On the Equilibrium of Heterogeneous Substances
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: TRANSACTIONS CONNECTICUT ACADEMY OF ARTS AND SCIENCES. VOX^XJJVIE III t7> NEW H AYEN PUBLISHED BY THE ACADEMY, 1874 to 1878. TulUr, Murehouse 4 Taylor, Printers, New Ha ———— CONTENTS. PAGE List of Additions to the Library,..,. v Art. I. —Report on the dredgings in the region of St. George's Banks, in 1872. By S. I. Smith and O. Harger. Plates 1 -8, 1 Art. IL—Descriptions op new and rare species of Hy- DROIDS from the NeW EnGLAND COAST. By S. F, Clark. Plates 9-10, 58 Art. III. On the Chondrodite from the Tilly-Foster IRON mine, Brewster, N. Y, By E. S. Dana, Plates 11-13, 67 Art. IV. On the Transcendental curves sin y sin iny=. a sin ic sin ??a; -|- ^''- By II. A. Newton and A. W. Philips. Plates 14-37, 97 Art. V. On the equilibrium of heterogeneous sub- stances. By J. W. Gibbs. First Part, 108 Art. VI. The Hydroids of the Pacific coast of the United States south of Vancouver Island, with a report upon those in the Museum of Yale Coixege. By S. F. Clark. Plates 38-41, 249 Art. VII. On the anatomy and habits of Nereis virens. By Y. M, Turnbull. Plates 42-44, 265 Art. VIII. Median and paired fins, a contribution to the history of vertebrate limbs. By J. K. Thacher, Plates 49-60, 281 Art. IX. Early stages op Hippa talpoida, with a note ON THE structure OF THE MANDIBLES AND MAXILLJE IN Hipp A and Remipes. By S. I. Smith. Plates 45-48, 311 Art. X. — On the equilibrium op heterogeneous sub- stances (concluded). By J. W. Gibbs, 343 OFFICERS OF THE ACADEMY. President, ELIAS LOOMIS. Vice-President, HUBERT A. NEWTON. /Secretary, OSCAR HARGER. Librarian, ADDISON VAN NAME. Treasurer, HENRY C. KINGSLEY. PtihlisJdng Committee, HUBERT A. NEWTON, ELIAS LOOMIS, GEORGE J. BRUSH, ADDISON E. VERRILL, CHESTER S. LYMAN, WILLIAM D. WHITNEY. Auditiiig Committee, HUBERT A. NEWTON, DANIEL C. EATON, HENRY T. BLAKE. V. On the Equilibrium of Heterogeneous Substances. Bt J. WlLLARD GiBBS. " Die Energie der Welt ist constant. Die Entropie der Welt strebt einera Maximum zu." CLAfsitrs.* The comprehension of the laws which govern any material system is greatly facilitated by considering the energy and entropy of the system in the various states of which it is capable. As the difference of the values of the energy for any two states represents the com- bined amount of work and heat received or yielded by the system when it is brought from one state to the other, and the difference of entropy is the limit of all the possible values of the integral / , [dQ denoting the element of the heat received from external sources, and t the temperature of the part of the system receiving it,) the varying values of the energy and entropy characterize in all that is essential the effects producible by the system in passing from one state to another. For by mechanical and theiTuodynamic con- trivances, supposed theoretically perfect, any supply of work and heat may be transformed into any other which does not differ from it either in the amount of work and heat taken together or in the value of the integral / . But it is not only in respect to the external relations of a system that its energy and entropy ai'e of predominant importance. As in the case of simply mechanical sys- tems, (such as are discussed in theoretical mechanics,) which are capable of only one kind of action upon external systems, viz., the perform- ance of mechanical work, the function which expresses the capability of the system for this kind of action also plays the leading part in the theory of equilibrium, the condition of equilibrium being that the variation of this function shall vanish, so in a thermodynamic system, (such as all material systems actually are,) which is capable of two different kinds of action upon external systems, the two functions which express the twofold capabilities of the system afford an almost equally simple criterion of equilibrium. *Pogg. Ann. Bd. cxxv (1865), S. 400; or Mechanische Warmetheorie, Abhand. ix., S. 44. : J. W. Gihhs—Equilibrium of Hetexogeneous Substances. 