Technology & Engineering
Thermodynamic Potentials
Thermodynamic potentials are functions used to analyze and predict the behavior of thermodynamic systems. They provide valuable information about the system's equilibrium and its ability to do work. Common thermodynamic potentials include internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy, each of which is useful for different types of thermodynamic processes and conditions.
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9 Key excerpts on "Thermodynamic Potentials"
eBook - ePub- Howard Reiss(Author)
- 2012(Publication Date)
- Dover Publications(Publisher)
VIIIThermodynamic Potentials
1. Concept of the Thermodynamic Potential
In our development, Equation (7.1) has emerged as a consequence of the inequality (4.16). It represents the condition that S shall have an extremum in the state of equilibrium determined by the constraints X 0 . Since S is maximized in this particular state, it behaves like the mechanical potential behaves in respect to mechanical equilibrium, except that potential energy is minimized rather than maximized. As a result, the entropy is sometimes referred to as a thermodynamic potential. In view of the useful relation (7.22), (7.23), and (7.24), all of which may be traced to Equation (7.1), it is clear that the thermodynamic potential is a valuable concept. It is therefore natural to inquire whether Thermodynamic Potentials other than entropy exist. The answer is affirmative, and the present chapter will be devoted to this problem.With the possible exception of entropy itself, no other concept in thermodynamics has occasioned so much confusion as the thermodynamic potential. The most distinguished, as well as less distinguished authors have propagated what is a manifestly false doctrine. The confusion is compounded by the fact that it is possible to derive the correct relations with the incorrect philosophy. As we shall see, part of the reason for this lies in the circumstance that the detailed character of the new equilibrium state, to which a system is displaced by a virtual variation, need not be specified. Another part of the reason is the high degree of symmetry found among thermodynamic relations so that new formulas are frequently derivable by analogy with old ones.Unfortunately, however, it is always possible to make mistakes through such reasoning-by-analogy, and some important errors have been committed. Since much of the value of thermodynamics lies in the almost absolute character of the information which it provides, one can hardly afford unnecessary sacrifices of rigor. One of the uses to which Thermodynamic Potentials are put is the derivation of formulas such as Equation (7.24), which specifies the condition under which matter will exhibit no tendency to flow between two phases in contact. If one only cared about deriving Equation (7.24), it would hardly be necessary to set in motion the rather sophisticated machinery of Thermodynamic Potentials. We shall demonstrate this in the next section, but before doing so it is important to assure the reader that there are other uses for such potentials. They make possible the discussion of “stability,” and the separate treatment of stable or unstable equilibriums (see Chapter 14); whereas the elementary method presented in the next section permits us to derive relations like Equation (7.24) which are valid for either
eBook - PDF- Lokesh Pandey(Author)
- 2020(Publication Date)
- Arcler Press(Publisher)
Laws of Thermodynamics and Thermodynamic Potentials CHAPTER 2 CONTENTS 2.1. Introduction ...................................................................................... 34 2.2. Fundamental Laws of Thermodynamics ............................................. 35 2.3. The Four Laws of Thermodynamics .................................................... 35 2.4. The Zeroth Law of Thermodynamics .................................................. 36 2.5. The First Law of Thermodynamics ..................................................... 37 2.6. The Second Law of Thermodynamics ................................................ 41 2.7. The Third Law of Thermodynamics .................................................... 46 2.8. Thermodynamic Potentials ................................................................ 47 2.9. Conclusion ....................................................................................... 52 References ............................................................................................... 53 Smart Thermodynamics 34 Thermodynamics basically deals with the relation between heat and other form of energy. The Laws of thermodynamics are important for understanding the thermodynamic principles. This chapter involves the study of the laws of thermodynamics that includes Zeroth law, first law, the second law and the third law. The chapter provides insights about the four potentials of the thermodynamics. The Thermodynamic Potentials are the measuring scale that measures the thermodynamic state of a system. 2.1. INTRODUCTION According to the Agency. Thermal energy is the energy a substance or system has due to its temperature, i.e., the energy of moving or vibrating molecules. The study of energy interactions between systems and the effect of these interactions on the system properties is known as Thermodynamics. There is a transfer of energy between systems that takes place in the form of heat or work.
