Page:Scientific Papers of Josiah Willard Gibbs.djvu/73

which unites these two points. As the pressure and temperature are evidently constant for this line, a single plane can be tangent to the derived surface throughout this line and at each end of the line tangent to the primitive surface.^[1] If we now imagine the temperature and pressure of the compound to vary, the two points of the primitive surface, the line in the derived surface uniting them, and the tangent

1. It is here shown that, if two different states of the substance are such that they can exist permanently in contact with each other, the points representing these states in the thermodynamic surface have a common tangent plane. We shall see hereafter that the converse of this is true,—that, if two points in the thermodynamic surface have a common tangent plane, the states represented are such as can permanently exist in contact; and we shall also see what determines the direction of the discontinuous change which occurs when two different states of the same pressure and temperature, for which the condition of a commong tangent plane is not satisfied, are brought into contact. It is easy to express this condition analytically. Resolving it into the conditions, that the tangent planes be parallel, and that they shall cut the axis of $\epsilon$ at the same point, we have the equations

   $p'=p'',$

   (α)

   $t'=t'',$

   (β)

   $\epsilon ^{\prime }-t'\eta ^{\prime }+p'v'=\epsilon ^{\prime \prime }-t''\eta ^{\prime \prime }+p''v'',$

   (γ)

   where the letters which refer to the different states are distinguished by accents. If there are three states which can exist in contact, we must have for these states,

   $p'=p''=p''',$

   $t'=t''=t''',$

   $\epsilon ^{\prime }-t'\eta ^{\prime }+p'v'=\epsilon ^{\prime \prime }-t''\eta ^{\prime \prime }+p''v''=\epsilon ^{\prime \prime \prime }-t'''\eta ^{\prime \prime \prime }+p'''v'''.$

   These results are interesting, as they show us how we might foresee wether two given states of a substance of the same pressure and temperature, can or cannot exist in contact. It is indeed true, that the values of $\epsilon$ and $\eta$ cannot like those of $v$, $p$, and $t$ be ascertained by mere measurements upon the substance while in the two states in question. It is necessary, in order to find the value of $\epsilon ^{\prime \prime }-\epsilon ^{\prime }$ or $\eta ^{\prime \prime }-\eta ^{\prime }$, to carry out measurements upon a process by which the substance is brought from one state to the other, but this need not be by a process in which the two given states shall be found in contact, and in some cases at least it may be done by processes in which the body remains always homogeneous in state. For we know by the experiments of Dr. Andrews, Phil. Trans., vol. 159, p. 575, that carbonic acid may be carried from any of the states which we usually call liquid to any of those which we usually call gas, without losing its homogeneity. Now if we had so carried it from a state of liquidity to a state of gas of the same pressure and temperature, making the proper measurements in the process, we should be able to foretell what would occur if these two states of the substance should be brought together,—whether evaporation would take place, or condensation, or wether they would remain unchanged in contact,—although we had never seen the phenomenon of the coexistence of these two states, or of any other two states of this substance.

   Equation (γ) may be put in a form in which its validity is at once manifest for two states which can pass either into the other at a constant pressure and temperature. If we put $p'$ and $t'$ for the equivalent $p''$ and $t''$, the equation may be written

   $\epsilon ^{\prime \prime }-\epsilon ^{\prime }=t'(\eta ^{\prime \prime }-\eta ^{\prime })+p'(v''-v').$

   Here the left hand member of the equation represents the difference of energy in the two states, and the two terms on the right represent severally the heat received and