Enthalpy and Internal Energy: Liquids, Solutions and Vapours
640
Enthalpy and Internal Energy: Liquids, Solutions and Vapours
640eBook
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| ISBN-13: | 9781788011969 |
|---|---|
| Publisher: | Royal Society of Chemistry |
| Publication date: | 09/08/2017 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 640 |
| File size: | 14 MB |
| Note: | This product may take a few minutes to download. |
About the Author
National Institute of Standards (NIST), USA
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CHAPTER 1
Internal Energy and Enthalpy: Introduction, Concepts and Selected Applications
EMMERICH WILHELM
Institute of Materials Chemistry & Research/Institute of Physical Chemistry, University of Wien, Wahringer Strasse 42, A-1090, Wien (Vienna), Austria Email: emmerich.wilhelm@univie.ac.at
1.1 Introduction
Life is girt all round with a zodiac of sciences, the contributions of men who have perished to add their point of light to our sky.
Ralph Waldo Emerson, Representative Men. Seven Lectures: I. Uses of Great Men, The Riverside Press, Cambridge, Mass., USA (1883).
This monograph is concerned with internal energy and enthalpy and related properties of fluids, pure and mixed, and their role in the physico-chemical description of systems ranging from pure rare gases to proteins in solution. In this introductory Chapter 1, I shall only consider nonreacting fluid equilibrium systems of uniform temperature T and pressure P (i.e., systems in thermal, mechanical and diffusional equilibrium) characterised by the essential absence of surface effects and extraneous influences, such as electric fields. However, the influence of the earth's gravitational field is omnipresent: though usually ignored, it becomes important near a critical point. Under ordinary conditions, the molar volumes V (or specific volumes V/mm, where mm denotes the molar mass) of homogeneous fluids in equilibrium states are functions of T, P and composition only. Such systems are known as PVT systems or simple systems. However, the generality of thermodynamics makes it applicable to considerably broader types of systems by adding appropriate work terms, i.e., products of conjugate intensive and extensive variables, such as surface tension and area of surface layer. Finally, there is a caveat concerning idealised concepts for systems and processes, such as isolated systems, isothermal and reversible processes, to name but a few. Fortunately, they can be well approximated experimentally, and while classical thermodynamics only treats the corresponding limiting cases, the ensuing restrictions are not severe: values of thermodynamic quantities obtained with different experimental techniques are expected to agree within experimental error. Classical thermodynamics deals only with measurable equilibrium properties of macroscopic systems. It is a formalised phenomenological theory of enormous generality in the following sense:
(I) Thermodynamic theory is applicable to all types of macroscopic matter, irrespective of its chemical composition and independent of molecule-based information, i.e., systems are treated as "black boxes" and the concepts used ignore microscopic structure, and indeed do not need it.
(II) Classical thermodynamics does not allow ab initio prediction of numerical values for thermodynamic properties. It provides, however, a mathematical network of equations (and a few inequalities) that yields exact relations between measurable equilibrium quantities and restricts the behaviour of any natural process.
The scope of chemical thermodynamics was succinctly summarised by McGlashan: W hat then is the use of thermodynamic equations to the chemist? They are indeed useful, but only by virtue of their use for the calculation of some desired quantity which has not been measured, or which is difficult to measure, from others which have been measured, or which are easier to measure.
This aspect alone is already of the greatest value for applications: augmenting the formal framework of chemical thermodynamics with molecule-based models of material behaviour, i.e., by using concepts from statistical mechanics, experimental thermodynamic data contribute decisively towards a better understanding of molecular interactions, and lead to improved descriptions of macroscopic systems. This field of molecular thermodynamics (the term was coined by Prausnitz more than four decades ago) is of great academic fascination, and is indispensable in (bio-)physical chemistry and chemical engineering. Its growth has been stimulated by the increasing need for thermodynamic property data and phase equilibrium data in the applied sciences, and it has profited from advances in experimental techniques, from modern formulations of chemical thermodynamics, from advances in the theory of fluids in general and from advances in computer simulations of model systems. In Subsection 1.2 I shall present thermodynamic fundamentals of relevance for the book's topic, while 1.3 is devoted to derived thermodynamic properties and relations of relevance for many of the book chapters. A few comments on nomenclature, a brief outlook and concluding remarks will be given in Subsection 1.4.
Experiments, molecule-based theory and computer simulations are the three pillars of science. Experiments provide the basis for inductive reasoning, known informally as bottom-up reasoning, which, after amplifying, logically ordering and generalising our experimental observations, leads to hypotheses and then theories, and thus to new knowledge. In contradistinction, deduction, informally known as top-down reasoning, orders and explicates already existing knowledge, thereby leading to predictions which may be corroborated experimentally, or falsified: a theory has no value in science unless it is possible to test it experimentally. As pointed out by Freeman Dyson,
Science is not a collection of truths. It is a continuing exploration of Mysteries ... an unending argument between a great multitude of voices.