eBook - PDFBiothermodynamics
Principles and Applications
- Mustafa Ozilgen, Esra Sorguven Oner(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
1 1 Energy, Entropy, and Thermodynamics 1.1 Energy Energy ( e) is the capacity for doing work. It may exist in a variety of forms and may be transformed from one type to another. Kinetic energy ( e k ) refers to the energy associated with the motion. It is proportional to the square of the system’s velocity. Potential energy ( e p ) refers to the energy that a system has because of its position or configuration. An object may have the capacity for doing work because of its position in a gravitational field (gravitational potential energy), in an electric field (electric potential energy), or in a magnetic field (magnetic potential energy). Internal energy ( u) refers to the energy associated with the chemical structure of the matter. It includes the energy of the translation, rotation, and vibration of the molecules. It is the energy associated with the static constituents of matter like those of the atoms and their chemical bonds. The internal energy of a matter changes with temperature and the pressure acting on the matter. Enthalpy (h) is the energy of a fluid in motion. Consider a fluid particle, which is originally at rest and has an internal energy u. If this fluid particle is set to motion, then its internal energy does not change, but its total energy increases because of the flow motion. The energy that the fluid particle possesses to push all the other fluid particles in front of it is called the flow energy, and it may be estimated as the multiplication of the particles’ pressure and specific volume, pv . The total energy of a flowing fluid particle is then the sum of its internal energy and flow energy. Accordingly, enthalpy is defined as h u pv = + Enthalpy of formation ( Δh f ) is the energy required for the formation of 1 mol of a com- pound from its elements. If all the substances are in their standard conditions, then it is called the standard enthalpy of formation, which is denoted with Dh f O .
eBook - PDF- Juan J. de Pablo, Jay D. Schieber(Authors)
- 2014(Publication Date)
- Cambridge University Press(Publisher)
3 Generalized Thermodynamic Potentials From the postulates in Chapter 2, we have already seen that from U(S, V , N 1 , . . . , N m ) or S(U, V , N 1 , . . . , N m ) we can derive all of the thermodynamic information about a system. For example, we can find mechanical or thermal equations of state. However, we also know that it is sometimes convenient to use other independent variables besides entropy and volume. For example, when we perform an experiment at room temperature open to the atmosphere, we are manipulating temperature and pressure, not entropy and volume. In this case, the more natural independent variables are T and P . Then, the following question arises: Is there a func-tion of ( T, P, N 1 , . . . , N m ) that contains complete thermodynamic information? In other words, is there some function, say G (T, P, N 1 , . . . , N m ) , from which we could derive all the equations of state? It turns out that such functions do exist, and that they are very useful for solving practical problems. In the first section we show how to derive such a function for any complete set of independent variables using something called Legendre transforms. In this book we introduce three widely used potentials: the enthalpy H (S, P, N 1 , . . . , N m ) , the Helmholtz potential F (T, V, N 1 , . . . , N m ) , and the Gibbs free energy G (T, P, N 1 , . . . , N m ) . These functions which contain complete thermodynamic information using independent variables besides S, V , and N 1 , . . . , N m are called generalized Thermodynamic Potentials . These quantities are essential for engineering or applied thermodynamics. For example, we already know that an isolated system attains equilibrium when the entropy is maximized. How-ever, a system in contact with a thermal reservoir attains equilibrium when the Helmholtz potential F is minimized. We show below (Examples 3.2.1 and 3.2.2) that the work necessary to compress any gas isothermally is just the change in F (T, V, N) .
eBook - PDF- Krystyna Jackowska, Paweł Krysiński(Authors)
- 2020(Publication Date)
- De Gruyter(Publisher)
1 Basic concepts 1.1 Structure of interfaces To begin with, it is necessary to establish a solid foundation for all topics presented in the following chapters. This foundation can be derived from the laws of thermo-dynamics that provide tools not only for qualitative and quantitative description of systems and processes but also capabilities to predict their further development. There are four state functions in thermodynamics, namely, the internal energy, U , enthalpy H , entropy, S , Gibbs ’ free energy, G (also called the thermodynamic poten-tial ), and the Helmholtz ’ free energy, F . Together with their parameters of state – V , p , T , n i , these functions describe precisely the state of a given system or process. For the purpose of this book, let us choose Gibbs ’ free energy G (the thermodynamic potential) for subsequent chapters of this book. As mentioned earlier, this function is the state function, meaning that its change depends only on the initial and final state of the system. From the mathematical point of view such extremely small change can be written as the total differential versus the state parameters of Gibbs ’ free energy: G = G(p, t, n i ) . Thus, dG = ∂ G ∂ p T , n i dp + ∂ G ∂ T p , n i dT + Σ ∂ G ∂ n i p , T dn i (1 : 1) Subscripts next to parentheses show that the remained parameters of state are con-stant. Thus, the meaning of this total differential is that we can sum partial differential of G versus p , keeping T and n i constant, controlling the change dp , and so on, and then sum all partial differentials to get the overall change of G , dG . As long as we do not assign the physicochemical meanings of all three partial differentials in the above equation, it remains purely mathematical.
