Page:Scientific Papers of Josiah Willard Gibbs.djvu/43

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THERMODYNAMICS OF FLUIDS.

7

W^{C}=H^{C}=

∑^C

{\frac {1}{\gamma }}\delta A.

(8)

We have thus an expression for the value of the work and heat of a circuit involving an integration extending over an area instead of one extending over a line, as in equations (5) and (6).

Similar expressions may be found for the work and the heat of a path which is not a circuit. For this case may be reduced to the preceeding by the consideration that $W=0$ for a path on an isometric or on the line of no pressure (eq. 2), and $H=0$ for a path on an isentropic or on the line of absolute cold. Hence the work of any path $S$ is equal to that of the circuit formed of $S$ , the isometric of the final state, the line of no pressure and the isometric of the initial state, which circuit may be represented by the notation $[S,v'',p^{0},v'].$ And the heat of the same path is the same as that of the circuit $[S,\eta '',t^{0},\eta '].$ Therefore using $W^{S}$ and $H^{S}$ to denote the work and heat of any path $S$ , we have

W^{S}=

∑^{[S, v'', p⁰, v']}

{\frac {1}{\gamma }}\delta A,

(9)

H^{S}=

∑^{[S, η'', t⁰, η']}

{\frac {1}{\gamma }}\delta A,

(10)

where as before the limits of the integration are denoted by the expression occupying the place of an index to the sign ∑.^[1] These equations evidently include equation (8) as a particular case.

It is easy to form a material conception of these relations. If we imagine, for example, mass inherent in the plane of the diagram with a varying (superficial) density represented by ${\tfrac {1}{\gamma }},$ then ∑ ${\tfrac {1}{\gamma }}\delta A$ will

↑ A word should be said in regard to the sense in which the above propositions should be understood. If beyond the limits, within which the relations of $v,p,t,\epsilon$ and $\eta$ are known and which we may call the limits of the known field, we continue the isometrics, isopiestics, &c., in any way we please, only subject to the condition that the relations of $v,p,t,\epsilon$ and $\eta$ shall be consistent with the equation $d\epsilon =td\eta -pdv$ , then in calculating the values of quantities of $W$ and $H$ determined by the equations $dW=pdv$ and $dH=td\eta$ for paths or circuits in any part of the diagram thus extendend, we may use any of the propositions or processes given above, as these three equations have formed the only basis of the reasoning. We will thus obtain values of $W$ and $H$ , which will be identical with those which would be obtained by the immediate application of the equations $dW=pdv$ and $dH=td\eta$ to the path in question, and which in the case of any path which is entirely contained in the known field will be the true values of the work and heat for the change of state of the body which the path represents. We may thus use lines outside of the known field without attributing to them any physical signification whatever, without considering the points in the lines as representing any states of the body. If however, to fix our ideas, we choose to conceive of this part of the diagram as having the same physical interpretation as the known field, and to enunciate our propositions in language based upon such a conception, the unreality or even the impossibility of the states represented by the lines outside of the known field cannot lead to any incorrect results in regard to paths in the known field.

[1] A word should be said in regard to the sense in which the above propositions should be understood. If beyond the limits, within which the relations of $v,p,t,\epsilon$ and $\eta$ are known and which we may call the limits of the known field, we continue the isometrics, isopiestics, &c., in any way we please, only subject to the condition that the relations of $v,p,t,\epsilon$ and $\eta$ shall be consistent with the equation $d\epsilon =td\eta -pdv$ , then in calculating the values of quantities of $W$ and $H$ determined by the equations $dW=pdv$ and $dH=td\eta$ for paths or circuits in any part of the diagram thus extendend, we may use any of the propositions or processes given above, as these three equations have formed the only basis of the reasoning. We will thus obtain values of $W$ and $H$ , which will be identical with those which would be obtained by the immediate application of the equations $dW=pdv$ and $dH=td\eta$ to the path in question, and which in the case of any path which is entirely contained in the known field will be the true values of the work and heat for the change of state of the body which the path represents. We may thus use lines outside of the known field without attributing to them any physical signification whatever, without considering the points in the lines as representing any states of the body. If however, to fix our ideas, we choose to conceive of this part of the diagram as having the same physical interpretation as the known field, and to enunciate our propositions in language based upon such a conception, the unreality or even the impossibility of the states represented by the lines outside of the known field cannot lead to any incorrect results in regard to paths in the known field.

[1]