A METHOD OF GEOMETRIC REPRESENTATION OF THE THERMODYNAMIC PROPERTIES OF SUBSTANCES BY MEANS OF SURFACES¹

J. W. Gibbs

The fundamental thermodynamic properties of a liquid or gas are determined by the relations existing among the volume, pressure, temperature, energy, and entropy of a given mass of liquid or gas in a state of thermodynamic equilibrium. This statement is also valid for solids with respect to properties manifested in processes in which the pressure about each point of the body is the same in all directions. But all the relations existing among these five quantities for any substance (three independent equations) may be derived from one single relation between the volume, energy, and entropy for the given substance. This may be done by means of the general equation

\[ d\varepsilon = t\,d\eta - p\,dv \tag{1} \]

or

\[ p = -\left(\frac{d\varepsilon}{dv}\right)_{\eta}, \tag{2} \]

\[ t = \left(\frac{d\varepsilon}{d\eta}\right)_{v}, \tag{3} \]

where \(v, p, t, \varepsilon\), and \(\eta\) denote, respectively, the volume, pressure, absolute temperature, energy, and entropy of the body under consideration. The subscript attached to the derivative indicates the quantity which is assumed constant during differentiation.

REPRESENTATION OF VOLUME, ENTROPY, ENERGY, PRESSURE, AND TEMPERATURE

This relation between volume, entropy, and energy may be graphically represented in the form of a surface, and most simply if the rectangular coordinates of the various points of the surface are taken to be equal to the volume, entropy, and energy of the body in its various states.

¹ Scient. Pap., 1906, Longmans, Green a. Co., V. of J. W. Gibbs. Translated from the English by E. P. Shubina. Edited by K. V. Astakhov.

It is of known interest to investigate the properties of such a surface, which we shall call the thermodynamic surface of the body for which it has been constructed1.

For definiteness let us choose the axes \(v\), \(\eta\), and \(\varepsilon\) along the directions usually assigned to the axes \(X\), \(Y\), and \(Z\) (i.e. so that \(v\) increases to the right, \(\eta\) forward, and \(\varepsilon\) upward). Then the pressure and temperature of the state represented by any point of our surface will be equal to the tangents of the angles of inclination of the surface to the horizontal plane at that point, measured in planes respectively perpendicular to the axes \(\eta\) and \(v\) (Figs. 2 and 3). It should be noted, however, that in the first case the angle of inclination is measured upward from the direction of decreasing \(v\), and in the second case upward from the direction of increasing \(\eta\). Thus the tangent plane at each point indicates the temperature and pressure of the state represented by that point. It is expedient to call the plane representing definite pressures and temperatures the plane whose tangents of the angles of inclination to the horizontal plane, measured in the manner indicated above, are equal to the given values of the pressure and temperature.

Before continuing our investigation, it will be useful to establish what, in a surface constructed in this way, is essential and what is arbitrary. The position of the plane \(v=0\) relative to the surface is, obviously, firmly determined; but the position of the planes \(\eta=0\) and \(\varepsilon=0\) is arbitrary, provided only that the directions of the axes \(\eta\) and \(\varepsilon\) remain unchanged. This follows from the very definition of the quantities entropy and energy, each of which includes an arbitrary constant. Since one may put \(\eta=0\) and \(\varepsilon=0\) for any state of the body whatever, we may place the origin of coordinates at any point of the plane \(v=0\). Further, from the form of equation (1) it is obvious that, in whatever way we change the units of measurement of volume, entropy, and energy, it will always be possible so to change the units of temperature and pressure that the equation remains valid in the same form, without the introduction of constants. It is easy to see how the change of the units of volume, entropy, and energy will be reflected on our surface. The projections of the distances between points of the surface onto the direction of any coordinate axis change inversely proportionally to the change of the corresponding unit. These considerations allow us, to a certain extent, to foresee the character of the general properties of the surface that we are about to investigate, namely, these

the properties must be such that none of the above-mentioned changes is reflected in them. For example, we may find properties of the surface that relate to the plane \(v=0\) (for example, the entire surface must necessarily be situated on the positive side of this plane), but we cannot calculate and find properties that relate to the planes \(\eta=0\) or \(\varepsilon=0\), as distinct from other planes parallel to them. It may also be added that, since the volume, entropy, and energy of a body are equal to the sum of the volumes, entropies, and energies of its parts, surfaces formed in the indicated manner for bodies differing from one another in quantity, but not in the nature of the substance, will be similar to one another, since their linear dimensions will be proportional to the quantity of substance.