eBook - PDF- H J Kreuzer, Isaac Tamblyn;;;(Authors)
- 2010(Publication Date)
- WSPC(Publisher)
Chapter 6 Thermodynamic Potentials So far we have based thermodynamics on two fundamental relations, namely the entropy representation, S = S ( U, V, N ), and the internal energy rep-resentation, U = U ( S, V, N ). Although they are completely equivalent we have seen in a number of examples that one or the other may some-times be more appropriate or convenient. It is not really surprising that we have two equivalent formulations of thermodynamics; other fields of physics do the same. Consider classical mechanics: we have (1) Newton’s equations of motion, (2) Lagrange equations, (3) Hamilton’s equations and (4) the Hamilton-Jacobi equation. Again, in which framework we work is a matter of convenience. Similarly in quantum mechanics we can work with Heisenberg’s matrix mechanics or with Schr¨ odinger’s wave mechanics; again the choice is one of convenience. Geometry also offers a multitude of approaches: you can construct a circle by (1) minimizing the circumference of a fixed area, or (2) by maximizing the area for a fixed circumference. Both extremal principles yield circles. In this chapter we will successively eliminate the extensive variables in favor of their intensive counterparts. Eliminating the entropy from the internal energy will give us the Helmholtz free energy F = F ( T, V, N ), further elimination of the volume will produce the Gibbs free energy G = G ( T, P, N ), etc. Let us then revisit the maximum entropy principle and deduce its impli-cations for the internal energy. We start with an isolated composite system of fixed internal energy U , volume V , and mole number N . We separate off a subsystem by a rigid, adiabatic, and impermeable wall; the subsystem has variables U 1 , V 1 , and N 1 . If we make the separating wall moveable, permeable, and diathermal, then these variables will adjust in such a way 103 104 Thermodynamics that the entropy is maximum, i.e.
eBook - PDF- Robert J. Silbey, Robert A. Alberty, George A. Papadantonakis, Moungi G. Bawendi(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Fortunately, however, more useful thermodynamic properties can be defined, based on the internal energy. 4.2 DEFINITIONS OF ADDITIONAL THERMODY- NAMIC POTENTIALS USING LEGENDRE TRANSFORMS The internal energy and other thermodynamic properties defined starting with the internal energy are referred to as Thermodynamic Potentials. To define new Thermodynamic Potentials, we use the method of Legendre transforms. We have already seen an example of this with the enthalpy H, defined by H = U + PV (see Section 2.7). We did not emphasize that this is a Legendre transform, but it is, and now we are going to use two more Legendre transforms to define two Thermodynamic Potentials. A Legendre transform is a linear change in variables that starts with a mathematical function and defines a new function by subtracting one or more products of conjugate variables. This is different from the usual change in variables in that a partial derivative of a thermodynamic potential becomes an independent variable in the new thermodynamic potential. As explained in Section 2.10 (Mathematical Treatise) at the end of the chapter, no information is lost in this process. Thus a new thermodynamic potential defined in terms of the internal energy contains all of the information that is in U (S, V ,{n i }). ∗ ∗ H. B. Callen, Thermodynamics, 2nd ed. Hoboken, NJ: Wiley, 1985; R. A. Alberty, J. M. G. Barthel, E. R. Cohen, R. N. Goldberg, and E. Wilhelm, Use of Legendre Transforms in Chemical Thermodynamics (an IUPAC Technical Report), Pure Appl Chem. 73:8(2001). 4.2 Definitions of Additional Thermodynamic Potentials Using Legendre Transforms 105 Now we can make a more complete treatment of the enthalpy H than in Section 2.7.
