Entropy

“ As, however, it is only differences of energy and of entropy that can be measured, or indeed, that have a physical meaning, the values of these quantities are so far arbitrary, that we may choose independently for each simple substance the state in which its energy and its entropy are both zero. The values of the energy and the entropy of any compound body in any particular state will then be fixed. ”

“ The amount of its change in a process would furnish a general measure for the irreversibility of the process."
"Now R. CLAUSIUS really found such a magnitude and called it entropy. Every bodily system possesses in every state a particular entropy, and this entropy designates the preference of nature for the state in question; in all the processes which occur in the system, entropy can only grow, never diminish. ”

“ The entropy of a state is measured by a reversible process which conducts the body from its present state to the zero state, but this ideal process has nothing to do with the changes of state that the body has experienced or is going to experience. ”

“ NEGATIVE CHANGE OF ENTROPY; SOME OF ITS PHYSICAL FEATURES OR NECESSARY ACCOMPANIMENTS
A negative transformation in any part of a system is the diminution of entropy which it experiences, and this we know means a diminution in the number of complexions of the part considered. But there are some features of such negative transformations which, while they do not in themselves constitute any additional principle, deserve special mention. ”

“ Growth of entropy is a passage from more to less available energy. By available is here meant energy which we can direct into any required channel. With the growth in the number of complexions we can readily see there is greater inability, on the part of the molecules, for that concerted and co-operative action which is necessary for the putting forth of the energy of a system. ”

“ It is an inference naturally suggested by the general increase of entropy which accompanies the changes occurring in any isolated material system that when the entropy of the system has reached a maximum, the system will be in a state of equilibrium. ”

“ Every reversible change in an isolated system thus maintains the entropy of that system unaltered; no possible spontaneous change can involve decrease of the entropy; while any defect of reversibility, arising from diffusion of matter or motion in the system, necessarily leads to increase of entropy. ”

“ Growth of entropy is a passage from a concentrated to a distributed condition of energy. Energy originally concentrated variously in the system is finally scattered uniformly in said system. In this aggregate [Pg 42] aspect it is a passage from variety to uniformity. ”

“ The last expression is called by CLAUSIUS the entropy of the body in state 2, referred to state 1 as the zero state. The entropy of a body in a particular state is, therefore, like energy, completely determined down to an additive constant depending on the choice of the zero state. ”

“ Growth of entropy is a passage from a somewhat regulated to a less regulated state. It represents, in a certain sense, Nature's escape from thralldom.
Growth of entropy is a passage from a somewhat ordered to a less ordered molecular arrangement.
Growth of entropy is an increase in the disorder of a system of mass points. ”

“ As the final state, like the initial state, is in thermal equilibrium, and possesses the same temperature, we can ascertain the total change in the number of complexions as we did when discussing isothermal and reversible changes and because of the accompanying increase in the volume of the air, find that here as there the number of complexions has increased and that therefore the entropy of the air has increased in this case. ”

“ The general result is:
"IT IS IN NO WAY POSSIBLE TO DIMINISH THE ENTROPY OF A SYSTEM OF BODIES WITHOUT HAVING CHANGES REMAIN IN OTHER BODIES."
If we admit to the system all the bodies participating in the process, this theorem becomes:
"Every physical and chemical process occurring in Nature takes place in such a way that there is an increase in the sum of the entropies of the bodies in any way participating in the process." ”

“ Now there are mechanical systems which possess the entropy property and it has been truly said that the Second Law and irreversibility do not depend on any special peculiarity of heat motion, but only on the statistical property of a system possessing an extraordinary number of degrees of freedom. ”

“ Now, after the principle of elementary disorder has been introduced and accepted by us as valid throughout nature, the fundamental question arises as to the calculation of the probability of a given state, and the actual derivation of the entropy therefrom. From the entropy all the laws of thermodynamic states of equilibrium, for material substances, and also for energy radiation, may be uniquely derived. ”

“ In the first place, if the system is in a state in which the entropy is greater than in any other state of the ame energy, it is evidently in equilibrium, as any change of state must involve either a decrease of entropy or an increase of energy, which are alike impossible for an isolated system. ”

“ In accordance with the method which has been developed, the intrinsic energy (i.e. the part of the energy which is independent of gravity) , the entropy, and the quantities of the several components must each be divided into two parts, one of which we regard as belonging to the surfaces which divide approximately homogeneous masses, and the other as belonging to these masses. The elements of intrinsic energy, entropy, etc., relating to an element of surface will be denoted by , etc., and those relating to an element of volume , by , etc. ”

“ The laws thus far deduced for ideal gases can in the same way be transferred to any other bodies, the main difference in general being that the expression for the entropy of any body cannot be written in finite magnitudes because the equation of condition is not generally known. ”