The most popular heuristic principle to guide hypothesis/theory testing is known as Occam's razor, named after the Franciscan friar William of Ockham (England, ca. 1285–1349). Also called the principle of parsimony or the principle of the economy of thought, it states that the number of assumptions to be incorporated into an adequate model should be kept minimal. While this is the preferred approach, the model with the fewest assumptions may turn out to be wrong. More elaborate versions of Occam's razor have been introduced by modern scientists, and for in-depth philosophical discussions see Mach, Popper, Katz and Sober.
1.2 Thermodynamic Fundamentals
Thermodynamics is a physical science concerned with energy and its transformations attending physical and chemical processes. Historically, it was developed to improve the understanding of steam engines, the focus being on the convertibility of heat into useful work. The concepts of work, heat and energy were developed over centuries, and the formulation of the principle of conservation of energy has been one of the most important achievements in physics. Work, w, and heat, q, represent energy transfers, they are both energy in transit between the system of interest and its surroundings. The transfer of energy represented by a quantity of work w is a result of the existence of unbalanced forces between system and surroundings, and w is not a system property. The transfer of energy represented by a quantity of heat q is a result of the existence of a temperature difference between system and surroundings, and q is not a system property: w and q are defined only for processes transferring energy across a system boundary.
A convenient way to present the fundamentals of the phenomenological theory of thermodynamics is the postulatory approach. A small number of postulates inspired by observation are assumed to be valid without admitting the existence of more fundamental relations from which the postulates could be deduced. The ultimate justification of this approach rests solely on its usefulness. Consider a homogeneous fluid in a closed PVT system, that is a system with a boundary that restricts only the transfer of matter (constant mass system). For such a system, the existence of a form of energy called total internal energy Ut is postulated, which is an extensive material property (for a definition see below) and a function of T, P and mass m or amount of substance n = m/mm. This designation distinguishes it from kinetic energy Ek and potential energy Ep which the system may possess (external energy).
Next, the first law of thermodynamics is introduced as a generalisation and abstraction of experimental results concerning energy conservation. It applies to a system and its surroundings: the law states that energy may be transferred from a system to its surroundings and vice versa, and it may be converted from one form into another, yet the overall quantity of energy is constant. Thus, for a closed PVT system, for any process taking place between an initial equilibrium state 1 and a final equilibrium state 2 of a homogeneous fluid at rest at constant elevation (no changes in kinetic and/or potential energy), eqn (1.1) is a statement of the first law:
Δ Ut = q + w. (1.1)
For a differential change it is written
dUt = δq + δw. (1.2)
Mathematically, dUt is an exact differential of the state function Ut: the change in the value of this extensive property,
[MATHEMATICAL EXPRESSION OMITTED] (1.3)
depends only on the two states. On the other hand, δq and δw are inexact differentials, i.e., they represent infinitesimal amounts of heat and work: q and w are path functions. When integrated, δq and δw give finite amounts q and w, respectively. The notation used in the first law, eqn (1.1), asserts that the sum of the two path functions q and w always yields an extensive state function change ΔUt between two equilibrium state points, independent of the choice of path 1 -> 2.
The exact (total) differential of a function f of n independent variables Xi,
f=f(X1, X2, ... Xn), (1.4)
is defined by
[MATHEMATICAL EXPRESSION OMITTED] (1.5)
[MATHEMATICAL EXPRESSION OMITTED] (1.5)
and the partial derivative ci, and its corresponding variable Xi are known as being conjugate to each other. For a non-pathological function f the order of differentiation in mixed second derivatives is immaterial, thus yielding the Euler reciprocity relation
[MATHEMATICAL EXPRESSION OMITTED] (1.7)
for any two pairs (ci, Xi) and (cj, Xj) of the conjugate quantities. Eqn (1.7) serves as a necessary and sufficient criterion of exactness. For n independent variables, the number of conditions to satisfy is n(n – 1)/2. As shown later on, this is the number of Maxwell relations. If df(X1, X2,..., Xn) is exact, we have
[MATHEMATICAL EXPRESSION OMITTED] (1.8)
independent of the integration path; in thermodynamics such a function f is called a state function.
