Thermodynamics of Condensed Phases
2.0 INTRODUCTION AND OBJECTIVES
Thermodynamics is the foundation of all the engineering and many of the natural science disciplines, and materials engineering and science is no exception. A certain familiarity with the tenants of thermodynamics—First and Second Laws, enthalpy and entropy—will be assumed in this chapter, since you have probably been exposed to them already in chemistry and physics and possibly in an introductory thermodynamics course. A review of these concepts is certainly in order—not only to reacquaint you with them, but to establish certain conventions (sign conventions, which letters rep- resent which quantities) that can vary between disciplines, instructors, and books. As we traverse through this review and begin applying thermodynamic principles to real material systems, you should be delighted to notice that some of the concepts you may have found confusing in other courses (e.g., Carnot cycles, compressibility factors, and fugacities) are not of concern here. There is even little need to distinguish between internal energy and enthalpy. This stems from the fact that we are dealing (almost) exclusively withcondensed systems—that is, liquids and solids. Though gases can cer- tainly behave as material systems, we will not find much utility in treating them as such.
By the end of this chapter, you should be able to:
ž Identify the number of components present, the number of phases present, the composition of each phase, and the quantity of each phase from unary, binary, and ternary phase diagrams—that is, apply the Gibbs Phase Rule.
ž Apply the Lever Rule to a two-phase field in a binary phase diagram.
ž Identify three-phase reactions in binary component systems.
ž Calculate the free energy and heat of mixing for a simple binary mixture.
ž Define surface energy, and relate it to thermodynamic quantities such as free energy.
ž Apply the Laplace equation to determine the pressure across a curved surface.
ž Apply the Young equation to relate contact angle with surface energies.
ž Identify the three stages of sintering, and describe how surface energy drives each process.
ž Differentiate between binodal and spinodal decomposition in polymer mixtures.
An Introduction to Materials Engineering and Science: For Chemical and Materials Engineers, by Brian S. Mitchell
ISBN 0-471-43623-2 Copyright2004 John Wiley & Sons, Inc.
136
INTRODUCTION AND OBJECTIVES 137
ž Differentiate between adhesion, cohesion, and spreading, and calculate the work associated with all three.
ž Describe cell–cell adhesion in terms of free energy concepts.
2.0.1 Internal Energy
The letter U will be used to represent the internal energy of a system. Recall that the internal energy of a system is the sum of the microscopic kinetic and potential energies of the particles. Kinetic energy is the energy due to motion of the particles, including translation, rotation, and vibration. The potential energy is the energy due to composition. We saw in Chapter 1 how there is potential energy stored in chemical bonds, and how the potential energy between two atoms or ions changes as a function of separation distance. The quantityU represents the change of internal energy for the system from some initial state to some final state.
The letter E will be used to represent the total energy of a system. The quantity E represents the change of total energy from an initial state to a final state. The total energy is comprised of the internal energy,U, the kinetic energy of the system, and the potential energy of the system. Do not confuse themacroscopickinetic and poten- tial energies, Ek, andEp, respectively, with the microscopic kinetic and potential energy contributions to the internal energy just described above. By way of analogy, the former has to do with a rock being pushed off a 10-meter-high ledge, while the latter has to do with the bonds between molecules in the rock and the movement of the individual atoms within the rock.
Already, you should be thinking to yourself “But the particles in solids really don’t move that much!” and you are certainly correct. They do move or “translate” in the liquid state of that same solid, however, and don’t forget about rotation and vibration, which we will see in subsequent chapters can be very important in solids. But along this line of thinking, we can simplify the First Law of Thermodynamics, which in general terms can be written for a closed system (no transfer of matter between the system and surroundings) as
U+Ek+Ep= ±Q±W (2.1)
whereQis theheat transferred between the surroundings and the system, andW is the work performed. The signs used in front ofQandW are a matter of convention. Each is taken as positive when the exchange is from the surroundings to the system—that is, by the surroundingsto the system. This is the convention we will use. This convention is universally accepted for Q, but not so for W. In this way, exothermic processes in which heat is transferred from the system to the surroundings have a negative sign in front of Q; and endothermic processes, in which heat is transferred from the surroundings to our system, have a positive sign. The sign convention for W is less problematic. We will see that in condensed, closed systems, the work term will not be of interest to us and will be ignored, since it mostly arises from pressure–volume (PV) work. Similarly, the macroscopic kinetic and potential energies,Ek, andEp, respectively, are not of importance, so that the functional form of the First Law is
U =Q (2.2)
or, in differential form for infinitesimal changes of state,
dU =dQ (2.3)
BothU andQhave units of joules, J, in the SI system, dynes (dyn) in cgs, and calorie (cal) in American engineering units.