CHARACTER OF THAT PART OF THE SURFACE WHICH DEPICTS STATES THAT ARE NOT HOMOGENEOUS

This method of representing the volume, entropy, energy, pressure, and temperature of a body is applicable both to the case in which different parts of the body are in different states (always assuming that the body as a whole is in a state of thermodynamic equilibrium), and to the case in which the state of the body is identical in all its parts. For the body as a whole has a definite volume, entropy, and energy, just as it has a definite pressure and temperature, and the applicability of the general equation (1) does not depend on whether the state of the several parts of the body is identical or different.\(^1\) It is therefore evident that the thermodynamic

\(^1\) In this equation, however, there is contained the assumption that changes in the state of the body to which the quantities \(dv\), \(d\eta\), and \(d\varepsilon\) refer are such that they can be carried out reversibly by means of expansion and compression, or by the addition and removal of heat. Therefore, when a body consists of parts that are in different states, these states must be such that any one of them could pass into another without a perceptible change of pressure or temperature. Otherwise it would be necessary to add to the differential equation (1) the supposition that the ratio between the parts of the body that are in different states remains unchanged throughout. But a restriction of this kind would make this equation unsuitable for the purpose we have set ourselves in its application to a system of different states. If, however, we exclude those cases in which one has to consider states as chemically different, which lie beyond the limits of the present work, then our assumption (that either of two coexisting states can pass into the other without a perceptible change of pressure or temperature) is justified by experimental data at least as approximately correct for the case in which one of the states is liquid or gaseous. But when both states are solid, the necessary mobility of the parts of the body is absent. It should therefore be borne in mind that the consideration given below of systems of several states cannot be extended without restrictions to those exceptional cases in which we are dealing with two different solid states of one and the same substance at the same pressure and temperature. It may also be added that thermodynamic equilibrium between two such

the surface can, at least for many substances, be divided into two parts, one of which represents homogeneous states, and the other—states that are not such. We shall see below that if the first part of the surface is given, then the latter can easily be constructed, as, indeed, was to be expected. We may therefore call the first part of the surface the primary surface, and the second part the derived surface.

To establish the character of the derived surface and its relation to the primary surface, it is enough to construct the former, if the latter is given, using only the proposition that the volume, entropy, and energy of the body as a whole are respectively equal to the sum of the volumes, entropies, and energies of its parts, while the pressure and temperature of the body are equal to the pressure and temperature of each of its parts separately. Let us begin with the case when in one of its parts the body is solid, in another—liquid, and in the third—vaporous. The position of the point determined by the volume, entropy, and energy of such a system will coincide with the position of the center of gravity of masses, proportional to the masses of the solid part, the liquid, and the vapor, placed at three points of the primary surface, representing respectively the states of an entirely solid, entirely liquid, and entirely vaporous body, each at the temperature and pressure of the whole system. Consequently, the part of the surface representing the system consisting of a solid body, a liquid, and a vapor is a plane triangle with vertices at the mentioned points. The fact that the surface is, in this case, a plane means that the pressure and temperature of the represented system are constant; their numerical values are determined by the inclination of this plane. Moreover, since these values are the same for the whole system and for the three different homogeneous states corresponding to its different parts, the plane of the triangle is tangent at all its vertices to the primary surface, namely, at one vertex—to that part of the primary surface which represents the solid body, at another—to the part representing the liquid, and at the third—to the part representing the vapor.

solid states of one and the same substance also differs greatly from the equilibrium that takes place when one of the states is liquid or gaseous, just as in statics an equilibrium maintained by friction differs from equilibrium in the machine without friction, where the active forces are so balanced that the slightest change of force causes motion in one direction or another.

The necessity of another restriction is caused by the circumstance that in the subsequent discussion no account is taken of the influence of the shape of the bodies bounding and separating its surface, so that the results obtained will, generally speaking, be strictly applicable only to those cases in which the influence of these factors may be neglected. Therefore, when we call two states of a substance coexisting, it must be understood that the surface separating them is plane. Consideration of the question in a more general form would require the introduction of considerations pertaining to the theories of capillarity and crystallization.