As recommended by the International Union of Pure and Applied Chemistry, the sign convention for heat and work is that the internal energy increases when heat "flows" into the system, i.e., q>0, and work is done on the system, i.e., w>0. Eqn (1.1) does not provide an explicit definition of internal energy. In fact, there is no known way to measure absolute values of Ut: the internal energy of a system is an extensive conceptual property. Fortunately, only changes in internal energy are of interest, and these changes can be measured. Consider now a series of experiments performed essentially reversibly on a homogeneous constant-composition fluid (closed system) along different paths from an initial equilibrium state at (T1, P1) to a final equilibrium state at (T2, P2). All measurements show that the experimentally determined sum qrev + wrev is constant, independent of the path selected, as it must be provided the postulate of internal energy being a material property is valid. Thus, one has
[MATHEMATICAL EXPRESSION OMITTED] (1.9)
as a special case of eqn (1.1), which for closed systems is generally applicable for reversible as well as irreversible processes between equilibrium states. In differential form this equation reads
[MATHEMATICAL EXPRESSION OMITTED] (1.10)
Hence the measurability of any change of the internal energy follows from
[MATHEMATICAL EXPRESSION OMITTED] (1.11)
regardless of the path 1 -> 2. Since all experimental evidence to date has shown this relation to be true, we may safely assume that it is generally true, though the possibility of falsification, of course, remains. No further "proof" exists beyond the experimental evidence.
Internal energy is a macroscopic equilibrium property, and the adjective internal derives from the fact that for homogeneous fluids in closed PVT systems, Ut is determined by the state of the system characterised by the independent variables T, P and m or n. Digressing from formal thermodynamics – where the existence of molecules is never invoked – it is instructive to briefly indicate the molecular interpretation of Ut. On a microscopic level, internal energy is associated with molecular matter, and for nonreacting systems at common T and P contributions to ΔUt essentially derive from changes of the molecular kinetic energy, the configurational energy and the intramolecular energy associated with rotational and vibrational modes.
Eqn (1.1) may be expanded to incorporate external energy changes resulting from changes of the closed system's macroscopic motion and/or position. Thus, when changes in kinetic energy and potential energy are to be considered, the first law becomes
[MATHEMATICAL EXPRESSION OMITTED] (1.12)
Work done on the system with mass m by accelerating it from initial speed v1 to final speed v2 is
[MATHEMATICAL EXPRESSION OMITTED] (1.13)
and the work done on the system by raising it from an initial height h1 to a final height h2 is
[MATHEMATICAL EXPRESSION OMITTED] (1.14)
where g is the acceleration of gravity. Since the zero of potential energy can be chosen arbitrarily, only differences in potential energy are meaningful. A similar comment applies to kinetic energy.
Thermodynamic properties may be classified as being intensive or extensive. Properties that are independent of the system's extent are called intensive; examples are temperature, pressure and composition, i.e., the principal thermodynamic coordinates for homogeneous fluids. A property that is additive for independent, noninteracting subsystems is called extensive; examples are mass and amount of substance. The value of an extensive property is either proportional to the total mass m or to the total amount of substance n, and the proportionality factor is known as a specific property or as a molar property, respectively. Thus, for the extensive total internal energy we have
Ut = mu, (1.15)
where u denotes the specific internal energy, an intensive property, or alternatively we have
Ut = nU, (1.16)
where U denotes the intensive molar internal energy. Evidently, the quotient of any two extensive properties is an intensive property. Hence an extensive property is transformed into an intensive specific property by dividing by the total mass, and into an intensive molar property by dividing by the total amount of substance; a density is obtained when dividing by the total volume
Vt = nV. (1.17)
As pointed out, classical thermodynamics is concerned with macroscopic properties and with relations among them. No assumptions are made about the microscopic, molecular structure, nor does it reveal any molecular mechanism. In fact, essential parts of thermodynamics were developed before the internal structure of matter was established, and a logically consistent theory can be developed without assuming the existence of molecules. However, since we do have reliable theories involving molecular properties and interactions, using appropriate molecule-based models statistical mechanics allows the calculation of macroscopic properties. Essentially because of this connection, molar properties are used together with the appropriate composition variable, the mole fraction xi of component i of a homogeneous system, i.e., of a phase:
[MATHEMATICAL EXPRESSION OMITTED] (1.18)
where [MATHEMATICAL EXPRESSION OMITTED] is the total amount of substance in the phase, and ni is the amount of component [MATHEMATICAL EXPRESSION OMITTED], and for a pure fluid xi = 1. Of course, use is limited to systems of known molecular composition. Thus, if Mt is taken to represent an extensive total property, such as Ut, of a homogeneous system, the corresponding intensive molar property M is defined by
M [equivalent] Mt/n. (1.19)
The overall molar property of a closed multiphase system, with p equilibrium phases α, β, ..., is
[MATHEMATICAL EXPRESSION OMITTED] (1.20)
(Continues…)
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