109 CRITERIA OF EQUILIBRIUM AND STABILITY. The criterion of equilibi-ium for a material system which is isohited from all external influences may be expressed in either of the follow- ing entirely equivalent forms I. For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations of the state of the system which do not alter its energy, the va.riatio7i of its entropy shall either vanish or be negative. If a denote the energy, and // the entropy of the system, and we use a subscript letter after, a variation to indicate a quantity of which the value is not to be varied, the condition of equilibrium may be written {^V)e ^0. (1) II. For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations in the state of the system which do not alter its entropy, the variation of its energy shall either vanish or be positive. This condition may be written m,= 0. (2) That these two theorems are equivalent will appear from the con- sideration that it is always possible to increase both the energy and the entropy of the system, or to decrease both together, viz., by imparting heat to any part of the system or by taking it away. For, if condition (1) is not satisfied, there must be some variation in the state of the system for which S}j > and 6e =0; therefore, by diminishing both the energy and the entropy of the system in its varied state, we shall obtain a state for which (considered as a variation from the original state) dTf=iO and d"£<0; therefore condition (2) is not satisfied. Conversely, if condition (2) is not satisfied, there must be a variation in the state of the system for which d£< and St/z^ 0; hence there must also be one for which (J// de=iO and > ; therefore condition (1) is not satisfied. The equations which express the condition of equilibrium, as also its statement in words, are to be interpreted in accordance with the general usage in respect to difterential equations, that is, infinitesimals 110 tf. W. Gibbs — Equilibrium of Heterogeneous Substances. of higher orders than the fii-st relatively to tliose which express the amount of change of the system are to be neglected. But to distin- guish the different kinds of equilibrium in respect to stability, we must have regard to the absolute values of the variations. We will use A as the sign of variation in those equations which are to be con- strued strictly, i. e., in which infinitesimals of the higher orders are not to be neglected. With this understanding, we may express the necessary and sufficient conditions of the different kinds of equi- librium as follows;—for stable equilil)rium (•^;/),<0, i. e., (A6),^>0: (3) for neutral equilibrium there must be some variations in the state of the system for which (A//)^ = 0, i. e., {Af),^ = 0, (4) while in general {^V)e ^0, i.e., (Af),^^0; (5) and for unstable equilibrium there must be some variations for which (A;/),>0, (6) i. e., there must be some for which (^f),<0, (V) while in general (d7),^0,i.e., (c^f),^0. (8) In these criteria of equilibrium and stability, account is taken only oi })ossible variations. It is necessary to explain in what sense tliis is to be understood. In the first place, all variations in the state of the system which involve the transportation of any matter through any finite distance are of course to be excluded from consideration, although they may be capable of expression by infinitesimal varia- tions of quantities which perfectly determine the state of the system. For example, if the system contains two masses of the same sub- stance, not in contact, nor connected by other masses consisting of or containing the same substance or its components, an infinitesimal increase of the one mass with an equal decrease of the other is not to be considered as a possible variation in the state of the system. In addition to such cases of essential impossibility, if heat can pass by conduction or radiation from every part of the system to every other, only those variations are to be rejected as impossible, which involve changes which are prevented by passive forces or analogous resist- ances to change. But, if the system consist of parts between which there is supposed to be no thermal communication, it will be neces- sary to regard as impossible any diminution of the entropy of any of J. W. Gihhs—Equilihrium of Heterogeneous Substances. Ill these parts, as sueli a change can not take place without the passage of heat. This limitation may most conveniently l>e ai»plied to the second of the above forms of the condition of equilibrium, which will then become ri\ ?/', etc., denoting the entropies of the various parts between which there is no communication of heat. When the condition of equi- librium is thus expressed, the limitation in respect to the conduction of heat will need no farther consideration. In order to apply to any system the criteria of equilibrium which have been given, a knowledge is requisite of its passive forces or resistances to change, in so far, at least, as they are capable of pre- venting change. (Those passive forces which only retard change, like viscosity, need not be considered.) Such properties of a system ai-e in general easily recognized upon the most superficial knowledge of its nature.