If the body is a system of two different homogeneous states, then the point representing the system coincides with the center of gravity of masses proportional to the masses of the parts of the body in the two different states, placed at those points of the primitive surface which represent these two states (i.e., at the points representing the volume, entropy, and energy of the body, on the assumption that the entire mass of the body is successively in the two states of its parts). The point sought will therefore lie on the straight line joining these two points of the primitive surface. Since the pressure and temperature along this line are, obviously, constant, one and the same plane can at the same time be tangent to the derivative surface along this entire line and to the primitive surface at the ends of the line1. If we now—

\[ p' = p'', \tag{α} \]

\[ t' = t'', \tag{β} \]

\[ \varepsilon' - t'\eta' + p'v' = \varepsilon'' - t''\eta'' + p''v'', \tag{γ} \]

where the primes indicate to which state the given quantity belongs. If we have three states that can coexist, then for these states the following equations must be satisfied:

\[ p' = p'' = p''', \]

\[ t' = t'' = t''', \]

\[ \varepsilon' - t'\eta' + p'v' = \varepsilon'' - t''\eta'' + p''v'' = \varepsilon''' - t'''\eta''' - p'''v'''. \]

The results obtained are interesting in that they show how we could predict whether or not the coexistence of two given states of a substance with equal pressure and temperature is possible. It is true, of course, that the values of \(\varepsilon\) and \(\eta\) cannot be determined as readily as \(v\), \(p\), and \(t\), through successive changes with the given substance when it is in the two states under consideration. To determine the value of the quantity \(\varepsilon''-\varepsilon'\) or \(\eta''-\eta'\), it is necessary to carry out measurements in the course of a process by means of which the substance is transferred from one state to the other; but this process need not necessarily be such that, in it, the two given states are in contact, and at least in some cases the measurements can be made in processes during which the body remains homogeneous in state throughout. Thus, we know from the experiments of Andrews (Phil. Trans,

Now suppose that the temperature and pressure of the system change; then the two mentioned points of the primary surface, the line joining them on the derivative surface, and the tangent plane will change their position, while preserving the above relations. We may imagine the motion of the tangent plane as its rolling over the primary surface, during which it remains at all times tangent to the latter at two points; and since at the same time it touches the derivative surface along the lines joining these points, it is evident that this surface is developable and forms part of the envelope surface for the successive positions of the rolling plane. As we shall see below, the form of the primary surface is such that the plane of double tangency does not intersect it, so that the rolling is physically possible.

From these relations one can, by simple geometrical considerations, derive one of the fundamental propositions that hold

† Vol. 159, p. 575), that carbon dioxide can be transferred from any of the states that we usually call liquid into any of the states usually called gaseous, without disturbing its homogeneity. If, however, we carry out such a transition from the liquid state to the gaseous state at the same pressure and temperature, making the appropriate measurements during the transition, then we shall be able to predict what will occur if these two states of the given substance are brought into contact—whether evaporation or condensation will take place, or whether the states will remain unchanged, even though we may never have seen the phenomenon of the coexistence of these two states, or of any two other states of this substance.

The equation \((\gamma)\) may be brought to a form in which its validity is immediately obvious for the case of two states that can pass into one another at constant pressure and temperature. If, instead of \(p'\) and \(t'\), we substitute the quantities \(p''\) and \(t''\), which are equal to them, then the equation can be written in the form

\[ \varepsilon''-\varepsilon'=t'(\eta''-\eta')-p'(v''-v'). \]

The left-hand side of this equation is the difference of the energies of the two states, and the two terms on the right-hand side represent, respectively, the heat absorbed and the work performed in the transition of the body from one state to the other. This equation may also be obtained directly from the general equation (1) by integration.

It is well known that when two liquid states are in contact along a curved surface, then instead of \((\alpha)\) we have

\[ p''-p'=T\left(\frac{1}{r}+\frac{1}{r'}\right), \]

where \(r\) and \(r'\) are the principal radii of curvature of the surface of contact at any point (the curvature is taken as positive when the concavity of the surface is turned toward the state to which the quantity \(p''\) belongs), and \(T\) is a quantity called the surface tension. Equation \((\beta)\) remains, however, valid for such cases, and it is not difficult to show that the same can be said of equation \((\gamma)\). In other words, the tangent planes at the points of the thermodynamic surface representing these two states intersect the plane \(v=0\) along one and the same line.

for such systems. Suppose that the tangent plane touches the primitive surface at two points \(L\) and \(V\) (Fig. 1), and, for definiteness, let us suppose that they represent liquid and vapor; through these points draw planes perpendicular respectively to the axes \(v\) and \(\eta\), and intersecting along the line \(AB\), which will be parallel to the axis \(\varepsilon\). Suppose that the tangent plane intersects this line at the point \(A\), and draw the straight lines \(LB\) and \(VC\) at right angles to \(AB\), parallel to the axes \(\eta\) and \(v\). It is now evident that the pressure and temperature represented by the tangent plane will be equal respectively to the ratios \(\dfrac{AC}{CV}\) and \(\dfrac{AB}{BL}\), and if we suppose that the tangent plane, in rolling over the primitive surface, has turned through an infinitely small angle about its instantaneous axis \(LV\) so that it intersects \(AB\) already at the point \(A'\), then \(dp\) and \(dt\) will be equal, respectively, to \(\dfrac{AA'}{CV}\) and \(\dfrac{AA'}{BL}\). Consequently,

