with slope m , it
also satisfies the equation. We can equate the two expressions for m to find the
interpolated value of V at P.
Example 1.2 Introduction to steam tables
For the following states, specify if water exists as vapor, liquid or a mixture: (a) 110° C and 0.12 MPa; (b) 200° C and 2 MPa; (c) 0.8926 MPa and 175° C.
Solution: (a) Looking in the saturation temperature table, the saturation pressure at 110 o^ C is 0.143 MPa. Below this pressure, water is vapor (steam). (b) From the saturation temperature table, the saturation pressure is 1.5549 MPa, therefore water is liquid. (c) This is a saturation state listed in the saturation temperature table. The water exists as satu- rated liquid, saturated vapor, or a mixture.
Linear interpola- tion is a necessary skill for problem solv- ing using thermody- namic tables.
P 1 P 2
V 1
V 2
P
V
Pressure
Volume
Figure 1.5 Illustration of linear interpolation.
m ∆ y ∆ x
V 2 – V 1
P 2 – P 1
m ∆ y ∆ x
V – V 1
P – P 1