eBook - PDF- Cemil Hakan Gur, Jiansheng Pan, Cemil Hakan Gur, Jiansheng Pan(Authors)
- 2008(Publication Date)
- CRC Press(Publisher)
E , H , A , and G are different Thermodynamic Potentials. Depending on the conditions of the system, Gibbs free energy ( G ), Helmholtz free energy ( A ), enthalpy ( H ), or entropy ( S ) is used to determine the stability of the system. Another frequently used auxiliary function is chemical potential, which will be introduced later in this chapter. The following is a summary of stability criteria under different situations. 70 Handbook of Thermal Process Modeling of Steels Condition Stability Criteria Constant T , P , n i , E , H , . . . G minimum Constant T , V , n i , E , H , . . . A minimum Constant S , P , n i , E , H . . . H minimum Constant S , V , n i , E , H , . . . U minimum Constant V , U , n i , E , H , . . . S maximum In the case of both open and closed systems, the following relations are often used. d U ¼ T d S P d V þ X i m i d n i þ B d M þ E d Q þ ( 2 : 25 ) d H ¼ T d S þ V d P þ X i m i d n i þ B d M þ E d Q þ ( 2 : 26 ) d A ¼ S d T P d V þ X i m i d n i þ B d M þ E d Q þ ( 2 : 27 ) d G ¼ V d P S d T þ X i m i d n i þ B d M þ E d Q þ ( 2 : 28 ) The term S i m i d n i includes the effect due to changes in the number of moles of the various species. The term B d M is the contribution from the magnetic work done on the system and E d Q is the contribution from the electrical work done on the system. In most problems, these terms may not be relevant and thus often omitted. From Equations 2.25 through 2.28, the following thermodynamic relationships de fi ning P , T , V , and S can be obtained. T ¼ @ U @ S V , composition ¼ @ H @ S P , composition ( 2 : 29 ) P ¼ @ U @ V S , composition ¼ @ F @ V T , composition ( 2 : 30 ) V ¼ @ H @ P S , composition ¼ @ G @ P T , composition ( 2 : 31 ) S ¼ @ F @ T V , composition ¼ @ G @ T P , composition ( 2 : 32 ) Maxwell ’ s equations : Maxwell ’ s relations are a set of large number of equations that provide interrelationships between thermodynamic parameters.
eBook - PDF- Brent Fultz(Author)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
Minimizing E with respect to S is a way to find conditions of thermal equilibrium, but the role of temperature appears through the functional relationship between E and S, i.e., T = (∂ E /∂ S) V ,N . It is usually easier to work with T directly, and T is a natural variable for the Helmholtz or Gibbs free energies, F (T , V , N) or G(T , P, N). In general, the Gibbs free energy is the most convenient thermodynamic potential if we have control over T and P. If G does not depend significantly on P, however, it is simpler to ignore effects of pressure and focus on the role of T by using the 17 1.6 Brief Review of Thermodynamics and Kinetics Figure.1.8 Thermodynamic square with mnemonics. Helmholtz free energy F . We will usually use F because, as explained in Chapter 8, for most solids a change of 100 K in T causes approximately the same change in G as does 5000 atmospheres in P, which is less common in practice. The thermodynamic square of Fig. 1.8 can guide the selection of a thermodynamic potential, and it has other uses. • The Thermodynamic Potentials are in the four boxes, with their natural variables to their sides. • Computing total differentials such as dF of Eq. 1.22 uses differentials of its natural variables (with positive signs), and quantities in the opposite corner, e.g., dF = −SdT − PdV . The term μdN can be added later. • Maxwell relations are formed by going down the sides with a − sign, (∂ S/∂ P) T = −(∂ V /∂ T ) P , or across the rows with a + sign, (∂ P/∂ T ) V = (∂ S/∂ V ) T . 1.6.2 Free Energy The historical, but still correct, meaning of “free energy” is the amount of energy available in a system to do external work such as pushing a piston in a steam engine. The Helmholtz free energy, F ≡ E − TS, is the internal energy minus the entropic contribution that cannot perform work. When two parts of a system are brought together and interact at the same temperature, one part will do work on the other until the total free energy is a minimum.
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