Fig. 1

\[ \frac{dp}{dt}=\frac{BL}{CV}=\frac{\eta''-\eta'}{v''-v'}, \]

where \(v'\) and \(\eta'\) are the volume and entropy at the point \(L\), and \(v''\) and \(\eta''\) at the point \(V\).

If, in place of \(\eta''-\eta'\), we substitute the equivalent quantity \(\dfrac{r}{t}\) (where \(r\) is the heat of vaporization), then we obtain the equation in its usual form

\[ \frac{dp}{dt}=\frac{r}{t(v''-v')}. \]

PROPERTIES OF THE SURFACE RELATING TO THE STABILITY OF THERMODYNAMIC EQUILIBRIUM

Let us now turn to the consideration of those geometrical properties of the surface which indicate whether the thermodynamic equilibrium of a body is stable, unstable, or indifferent. In this connection we shall have to touch to some extent on the nature of the processes that occur when equilibrium is absent. We shall suppose that the body is placed in a medium with constant pressure and temperature; but in the case when the pressure or the temperature on the surface of the body differs from the corresponding values in the medium, the immediate contact of the body and the medium can scarcely be reconciled with our assumption of the invariability of the initial pressure and temperature of the medium, and we shall suppose that the body is separated from the medium by an envelope which is capable of yielding to the slightest changes of pressure between the body and the medium, but only very gradually, and which at the same time is a very poor conductor of heat. For the further reasoning it will be convenient also

it is permissible to restrict the properties of the envelope by the above-mentioned conditions and to suppose that the envelope occupies no space and absorbs no heat whatever, but only transmits it, i.e., to set its volume and specific heat equal to zero. In the presence of an envelope of this kind we have the right to suppose that the action of the body on the medium will be so slow that it cannot appreciably affect the uniformity of the pressure and temperature of the medium.

When a body is not in a state of thermodynamic equilibrium, its state is not among the number of states represented by our surface. However, the body as a whole possesses a definite volume, entropy, and energy, which are equal to the sum of the volumes, entropies, and energies of its parts1. Therefore, if we suppose that the points corresponding to masses proportional to the masses of the various parts of the body, which are in different thermodynamic states, are placed in positions determined by these states and by the motion of the parts of the body (i.e., so that their coordinates are equal to the volume, entropy, and energy of the whole body under the supposition that it is successively in the same states and has the same velocities as its parts), then the center of gravity of these points will, evidently, have as its coordinates the volume, entropy, and energy of the whole body. When all the parts of the body are at rest, the point representing the volume, entropy, and energy of the body will be the center of gravity of several points on the primary surface. A consequence of the presence of motion in the parts of the body will be the displacement of the corresponding points parallel to the axis \(\varepsilon\) by distances equal in each case to the vis viva which the whole body would possess if it had the velocity of the represented part of the body; the center of gravity of the points thus determined will represent the volume, entropy, and energy of the whole body.

Let us now suppose that a body possessing the initial volume, entropy, and energy \(v'\), \(\eta'\), and \(\varepsilon'\) (enclosed in the envelope mentioned above) is placed in a medium with constant pressure \(P\) and temperature \(T\), and that, as a result of the action of the medium and the interaction of its own parts, it arrives at a final state of rest in which its volume, entropy, and energy are equal to \(v''\), \(\eta''\), and \(\varepsilon''\); we wish to find the relation between these quantities. If we regard the medium as a very large body (which is entirely admissible), so that the communication of heat to it or compression within moderate limits has no appreciable effect on its pressure and temperature, then, denoting the volume, entropy, and energy of the medium by \(V\), \(H\), and \(E\), we can write equation (1) in the form

\[ dE = T dH - P dV , \]

where we can carry out the integration, regarding $P$ and $T$ as constant quantities, and obtaining

\[ E''-E'=TH''-TH'-PV''+PV', \tag{a} \]

where one prime denotes the initial state, and two primes the final state of the medium. Further, since the sum of the energies of the body and the surrounding medium may become smaller, but cannot increase (as follows from the character of the envelope assumed by us), we have

\[ \varepsilon''+E''\leq \varepsilon'+E', \tag{b} \]

and since the sum of the entropies can only increase, but cannot decrease,

\[ \eta''+H''\geq \eta'+H'. \tag{c} \]

