Page:Scientific Papers of Josiah Willard Gibbs.djvu/76

velocity of the parte represented;—the center of gravity of points thus determined wil give the volume, entropy, and energy of the whole body.

Now let us suppose that the body having the initial volume, entropy, and energy, $v'$, $\eta ^{\prime }$, and $\epsilon ^{\prime }$, is placed (enclosed in an envelop as aforesaid) in a medium having the constant pressure $P$ and temperature $T$, and by the action of the medium and the interaction of its own parts comes to a final state of rest in which its volume, etc., are $v''$, $\eta ^{\prime \prime }$, $\epsilon ^{\prime \prime }$;—we wish to find a relation between these quantities. If we regard, as we may, the medium as a very large body, so that imparting heat to it or compressing it within moderate limits will have no appreciable effect upon its pressure and temperatre, and write $V$, $H$, and $E$, for its volume, entropy, and energy, equation (1) becomes

$dE=TdH-PdV,$

which we may integrate regarding $P$ and $T$ as constants, obtaining

$E''-E'=TH''-TH'-PV''+PV',$

(a)

where $E'$, $E''$, etc., refer to the initial and final states of the medium. Again, as the sum of the energies of the body and the surrounding medium may become less, but cannot become grater (this arises from the nature of the envelop supposed), we have

$\epsilon ^{\prime \prime }+E''\leqq \epsilon ^{\prime }+E'.$

(b)

Again as the sum of the entropies may increase but cannot diminish

$\eta ^{\prime \prime }+H''\geqq \eta ^{\prime }+H'.$

(c)

Lastly, it is evident that

$v''+V''=v'+V'.$

(d)

These four equations may be arranged with slight changes as follows:

$-E''TH''-PV''=-E'+TH'-PV''$

$\epsilon ^{\prime \prime }+E''\leqq \epsilon ^{\prime }+E'.$

$-T\eta ^{\prime \prime }-TH''\leqq -T\eta ^{\prime }-TH'$

$Pv''+PV''=Pv'+PV'.$

By addition we have

$\epsilon ^{\prime \prime }-T\eta ^{\prime \prime }+Pv''\leqq \epsilon ^{\prime }-T\eta ^{\prime }+Pv'.$

(e)

Now the two members of this equation evidently denote the vertical distances of the points $(v'',\eta ^{\prime \prime },\epsilon ^{\prime \prime })$ and $(v',\eta ^{\prime },\epsilon ^{\prime })$ above the plane passing through the origin and representing the pressure $P$ and temperature $T$. And the equation expresses that the ultimate distance is less or at most equal to the initial. It is evidently immaterial whether the distances be measured vertically or normally, or that the fixed plane representing $P$ and $T$ should pass through the origin; but distances must be considered negative when measured from a point below the plane.