E. P. Gyftopoulos Massachusetts Institute of Technology Cambridge, MA 02139 G. P. Beretta Massachusetts Institute of Technology Cambridge, MA 02139 What is the Second LAW?* The thesis of this article is that thermodynamics is a rigorous science, and that the first law and the second law can be stated in an unambiguous and general way so that their implications are concrete and valid for both equilibrium and nonequilibrium states. In this light, we summarize the principles of thermodynamics, and introduce a graphical tool, the energy versus entropy diagram, that is very helpful to explain and grasp the general implications of these principles, especially in the nonequilibrium domain. [DOI: 10.1115/1.4026379] Premise In a recent thermodynamics text, Truesdell [1] identifies several different “second laws”. In a review in 1986, the physicist- philosopher Bunge [2] compiles a list of about twenty ostensibly inequivalent but equally vague formulations of the “second law”. In a manuscript published in 1983, Lindblad [3] gives a large number of different expressions for entropy. No wonder scientists and engineers are puzzled about the foundations of thermodynam- ics in general, and the second law in particular. The thesis of this presentation is that thermodynamics is a rig- orous science, that its principles can be stated in an unambiguous and general way, and the implications of these principles are con- crete and valid for both equilibrium and nonequilibrium states. In this article, we present a concise summary of the principles of thermodynamics. The summary provides evidence in support of our thesis. Most of the definitions, statements, and observations presented here, as well as the graphical representation by means of the energy versus entropy diagram, are well familiar to the M.I.T alumni who took the graduate course taught by the first author during the last twenty years (jointly with the second author during the last six), but are published here for the first time, and cannot be found in any of the hundreds of textbooks on thermodynamics published to date. Introduction Thermodynamics is concerned with the instantaneous condition that any material may assume, and the time-dependent evolution of this condition that may occur either spontaneously or as result of interactions with other materials, or both. It is a science with the same objective as the whole of physics and, therefore, sub- sumes each special branch of physics, such as the theory of mechanics, electromagnetism, and classical thermodynamics, as a special case. Because of the breadth and depth of its scope, the exposition of thermodynamics requires rigorous consideration of many ba- sic concepts. Some of these concepts are very well known from introductory courses in physics and, for this reason, we assume that ideas such as space, time, velocity, acceleration, mass, force, kinetic energy, and potential energy are well understood and need not be reemphasized. On the other hand, other concepts such as those represented by the terms system, property, state, process, energy, and entropy are sometimes not clearly defined and need special emphasis. In this article, we provide a brief summary of the key concepts, and graphical illustrations of the results. Systems, Properties and States A system is a collection of constituents which is defined by the following specifications: (a) the type and the range of values of the amount of each constituent; for example, 1 kg of water mole- cules, or between 5 and 10 kg of atmospheric air; (b) the type and the range of values of the parameters which fully characterize the external forces that are exerted on the constituents by bodies other than the constituents; for example, the parameters that describe the geometrical shape of an airtight container; and (c) the internal forces between constituents such as the forces between water molecules, the forces that promote or inhibit a chemical reaction, the partitions separating constituents in one region of space from constituents in another region, or the interconnections between separated parts. Everything that is not included in the system is called the environment or the surroundings of the system. For a system consisting of r different types of constituents, we denote their amounts by the vector n ¼ {n 1 ,n 2 ,…,n r } where n 1 , stands for the amount of the first type of constituent, n 2 for the amount of the second, and so on. For example, the different types of constituents could be: three specific molecules, such as the H 2 , O 2 , and H 2 O molecules, with amounts denoted, respectively, by n 1 ,n 2 , and n 3 , two specific ions, such as H þ ,O  ,H 3 O þ , and OH  ions, with amounts denoted, respectively, by n 1 ,n 2 ,n 3 and n 4 ; four specific ions, such as the H þ ,O  ,H 3 O þ , and OH  ions, with amounts denoted, respectively, by n 1 ,n 2 ,n 3 , and n 4 ; three specific elementary particles such as the electron, proton, and neutron par- ticles, with amounts denoted by n 1 ,n 2 , and n 3 ; or a single specific field such as the electromagnetic radiation field, with amount denoted by n and equal to unity, n ¼ 1. It is clear that for each set of different types of constituents there are different internal forces between constituents. For exam- ple, if only H 2 O molecules are considered then the only internal force is that between H 2 O molecules. Again, if H 2 O, H 2 and O 2 molecules are considered, and the chemical reaction H 2 þ 1 = 2 O 2 ¼ H 2 O occurs, the intermolecular forces between ail types of molecules must be specified as well as the forces that control the chemical reaction. For a system with external forces described by s parameters, we denote the parameters by the vector b¼fb 1 ; b 2 ; … b s g where b 1 stands for the first parameter, b 2 for the second, and so on. For example, one of the parameters could be the side L or the volume V of a three-dimensional cubic region in space which is enclosed by either the walls of a container or a geometric (as opposed to material) surface chosen to separate the constituents that belong to the system from all the others that do not and are outside the en- closure. Again, another parameter could be the potential  of a uniform gravitational field in which the constituents are immersed, the potential w of an electromagnetic field in which the constituents are floating, or the area a of a two-dimensional sur- face in space on which the constituents are constrained. At any instant of time, the amount of each type of constituent and the parameters of each external force have specific values within the corresponding ranges of the system. By themselves, * Published in the Proceedings of the Fourth International Symposium on Second Law Analysis of Thermal Systems (Rome, Italy, May 25–29, 1987), Edited by M. J. Moran and E. Sciubba, ASME book I00236, pp. 155-170 (1987). Reprinted with permission. Journal of Energy Resources Technology MARCH 2015, Vol. 137 / 021003-1 Copyright V C 2015 by ASME Downloaded From: http://energyresources.asmedigitalcollection.asme.org/ on 01/09/2015 Terms of Use: http://asme.org/terms