Finally, it is evident that

\[ v''+V''=v'+V'. \tag{d} \]

These 4 equations may, with slight modification, be rewritten in the following form:

\[ -E''+TH''-PV''=-E'+TH'-PV', \]

\[ \varepsilon''+E''\leq \varepsilon'+E', \]

\[ -T\eta''-TH''\leq -T\eta'-TH', \]

\[ Pv''+PV''=Pv'+PV'. \]

By addition we obtain

\[ \varepsilon''-T\eta''+Pv''\leq \varepsilon'-T\eta'+Pv'. \tag{e} \]

It is easy to see that the left- and right-hand sides of this equation represent the heights of the points $v'', \eta'', \varepsilon''$ and $v', \eta', \varepsilon'$ above the plane passing through the origin of coordinates and representing the pressure $P$ and the temperature $T$. The equation itself means that the final distance of the point from the plane is less than the initial one, or at least equal to it. Clearly, it is immaterial whether these distances are measured vertically or along the normal to the plane, and also whether the plane representing $P$ and $T$ passes through the origin of coordinates; but the distances should be counted as negative if they are measured from a point lying below the plane.

It is clear that the sign of the inequality in (e) is valid in the case when it holds either in (b) or in (c); consequently, (e) is an inequality when there exist any differences of pressure or temperature between different parts of the body, or between the body and the medium, or when some part of the body has appreciable motion (in the latter case there will be an increase of entropy as a result of the transformation of motion into heat). But even if the body initially has no appreciable motion and the pressure and tem-

if in all its parts they are the same as in the medium, the sign \(<\) is nevertheless valid if the different parts of the body are in states represented on the thermodynamic surface by points lying at different distances from the fixed plane representing \(P\) and \(T\), since it is undoubtedly valid when, owing to such initial conditions, differences of pressure or temperature or appreciable velocities arise. Further, the sign of inequality will necessarily occur in \((e)\) if one part of the body, without causing a change of pressure or temperature or appreciable velocities, passes into the state of another part, which is represented by a point lying at a different distance from the fixed plane representing \(P\) and \(T\). But what has been enumerated is the only possibility for the case under consideration, unless we suppose that there exists an equilibrium in which the mentioned points must have a common tangent plane (see above), whereas according to our assumption the tangent planes at the different points are parallel but do not coincide with one another.

The results of the preceding reasoning may be summarized as follows: if the body initially has no appreciable motion, and if its state, being homogeneous, is such as is represented on the primitive surface by a point at which the tangent plane is parallel to the fixed plane representing \(P\) and \(T\), or if the body is nonhomogeneous in state, and the points on the primitive surface representing the states of its parts do not have a common tangent plane parallel to the fixed plane representing \(P\) and \(T\), then such changes will take place that the distance of the point representing the volume, entropy, and energy of the body from this fixed plane will decrease (the distance is considered negative if it is measured from points lying below the plane). Let us apply this result to the question of the stability of a body, if it is surrounded, as we supposed earlier, by a medium with constant temperature and pressure.

The state of a body in equilibrium will be represented by a point on the thermodynamic surface, and since the pressure and temperature of the body are the same as those of the surrounding medium, we may take the tangent plane at this point to be the fixed plane representing \(P\) and \(T\). If the body is nonhomogeneous in state, although it is in equilibrium, then, in discussing the question of stability, we may either consider a point on an arbitrary surface as representing the state of the body, or consider points on the primitive surface representing the states of the different parts of the body. These points, as we have already seen, have a common tangent plane identical with the tangent plane at the indicated point of the arbitrary surface.

Thus, if the form of the surface is such that the surface lies above this tangent plane, with the exception only of the point of tangency, then the equilibrium is necessarily stable; for if the state of the body is slightly changed, or if an appreciable motion is imparted to some part of it, or if it is slightly changed...

state of one of them, or by transferring some [[unclear: word]] part of the body into any other thermodynamic state, or, finally, by means of all these methods simultaneously, the point representing the volume, entropy, and energy of the entire body will be located above the original tangent plane; and in this case, according to the proposition we have derived, processes must follow that will decrease the distance between the point and the plane, and which cannot cease until the body has been brought to its initial state, after which they will inevitably cease as a result of the assumed form of the surface.

