Abstract
This chapter begins with a discussion intended to motivate the introduction of the property entropy and continues with an axiomatic presentation of it and a differential equation that governs its time evolution for a continuous system of fixed mass. A kinetic theory explanation of entropy is presented in which it has a physical, statistical, meaning. The entropic equation of state is then developed during which a number of experimentally verified results are produced that give practical importance to this theoretical construct. Its consequences are further developed in a discussion of fundamental relations, and thermodynamic derivatives. The entropic equation of state is specialized as the mechanical and energetic equations of state were in Chaps. 2 and 4 to obtain all the simple known state equations for this property. The second law is motivated and introduced by an axiom that puts a lower bound on the time increase of the entropy of a system of fixed mass. The lower bound, entropy transfer is evaluated for a number of special, but interesting cases, then the second law is extended to open systems. The entropy bound is used to develop several important results including the Carnot efficiency. This is used to discuss thermodynamic temperature, the Clapeyron equation, refrigerators and heat pumps, and thermodynamic efficiency.
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Notes
- 1.
Newton’s law of Universal Gravitation, Eq. (1.26), is an example. The introduction of a “force of gravity,” that acted over vast regions of empty space, together with the law of motion, Eq. (1.27), not only described celestial motion quantitatively and produced Kepler’s 3 laws, but it also showed how the planetary orbits were related to projectile motion in the vicinity of the earth.
- 2.
This leads to the third law of thermodynamics first stated by Walther Nernst in 1906. It can be expressed as follows: limT→0 S = S O. Accordingly S is an absolute, not a relative, quantity. However, as with time, this knowledge is only useful near T = 0. Like calendar time, we use entropy like a relative quantity.
- 3.
Because of what will follow we need to specify here which variable is held constant in the partial differentiation.
- 4.
Note the difference between ∂ T s|p = c p∕T and ∂ T s|v = c v∕T, which is why we must distinguish between them.
- 5.
The absolute temperature defined here as the slope of the u vs s projection curve can be either positive or negative. However, negative absolute temperatures, which can occur in some special systems, lasers, for example, are not colder than T = 0, but hotter. This can be understood by using the variable τ = 1∕T called the coldness. Then absolute cold, T = 0, corresponds to τ = ∞ and increasing temperature corresponds to decreasing τ. Finally τ passing through 0 corresponds to T going from + ∞ to −∞ and on to even higher temperatures as τ gets larger negative and approaches −∞ while T gets smaller negative and approaches 0 from below.
- 6.
Using the full Eq. (2.33) here simply adds smaller terms.
- 7.
- 8.
- 9.
However, physicists and philosophers regard the second law as an important law of physics, because it is the only one that identifies the direction of time. For an isolated system such as the universe it produces S future > S present > S past.
- 10.
As noted in connection with Axiom 5.2 the impossibility of the reverse of Theorem 5.2 (cold bodies heat hot bodies) was adopted by Clausius as the second law.
- 11.
A more nuanced view of the second law is that it places constraints on non-equilibrium properties, like h t here and μ in Eq. (3.8), or more generally on all material constitutive relations.
- 12.
As noted in connection with Axiom 5.2 the impossibility of the reverse of Corollary 5.9 was adopted by both William Thomson (Lord Kelvin) and Max Plank as the second law.
- 13.
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Whitman, A.M. (2023). Entropy and the Second Law. In: Thermodynamics: Basic Principles and Engineering Applications. Mechanical Engineering Series. Springer, Cham. https://doi.org/10.1007/978-3-031-19538-9_5
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DOI: https://doi.org/10.1007/978-3-031-19538-9_5
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