-Proceedings of the Asiatic Society of Bengal, February 1878.

ON MOLECULAR ATTRACTION IN ITS RELATIONS WITH THE TEMPERATURE OF BODIES. BY M. LÉVY.

The demonstration which we have given, in our last communication, of a general law upon the dilatation of bodies rests on the two fundamental propositions of thermodynamics, and upon this other proposition that the mutual actions of the molecules of a body are independent of their temperatures.

This last proposition we have assumed as an hypothesis; we wish now to prove that it flows from the first proposition of thermodynamics, so that our law itself will be found to be built solely upon the two propositions which serve as a foundation for that science.

To justify this assertion, let us conceive any body in motion under the influence of:-(1) external forces, F; (2) mutual actions, ƒ, on the nature of which we will make no hypothesis; (3) a certain quantity of heat received from without.

Let d'Q be the quantity, positive or negative, of heat received during an infinitely short interval of time dt (we will employ the characteristic d' for the infinitely small quantities which are not exact differentials or which are not known a priori to be so): a portion d'q of this heat is employed for increasing the temperatures of the various points of the body; the surplus, or d'Q-d'q, is transformed into work, and gives rise to a quantity of work E(d'Q-d'q), E being the mechanical equivalent of heat.

Suppose that the body describes any complete cycle, which means not only that all its points describe closed curves and resume their velocities at the end of the orbit, but also that they resume their temperatures. If to this cycle we apply the theorem of the vires vivæ, we get

0= = S & T F + S & T & ƒ + ES d'Q — ES d'q,

e

C. denoting an elemental work.

e.

But, in virtue of the first proposition of thermodynamics,

whence

'e

S&T, F+E S&Q=0,

S(±T,ƒ—Ed'q)=0,

(a)

which is equivalent to saying that the quantity under the symbol is the total differential of a certain function of all the variables, which resume their values at the end of the cycle-that is, not only of the coordinates x, y, zi of the various points of the material system considered, and which we suppose to be n in number (so that i=1, 2, 3,...n), but also of the temperatures T, of those

points. Thus

ΣTef-Ed'q=— EdU,

(b)

U being a function of the 4n variables xi, yi, zi, T. This function is no other than that which is called the internal heat.

The equation (a), or its equivalent (b), is the only one that can be directly deduced from the first proposition of the mechanical theory of heat, if no preconceived idea on the nature of heat be admitted; and we do not understand the reasonings by which some have attempted to deduce from it that Tef is a differential. It has certainly been proved that, for certain particular cyles, during which the temperature or the quantity of heat received remains

invariable, we have fTef=0; but from this it is not permissible to conclude that ΣTef is a differential.

I now say that, whatever idea may be formed of the nature of heat, the quantity of heat d'q employed to raise the temperatures of the various points of the body, without displacement of those points, is necessarily the exact differential of a function of the n variables T¡.

In fact, the quantity of heat necessary for raising by dT; the temperature of a molecule of mass m; is necessarily an expression of the form my,dTi, as y; can only depend on the temperature T¡ of the molecule and on the specific constants relating to the material of which it is composed.

Therefore the total quantity of heat remaining in the sensible state is

d'q=Σm;y;dT;=dΣmi SvedT1:

d'q being thus a differential, so also is Tef, in virtue of (a); and as this sum is an expression of the form

containing no term in dT, it cannot but be the differential of a function not containing the variables T,, consequently containing only the coordinates xi, Yi, Zi.

It follows from this :-first, that molecular attractions admit a function of the forces; secondly, that this function remains the same whatever may be the temperatures of the various points of the body; and, thirdly, that consequently the mutual action of two molecules of a body is quite independent of the temperature--which completely justifies the law laid down in our last communication, and places it among the necessary consequences of the two propositions of thermodynamics.

That law, that the pressure of a body heated under constant volume varies linearly with the temperature, proves that the empiric definition of temperature adopted by Dulong and afterwards by Regnault, viz. the pressure of a gaseous mass with constant volume, might be easily extended to the case in which, instead of a gaseous mass, any other body was in question.

Finally, without wishing here to draw from this law all the consequences which it admits of, we will nevertheless make the following remark :-

In a previous communication we have sought to discover what are the data strictly necessary to be derived from experiment to enable one to study a body from the thermodynamic point of view;

and the importance of this question will be especially apparent if we observe that in the best treatises superabundant data are taken from observation, even for constructing the simplest theory of all (that of gases). We then arrived at a result which can in brief be enunciated thus:-To know all the isothermal lines of a body, and one of its adiabatic lines, is sufficient.

The law which forms the object of the present investigation conducts to the following much more satisfactory and quite unexpected result:-In order to know all the isothermal lines and all the adiabatic lines of a body, and consequently to be able to study it completely, it is necessary and sufficient to know two of its isothermal lines and one only of its adiabatic lines.

In physical terms, one may say that it is sufficient to observe :1st, the dilatation of a body under two different pressures, or, more generally, for two series of states answering to two curves arbitrarily traced in the plane of the (pv)'s (which is equivalent to saying that the observations, of which we spoke at the outset of our previous communication, are replaced by two simple infinities of observations); 2ndly, one of the specific hea ts, or one particular pressure only, or, more generally, for a single series of states of the body corresponding to a curve arbitrarily traced in the plane.

If we admit, with MM. Clausius and Hirn, that the thermal capacity of every substance is a constant, this second series of observations reduces itself to a single observation.-Comptes Rendus de l'Académie des Sciences, Sept. 30, 1878, t. lxxxvii. pp. 488-491.

THE SONOROUS VOLTAMETER. BY THOMAS A. EDISON, PH.D. The sonorous or bubble voltameter consists of an electrolytic cell with two electrodes-one in free contact with a standard decomposable solution, and the other completely insulated by vulcanized rubber except two small apertures, one of which gives the solution free access to the insulated electrode, and the other allows the escape of bubbles of hydrogen as they are evolved by electrolysis. With a given current and a given resistance a bubble is obtained each second, which is seen at the moment of rising, and which at the same time gives a sound when it reaches the air. The resistance may be reduced so as to give one bubble in one, five, ten, or fifty seconds, or in as many hours. I have compared this instrument with the ordinary voltameter, and find it much more accurate. By the use of a very small insulated electrode and but one aperture, through which both the gas and water current must pass, great increase of resistance takes place at the moment when the bubble is forming; and just before it rises, a Sounder magnet included within the battery-circuit opens, closing again when the bubble escapes, thus allowing by means of a Morse register the time of each bubble to be recorded automatically. This apparatus, when properly made, will be found very reliable and useful in some kinds of work, such as measuring the electromotive force of batteries &c. By shunting the voltameter and using a recorder it becomes a measurer, not only of the current passing at the time, but also of that which has passed through a circuit from any source during a given interval.-Silliman's American Journal, November 1878.

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INDEX TO VOL. VI.