Since the energy of the heat: KTE is proportional to the temperature T, we can also say that the area of the rectangle MO is proportional to the temperature of the substance at the state M. When a substance undergoes a transformation, its temperature being maintained constant, the second member of the equation os KTE remains constant. Hence, the general equation of the isothermal lines is: = P constant which is the equation of equilateral hyperbolas, whose asymptotes coincide with the axes of coördinates. The isothermal lines are the same for all substances, since the equation: 8 = KTE applies to any substance. a2 Remark: As 48 = π2; = KTE, we see that the ratio of the square of the amplitude to the square of the period of the vibratory motion of the heat, is proportional to the temperature; so that this ratio remains constant as long as the temperature of the body is maintained constant. 9. The amount of heat, dH, absorbed during an elementary transformation, is composed of two parts: 1st, the variation of the actual amount of heat contained in the substance, which 1 E amount equals 98; 2d, the heat absorbed by the total work done during the elementary transformation, which is dR = qds. 2. We see from this equation, that the amount of heat necessary to let the substance describe a certain path AB (fig. 2) is proportional to the area AaßB plus twice the area AabB (both of these areas being determined by the path β AB). When a substance undergoes a transformation without transmission of heat, the path described is called an "adiabatic" or "isentropic line." This path is determined by the condition: a Whence by integration: ps constant Such is the general equation to the adiabatic lines for any substance. These lines are of the third degree and belong to the hyperbolic species. 10. Clausius Theorem.-Equation (9) can be written: EdHqs dp ds +2 = Or again, by the aid of the relation: s KTE: ds ан [dq +21) T = K( Whence, for a finite transformation: dH T = K(log 4,8-log 9,8) When the path is a closed cycle, A = B; in this case: = 0 Graphical representation of the Specific Heat. 11. Let M be the initial state of the substance (fig. 3); and suppose an elementary path MM' to be described in a certain direction. Amongst the infinite number of directions around M, some are of special interest : P n m A 0 3. we shall obtain: a M = 1st. The direction of the isothermal, whose equation is: 484.8.; 4, and 8. denoting the coördinates of point M. This equation has been derived from the condition: T = constant or dT= 0. 2d. The direction of the adiabatic, whose equation is: s2=s, derived from the condition: H = 0. All the other directions of special interest are found in the same way by equating to zero one of the differential quantities entering the equations. Thus, 3d. The direction and equation of the path described by the substance when its volume is maintained constant, by putting dV 0 or V = constant; substituting to Vits value in terms of and 8 as given in equations (8), gives for the equation to Ф the curve of constant volume passing through M: = g(P, 8) = g(P, $) 4th. The equation of the path described when the pressure is maintained constant, by putting dP = 0 or P= constant, or again by the same substitution : f(P, 8) = f(P., 8.) 5th. The equation to the curve of constant symbolical volume; since s = ña', the amplitude a is constant when s is constant, hence this curve can be defined as the path described by the substance when the amplitude of the vibratory motion is maintained constant and its equation is found by putting ds= 0, whence: π 8 = 80 which is the equation to a straight line parallel to the axis of 4. 6th. The curve of constant symbolical pressure, which, since y=, can be defined as the path described when the period of the vibratory motion of the heat is maintained constant. The equation of this curve is derived from the condition: dy= 0, whence: which represents a straight line parallel to the axis of 8. 7th. The path along which the total work dR = 0, or çds = 0, or: which is the same as the equation to the curve of constant symbolical volume. 8th. The path along which the external work dr = 0 or PdV0 or: V constant We see here again that the symbolical volume is in the same relation with the total work as is the ordinary volume with the external work. We might also remark that since dR = 0 along the curve of constant symbolical volume, this curve may be defined as representing a transformation in which all the heat furnished is used in raising the temperature of the substance (or increasing its heat-energy), no part of it being transformed into work. 9th. The path along which the internal work, dI = 0 or: dR-dT = 0 or again: ds-PdVO. Expressing P and V in terms of and s, with the aid of equations (8): pds — ƒ (9,8)d[g(P, 8)] = 0 which will furnish by integration the equation of the required curve. Such are the principal curves passing through any point M. When M describes an elementary path MM', the direction of MM' determines the value of the quotient аф ds The heat absorbed in this elementary transformation is given by: and the corresponding variation of temperature, by : KEdT = sdq+pds Dividing member to member, and putting tion, we have: γ sdp+2pds (10) dH The quotient r = K sdp+pds dT may be called the specific heat of the substance at the state Mand for the direction MM' (since 7 varies with the value of do ds' i. e., the direction of MM'). has such a value that the direction MM' coincides with that of the tangent to the curve of constant volume, the corresponding value of 7 is evidently the "specific heat at constant volume" at the state M, which specific heat we shall denote by the letter c. when has such a value that MM' coin dp ds In the same way, cides with the tangent to the curve of constant pressure, the corresponding value of 7 is by definition the "specific heat at constant pressure" of the substance at the state M; this specific heat will be denoted by the letter C. The value of 7 corresponding to a direction parallel to the axis of may be called for the same reason: the "specific heat at constant symbolical volume;" and that corresponding to a direction parallel to the axis of s: the "specific heat at constant symbolical pressure." Now, equation (10) can be written: аф αφ 8 +ዋ ds Let us produce MM' until it intersects the axis of at point A, and measure on this axis three equal lengths: om, mP and PQ, each one of them equal to the ordinate of point M; then, we shall have: Thus we see that the specific heat corresponding to any direction MA is to the absolute specific heat of the substance as the distances from point A (determined by the direction MA itself) to the stationary points P and Q. The specific heats c and C can then be easily obtained graphically by tracing the tangents at point M to the curve of constant volume and to the curve of constant pressure; or, if desired, we can also make use of the specific heats to find these tangents. Let us study now the variation of the specific heat 7, when the direction of the element MM' changes, by revolving around point M. 1st. Suppose that MM or MA be at first parallel to the axis of point A is then removed to an infinite distance in the negative direction of the axis of and we have: =1 or 7 K. In other words, the specific heat at con = stant symbolical volume is equal to the absolute specific heat of the substance, and therefore does not depend on its physical state. 2d. When MA, by revolving ninety degrees around M, coin the specific heat at constant symbolical pressure is equal to twice the absolute specific heat of the substance and remains therefore also constant, when the state of the substance changes. 3d. When MA has reached the position Mn (n being the γ AQ nQ center of mP), = K = AP nP = = 3, or r= 3K. This value is of some interest, since the specific heat at constant volume of certain solid substances has been found to be equal to three times their absolute specific heat; hence, for those substances, Mn is the tangent at M to the curve of constant volume. 4th. When MA coincides with MP: ∞ordT=0 or again T = constant. This AM. JOUR. SCI.-THIRD SERIES, VOL. XLIX, No. 289.-JAN., 1895. |