If, on the contrary, the surface has such a form that some part of it lies below the fixed tangent plane, then the equilibrium is unstable. Indeed, it is obvious that by means of a slight change in the initial state of the body (the state of equilibrium with the surrounding medium, represented by the point or points of tangency) one can move the point representing the volume, entropy, and energy of the body into a position below the fixed tangent plane, when, as we know from the preceding, processes will begin that will carry this point still farther from the plane and that cannot cease until the entire body has passed into some state completely different from the original one.

It remains to consider the case when the surface, although it does not lie everywhere below the fixed tangent plane, touches it at more than one point. In this case, as one might expect, judging by its intermediate character between the two cases already considered, the equilibrium will be indifferent. Indeed, if any part of the body is transferred from the original state into a state represented by another point of the thermodynamic surface lying on the same tangent plane, then equilibrium will still exist. For, according to our assumption concerning the form of the surface, the temperature and pressure of all parts of the body as a result of such a transition will still be the same, and the body will not have any necessary tendency either to pass completely into the second state or to return to the initial state, since it will clearly be sufficient to change the values of \(T\) and \(P\) by an arbitrarily small amount in order to reverse such a tendency, if one exists, because any point, at will, can by means of such an infinitely small change of \(T\) and \(P\) be brought nearer to the plane representing \(T\) and \(P\).

It is necessary to note that in the case when the thermodynamic surface near some point is concave upward in both its principal directions, but somewhere else passes below the tangent plane drawn through this point, the equilibrium, although unstable with respect to jump-like changes of state, is stable with respect to continuous changes, as can be verified by applying

criterion of stability with respect to a neighborhood of such a point; this means that, when a body is in the state represented by such a point, then, although the equilibrium will prove unstable if we introduce into the body a small quantity of the same substance in one of the states represented by points of the surface below the tangent plane, it will be stable if the conditions necessary for such an abrupt change are absent. A generally known illustration of this proposition may be water in the liquid state, heated at any pressure above the boiling temperature of water at that pressure1.

BASIC FEATURES OF THE THERMODYNAMIC SURFACE FOR SUBSTANCES IN THE SOLID, LIQUID, AND GASEOUS STATES

We can now form an idea of the general character of the primary and secondary surfaces and of their mutual relations for a substance that assumes the forms of a solid body, a liquid, and a vapor. The primary surface will possess a plane of triple tangency, touching it at three points representing three states that can exist in contact with one another. With the exception of these three points, the primary surface lies entirely above the tangent plane. That part of the plane which has the form of a triangle with vertices at the three points of contact is a derived surface representing a system of three states of the given substance. We may now suppose that the plane rolls along the lower side of the surface, continuing all the time to touch the surface at two points and not intersecting it. This rolling can occur in three ways, namely: the plane may begin to rotate about any one of the three sides of the triangle just mentioned. Each pair of points that the plane touches simultaneously represents states that can continue to exist in contact with one another. Thus six lines are traced on the surface. Each of these lines has, in the general case, the property that the tangent plane at any of its points also touches the surface at some other point. We have had to say “in the general case,” because, as we shall see below, this propo-

\[ \delta(\varepsilon - T\eta + Pv)=0, \]

where \(\delta\) denotes a variation that is the result of any changes in the state of parts of the body, and moreover (if different parts of the body are in different states) in the ratio in which the distribution of the body among the different states corresponds. The condition of stable equilibrium reduces to the requirement that the value of the expression in parentheses be minimal.

…ceases to be valid at the critical point. The tangent plane to any point of the surface outside these lines lies wholly beneath the surface, except for the single point of contact. The tangent plane to any point of the primary surface inside these lines will intersect the surface. All these lines taken together may be called the boundary of absolute stability, and the surface outside the lines—the surface of absolute stability. That part of the envelope surface of the rolling plane which is enclosed between the pairs of lines traced by the plane on the primary surface is part of the derived surface and represents a system of two states of the given substance.

The mutual arrangement of all these lines and surfaces is shown schematically in horizontal projection1 in Fig. 2, where

Fig. 2

solid lines represent lines on the primary surface, and dashed lines—lines on the secondary surface. \(S\), \(L\), and \(V\) are points having a common tangent plane and representing the solid, liquid, and vapor states, which can exist in contact with one another. The plane triangle \(SLV\) is the derived surface representing systems composed of these states. \(LL'\) and \(VV'\) are a pair of lines traced in the rolling of the plane of double tangency, between which is enclosed the derived surface representing systems composed of liquid and vapor. \(VV''\) and \(SS'\) are another pair of lines, between which is placed the surface representing systems composed of vapor and solid. \(SS''\) and \(LL'''\) are a third pair of lines, between which lies the derived surface representing systems composed of solid and liquid. The lines \(L'''LL'V'VV''\) and \(S''SS'\) are the boundaries of the surfaces, …

representing, respectively, absolutely stable states of the liquid, vapor, and solid.