2.0.2 Enthalpy
A closely related quantity to the internal energy is the enthalpy, H. It, too, has SI units of joules and is defined as the internal energy plus the pressure–volume product, PV. As in most cases, we are concerned with changes in internal energy and enthalpy from one state to another, so that the definition of enthalpy for infinitesimal changes in state is
dH =dU+d(P V ) (2.4)
Substitution of Eq. (2.3) into (2.4) and recognition that for condensed systems the d(PV) term is negligible leads to
dH =dQ (2.5)
which in integrated form is
H =Q (2.6)
2.0.3 Entropy
Entropy will be represented by the letter S. Entropy is a measure of randomness or disorder in a system and has SI units of J/K. Recall that the Second Law of Thermo- dynamics states thatthe entropy change of all processes must be positive. We will see that the origins of entropy are best described from statistical thermodynamics, but for now let us concentrate on how we can use entropy to describe real material systems.
The differential change in entropy for a closed system from one state to another is, by definition, directly proportional to the change in reversible heat,dQrev, and inversely proportional to the absolute temperature,T:
dS = dQrev
T (2.7)
Primarily reversible processes will be studied in this chapter, so that it is not necessary to retain the subscript on Q. Also, equilibrium transformations will be the primary focus of this chapter, such that we will be concerned mostly with constant-temperature processes. These facts allow us to simplify Eq. (2.7) by integration
S= Q
T (2.8)
Substitution of Eq. (2.6) into (2.8) gives as a useful relationship between the entropy and enthalpy for constant pressure and temperature processes:
S= H
T (2.9)
INTRODUCTION AND OBJECTIVES 139
Entropy also plays a role in the Third Law of Thermodynamics, which states thatthe entropy of a perfect crystal is zero at zero absolute temperature.
2.0.4 Free Energy
The most useful quantity for this chapter is theGibbs free energy,G. The Gibbs free energy for a closed system is defined in terms of the enthalpy and entropy as
G=H−TS (2.10)
which in differential form at constant pressure and temperature is written
dG=dH−TdS (2.11)
or in the integrated form
G=H−T S (2.12)
Recall that the important function of the free energy change from one state to another is to determine whether or not the process isspontaneous—that is, thermodynamically favored. The conditions under which a process is considered spontaneous are summa- rized in Table 2.1. The “processes” described here in a generic sense are the topic of this chapter, as are the implications of “equilibrium” to material systems. The issue of
“spontaneity” and what this means (or does not mean) to the rate at which a process occurs is the subject of Chapter 3.
HISTORICAL HIGHLIGHT Josiah Willard Gibbs was born in New
Haven, CT on February 11, 1839 and died in the same city on April 28, 1903.
He graduated from Yale College in 1858, received the degree of doctor of philosophy in 1863 and was appointed a tutor in the college for a term of three years. After his term as tutor he went to Paris (winter 1866/67) and to Berlin (1867), where he heard the lectures of Magnus and other teachers of physics and mathematics. In 1868 he went to Heidelberg where Kirchhoff and Ostwald were then stationed before returning to New Haven in June 1869. Two years later he was appointed Professor of mathematical physics in Yale College, a position he held until the time of his death. In 1876 and 1878 he published the two parts of the paper “On the Equilibrium of Heterogeneous Substances,” which is generally considered
his most important contribution to physical sciences. It was translated into German in 1881 by Ostwald and into French in 1889 by Le Chatelier.
Outside his scientific activities, J. W.
Gibbs’s life was uneventful; he made but one visit to Europe, and with the exception of those three years and of summer vacations in the mountains, his whole life was spent in New Haven. His modesty with regard to his work was proverbial among all who knew him; there was never any tendency to make the importance of his work an excuse for neglecting even the most trivial of his duties, and he was never too busy to devote as much time and energy as might be necessary to any of his students who sought his assistance.
Source: www.swissgeoweb.ch/minpet/groups/
thermodict/notes/gibbspaper.html
Table 2.1 Summary of Free Energy Effects on Process Spontaneity
G <0 Process proceeds spontaneously G >0 Process not spontaneous
G=0 Process at equilibrium
2.0.5 Chemical Potential
The final thermodynamic quantity for review is thechemical potential, which is rep- resented with the Greek letter mu,µ. The chemical potential can be defined in terms of the partial derivative of any of the previous thermodynamic quantities with respect to the number of moles of speciesi,ni, at constantnj (wherej indicates all species other thani) and thermodynamic quantities as indicated:
µi = ∂U
∂ni
S,V ,nj
= ∂H
∂ni
P ,S,nj
= ∂G
∂ni
T ,P ,nj
(2.13) The advantage of the chemical potential over the other thermodynamic quantities, U, H, andG, is that it is an intensive quantity—that is, is independent of the number of moles or quantity of species present. Internal energy, enthalpy, free energy, and entropy are all extensive variables. Their values depend on the extent of the system—that is, how much there is. We will see in the next section that intensive variables such asµ, T, andP are useful in defining equilibrium.