“ In the theory of heat the first mode is represented by the application of the atomic theory to the study of heat phenomena, and the second mode is represented by what is called thermodynamics." In solving the special problem before us, as to the physical meaning of entropy and of the Second Law, our main dependence must be on the first mode of attack. ”

“ Heterogeneous Equilibrium.—In a system, as distinguished from a homogeneous substance, consisting of two or more states or phases, a similar condition of equilibrium applies. In an spontaneous irreversible change, if the system is heat-isolated, there must be an increase of entropy. The total entropy of the system is found by multiplying the entropy per unit mass of the substance in each state by the mass existing in that state, and adding the products so obtained. ”

“ In chemical reactions, for example, the entropy increases without the substances borrowing any heat. When a perfect gas dilates in a vacuum its entropy increases, and yet the temperature does not change, and the gas has neither been able to give nor receive heat. We thus come to conceive that a physical phenomenon cannot be considered known to us if the variation of entropy is not given, as are the variations of temperature and of pressure or the exchanges of heat. The change of entropy is, properly speaking, the most characteristic fact of a thermal change. ”

“ A similar proposition can be established for a system of any number of gases by successively treating the gases in pairs as above. The theorem then reads: "If the gas system as a whole possesses the same entropy in two different states then the system can be brought from one state to the other in a reversible manner without changes remaining in other bodies."
(5) We know that the expansion of an ideal gas without doing external work and receiving any heat supply is an irreversible process. The consequence is that the entropy of this gas increases. ”

“ As the latter quantity is the greater, the entropy of the whole will have increased. By itself, we know, heat always passes, either by radiation or by conduction, from bodies of higher temperature to bodies of lower temperature. That evidently implies an increase in entropy, and it is in agreement with the law of Clausius that entropy tends to increase. ”

“ The same considerations will evidently apply to any case in which a system is in such a state that for any possible infinitesimal variation of the state for which , even if the entropy is not the greatest which the system is capable with the same energy. ”

“ In the settled state there is an equal distribution of density throughout all the elementary spaces, there are no different streams (whirls or eddies) present, and an equal partition of energy exists for all the elementary spaces. For it thermal equilibrium exists, the entropy is a maximum, and temperature of the state has now a definite meaning, because temperature is the mean energy of the molecules for this state of equilibrium. ”

“ There will then, by our precise definition, be a corresponding increase in the total entropy of the two systems. It is foreign to our present purpose to prove in an independent, purely mechanical way, that such excess does finally exist and will here content ourselves with the well-known and simple thermodynamic expression for this excess, where is the heat energy thus directly transferred from the hot to the cold body, the absolute temperature of the hot body and that of the cold body. ”

“ In particular, we will now determine the entropy of a gas in a state of equilibrium, and inquire first as to that form of the law of distribution which corresponds to thermodynamic equilibrium. In accordance with the second law of thermodynamics, a state of equilibrium is characterized by the condition that with given values of the total volume and the total energy , the entropy assumes its maximum value. ”

“ Entropy.—The study of steam-engine problems is greatly assisted by introducing the idea of entropy and making use of diagrams in which the two co-ordinates are entropy and temperature. Entropy is a condition of the working substance defined by the statement that when any quantity of heat δQ is received by, or generated in, or rejected by the substance, when its absolute temperature is τ, the substance gains or loses entropy by the amount δQ/τ. Thus ΣδQ/τ measures the whole change of entropy in a process which involves the taking in or rejection of heat at more than one temperature. ”

“ All other bodies used in the process must by its close be restored to their original state, except those used to supply the work, which must be used only as the source of the work. For, in a reversible process, when the entropies of other bodies are not altered, the entropy of the substance and mass taken together will not be altered. ”

“ We have numbered the following statements of the second law, for convenience of reference:
(1) J. W. GIBBS. "The impossibility of an uncompensated decrease in entropy seems to be reduced to an improbability." This of course considers all the participating bodies of the system.
(2) All changes in nature involve a net growth in entropy; when such a change is measured in reversible ways, the growth is indicated by the summation: , when the sign refers to processes which on the whole are completely . ”

“ If the entropy of the mass which occupies any one of the spaces considered is not in the nature of things determined without reference to the surrounding masses, we may suppose a similar method to be applied to the estimation of entropy. ”

“ A somewhat different procedure, in terms of entropy as fundamental, has been adopted and developed by Planck. In an isolated system the trend of change must be in the direction which increases the entropy φ, by Clausius’ form of the principle. ”

“ Even if the surface separating the masses were determined, the energy to be attributed to the masses separated would be partly arbitrary, since a part of the total energy depends upon the mutual action of the two masses. We ought perhaps to consider the case the same in regard to the entropy, although the entropy of a system never depends upon the mutual relations of parts at sensible distances from one another. ”