The geometrical interpretation of the results obtained by Dr. Andrews in his experiments with carbonic acid (Phil. Trans., vol. 159, p. 575) reduces to the following: at least for the given substance, the derived surface ends in the following manner: when the tangent plane is rolled along the primary surface, the two points of tangency approach one another and in the end coincide. The rolling of a plane of double tangency thus necessarily comes to an end. The point at which the two points of tangency coincide is a critical point. Before considering further the geometrical properties of this point and their physical meaning, it will be useful to investigate the character of the primary surface enclosed between the lines that form the boundary of absolute stability.

Between two points of the primary surface that have a common tangent plane—as, for example, the points \(L'\) and \(V'\) in Fig. 2—if there is no break in the primary surface, there must exist a portion of the surface where it is concave in the direction toward the tangent plane in at least one of its principal directions, and which therefore represents states of unstable equilibrium with respect to both discontinuous and continuous changes1 (see above). If on the primary surface we draw a line dividing it into parts that represent, respectively, states of stable and unstable equilibrium with respect to continuous changes, i.e. separating the surface concave upward in both principal directions from the surface concave downward in one or both principal directions, then this line, which may be called the boundary of essential instability, must in form resemble the line \(ll'Cvv'ss'\) in Fig. 2. This line touches the boundary of absolute stability at the critical point \(C\). Indeed, if we choose a pair of points, arbitrarily close to \(C\), on the lines \(LC\) and \(VC\), which have a common tangent plane, then the line connecting them on the primary surface, which is the section of the surface by a plane perpendicular to the tangent plane, will necessarily pass through a region of instability.

The geometrical properties of the critical point on our surface will become clearer if on the surface we draw lines of curvature for one of the principal directions, namely for that whose curvature has different signs on different sides of the boundary of essential instability. The lines of curvature that meet this line will, generally speaking, intersect it. Since at each point where such an intersection occurs the sign of the curvature of the lines changes, they evidently intersect here the plane—

a plane tangent to the surface, and, consequently, the surface itself intersects the tangent plane. But where one of these lines of curvature touches the boundary of essential instability without crossing it, so that the curvature of the line remains positive all the time (the curvature being regarded as positive when the concavity is on the upper side of the surface), the surface, obviously, does not intersect the tangent plane, but has contact with it of the third order in the section of least curvature. Therefore the critical point must be a point at which the line of that principal curvature which changes its sign is tangent to the line separating the regions of positive and negative curvature.

From the foregoing we may derive the following physical property of the critical state: although the critical state is a limiting one between states stable and unstable with respect to continuous changes of state, and although such limiting states, generally speaking, are unstable with respect to such changes of state, it is nevertheless stable with respect to them. An analogous proposition is also valid with respect to absolute stability, i.e., if one neglects the distinction between discontinuous and continuous changes, namely: although the critical state is a limiting one between states of stability and instability, and although equilibrium of such limiting states is, generally speaking, indifferent (on the assumption that the substance is surrounded by a medium with constant pressure and temperature), the critical point is, however, stable.

From what has been said about the curvature of the primitive surface near the critical point it follows that if we choose a point on this surface at an infinitely small distance from the critical point, in such a way that the tangent planes for these two points intersect along a line perpendicular to the section of least curvature at the critical point, then the angle between the two tangent planes will be an infinitely small quantity of the same order as the cube of the distance between these points. Consequently, at the critical point

\[ \left(\frac{dp}{dv}\right)_t = 0,\quad \left(\frac{dp}{d\eta}\right)_t = 0,\quad \left(\frac{dt}{dv}\right)_p = 0,\quad \left(\frac{dt}{d\eta}\right)_p = 0, \]

\[ \left(\frac{d^2p}{dv^2}\right)_t = 0,\quad \left(\frac{d^2p}{d\eta^2}\right)_t = 0,\quad \left(\frac{d^2t}{dv^2}\right)_p = 0,\quad \left(\frac{d^2t}{d\eta^2}\right)_p = 0, \]

and if on the primitive surface we draw the isotherm and the isobar for the critical point, then these lines will have contact of the second order.

