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properties depend essentially upon the choice of the two auxiliary variables a and i ; therefore, they ought to be chosen in such a way as to give the best possible graphical representation of the cycles of transformation, i. e., in such a way as to enable us to determine graphically the greatest possible number of the physical elements depending upon the transformation, by means of geometrical magnitudes depending only upon the form and position of the path described by the body in the adopted system of coördinates. Before defining this system, let us examine what conditions must be fulfilled by the functions 2, M, u, in order that the auxiliary variables a and i be respectively the amplitude and the period of the vibratory motion.
4. Denoting by m the mass of one of the particles composing the substance and by u the mean velocity of the vibratory motion, the expression Imu? is the actual kinetic energy of the heat (the sum ? being extended to all the particles). Dividing this sum by the mechanical equivalent of heat E, the result is equal to the amount of heat actually contained in the body.
This amount of heat is proportional to the absolute temperature, hence: Emu'= KTE
(2) K being a constant.
Denoting by f the mean value of the force producing the vibratory motion, the formulæ : fumu'
(3) can be established without difficulty, since in all vibratory motions of small amplitude, the force producing the vibration is proportional to the displacement of the particles.
Combining equations (2) and (3) and noticing that Em=1 and that the mean velocity u is the same for all the particles of the substance, we shall obtain :
no a T=
КЕ? which is the expression of Tin terms of a and i, and is therefore identical to the third of equations (1); in other words, when the two auxiliary variables a and i denote the amplitude and the period of the vibratory motion, the function v is no longer arbitrary, and the equation to the thermodynamic surface is :
(5) KE ¿?
the functions à and being still submitted to the condition that the result of eliminating a and i between equations (5) be: F(P, V, T)= 0.
When the functions , and ye have been determined for a particular substance, equations (5) do not only represent the thermodynamic surface, but also the value of the two elements (a and 2) of the vibratory motion, corresponding to any state of the substance defined by experimental data (P,V, T').
The last of equations (5) is the same for all substances, except that the value of the constant K changes from one substance to another. The determination of the functions à and will be investigated after we shall have studied the properties of the graphical representation, which properties can be found by assuming that these functions are known.
5. When a substance undergoes an elementary and reversible transformation, * the amount of heat dll, absorbed by the unit of mass, is composed of two parts: the variation of the actual energy of the heat contained in the substance, and the amount of heat absorbed by the total work (external and internal).
The first part is the elementary variation of the expression 1 E
Elmu', as found above; the second is the heat absorbed by the work done by the force f for a variation da of the amplitude. Hence :
EJH = dimu' + Efda But, by differentiating equation (2):
dzimu= KEAT This relation shows that the constant K is the quotient of the variation of the actual amount of heat contained in the substance, by the corresponding variation of temperature, so that K is by definition the absolute specific heat of the substance. We have also, from preceding formulæ :
da Efda = Emu? = 2KTE Whence finally
dH = KIT +- 2KT
Such is the expression of all in terms of land a; equation (4) gives by differentiation :
da di IT
* The formulæ contained in this paragraph have been already established in "La Thermodynamique et ses principales applications,” by J. Moutier, Paris, 1885; we recall them here, as we will have to use them in some of the demonstrations.
By the aid of tuis equation and of the preceding one, the value of dH can also be obtained in terms of T and i or of a and i:
dH dT da
T i dH
da dil = 2K 2
T 6. The two variables, which we intend to take as coördinates in this graphical study, depend directly upon the amplitude a and the period i of the vibratory motion of heat; denoting these variables by y and 8, we shall define them by the equations : =
(7) S = na By the aid of equation (7), any of the formulæ given above and involving a and i, can be transformed into corresponding formulæ involving o and 8.
For instance, by solving equations (7) with respect to a and i, and substituting the result in equations (5), we shall obtain the equation to the thermodynamic surface in terms of u and 8, as follows:
ps All the other equations can be transformed in the same manner, so that it is understood that the two independent variables are now y and s, and that these two quantities shall be taken as the coördinates of the point representing the physical state of the substance.
As these variables have been defined in an arbitrary manner, let us first investigate their physical nature. To reach this end, we must consider the vibratory motion of the particles as the projection of a uniform circular motion on one of its diameters; the radius of the circle is then equal to the amplitude of the vibration, and the velocity of the uniformn motion is equal to the maximum velocity of the vibratory motion. It can readily be seen, that the centripetal force of the circular motion is equal to the mean value f of the force supposed to produce the vibratory motion.
The total work (external and internal) absorbed by the substance during an elementary transformation is Efda as seen above. We can write identically :
fda = (*) (2nada)
= 4m. Since 2na is equal to the length of the circum2πα ference, &m represents geometrically the value of the centripetal force referred to the unit of length, i. e., a pressure, of so many ponnds per foot, supposed to be acting on the circumference of the circle.
Let 2nada = ds or na’= s. Then s is the area of the circle. According to these definitions :
Efda = Eq„ds
$ If now in the equation Om
f be replaced by its value as given in eqnation (3), the result is :
The last two equations are precisely the ones by which and & have been first defined. Since = Øm when m = 1, and since
is the pressure supposed to be acting on the circumference of the circle corresponding to the particle of mass m, we can define the physical nature of and s as follows: If the unit of mass of a substance be represented geometrically or symbolically by a circle, the physical state of said substance can be completely defined by the area 8 of the circle and by a pressure y, supposed to be acting on the circumference of the circle. The two data, thus defining the state of the substance, are precisely the coördinates o and s, which determine the position of the point representing this state. For this reason, the abscissa s shall be called the “symbolical volume" and the ordinate y the “symbolical pressure” of the substance.
As the considerations developed in this paragraph are somewhat abstract, it must not be forgotten that the graphical method, which is the object of this study is quite independent of these theoretical considerations, since the two variables y
and 8 can always be regarded as two variables defined by the equations : 4 and s = ma’, whatever be their physical
ia nature; moreover, we shall find other reasons for regarding y as a pressure and s as a volume.
a = 1
Properties of the graphical method. 7. Let M be the point representing any physical state of a substance, and y and s its coördinates, then according to the previous definitions :
P So that a and i, hence the state of the vibratory motion, are readily obtained from the actual value of the coördinates of point M.
Denoting by R the total work absorbed during a transformation, we have found that:
dR = Efia = ods Whence by integration :
i. e., if AB be the curve (fig. 1) representing the path of the substance referred to the coördinates y and 8, the total work
(external and internal) absorbed during 1.
the transformation is equal to the area
Aab B limited by the path, the axis of s P
and the two extreme ordinates.
Comparing this result with the property of Clapeyron's graphical method, we see that the symbolical pressure and the symbolical volume are in the same relation with the total work, as the ordinary pressure and volume are with the external
work. 8. The last of equations (8):
holding true for all substances, shows that the area of the rectangle Mm On formed by the coördinates y and s, is equal to the actual amount of energy contained in the substance at the physical state M. Thus, if AB represents the path of the substance the area of the rectangles Aa0a and BÌ03 is equal to the energy of the heat contained in the substance at its initial and final states.