But the elasticity of a substance at constant temperature and its specific heat at constant pressure may be determined by the equations

\[ e = -v\left(\frac{dp}{dv}\right)_t;\quad c_p = t\left(\frac{d\eta}{dt}\right)_p; \]

therefore, at the critical point,

\[ e=0,\qquad \frac{1}{s}=0, \]

\[ \left(\frac{de}{dv}\right)_t=0,\qquad \left(\frac{de}{d\eta}\right)_t=0,\qquad \left(\frac{d\frac{1}{s}}{dv}\right)_p=0,\qquad \left(\frac{d\frac{1}{s}}{d\eta}\right)_p=0 . \]

The last four equations will also be valid if the indices \(p\) and \(t\) are interchanged.

We have seen that, for such substances as can pass continuously from the liquid state into the vapor state, if the primary surface does not break off suddenly and, moreover, along a line that passes through the critical point, a part of this surface must represent states that are essentially unstable (i.e., unstable with respect to continuous changes), and which therefore can continue to exist only in very limited regions. This does not mean that such states are altogether unrealizable. It is quite probable that a substance in the critical state may be allowed to expand so rapidly that the time of the process will be too short for any appreciable transfer of heat to occur, and the substance will pass into one of these states of essential instability. Only such a result is possible under the assumption of the absence of heat transfer, according to which the points representing the states of all parts of the body must lie on the isentropic line (adiabat) drawn through the critical point on the primary surface. It is not difficult to see that there is no instability with respect to changes of state limited in this way, since this line (the section of the primary surface by a plane perpendicular to the \(\eta\)-axis) is turned with its concavity upward, as follows from the fact that the primary surface lies entirely above the tangent plane at the critical point.

We may suppose that in a substance which is initially in the critical state, waves of compression and expansion are propagated. The velocity of propagation of these waves will depend on the magnitude \(\left(\frac{dp}{dv}\right)_\eta\), and hence also on the magnitude \(-\left(\frac{d^2\varepsilon}{dv^2}\right)_\eta\). But for a compression wave the value of these expressions is determined by the form of the isentropic line on the primary surface. If the expansion wave has approximately the same velocity as the compression wave, then one may conclude that the substance, while expanding under the given conditions, remains in a state represented by the primary surface, and this means the realization of states of essential instability. The magnitude \(\left(\frac{d^2\varepsilon}{dv^2}\right)_\eta\) on the derived surface evidently has a value entirely different from that on the primary surface, since the curvature of these surfaces at the critical point is different.

Otherwise the situation is with respect to that part of the surface which lies between the boundary of absolute stability and the boundary of essential instability. Here we have experimental data on some of the states depicted. It is well known, for example, that for water liquid states can exist beyond the boundary of absolute stability, both beyond that portion of the boundary where evaporation usually begins (\(LL'\) in Fig. 2), and beyond that portion of it where freezing usually begins (\(LL'''\)). The possibility of the existence of vapor beyond the boundary of absolute stability, i.e. at a given temperature under pressures greater than the pressure in the state of equilibrium between the vapor and its liquid in contact along a plane surface at that temperature, was indisputably proved by Sir W. Thomson in his paper “On the equilibrium of vapour at a curved surface of liquid” (Proc. Roy. Soc. Edinb., Session 1869–1870 and Phil. Mag., vol. XLII, p. 448). By means of experiments similar to those proposed by Prof. J. Thomson in the work already mentioned above, we could bring vapors into states lying far beyond the boundary of absolute stability1. Owing to the fact that the resistance to deformations characteristic of solid bodies evidently tends to prevent the occurrence within them of discontinuous changes of state, substances, undoubtedly, can exist in solid states very far from the boundary of absolute stability.

The surface of absolute stability, together with the triangle representing a system of three states and the three developable surfaces representing, according to what was said above, systems of two states, forms one continuous surface which, with the exception of a plane portion, is everywhere concave upward and has only one value of \(\varepsilon\) for any given values of \(v\) and \(\eta\). Since \(t\) is always positive, this surface

has only one value of \(\eta\) for any given values of \(v\) and \(\varepsilon\). If evaporation can take place at any temperature, with the exception of \(0\), then \(p\) is everywhere positive, and the surface has only one value of \(v\) for any given values of \(\eta\) and \(\varepsilon\). It is the surface of dissipated energy. If we consider all points representing the volume, entropy, and energy of the body in all possible states, whether states of equilibrium or not, then these points form a three-dimensional figure which in some directions will be unbounded, but in other directions will be bounded by this surface1.

The lines traced on the primary surface by the rolling of the plane of double contact, which we have called the boundary of absolute stability, do not end at the vertices of the triangle representing the system of these states. Indeed, when

However, this surface will not include regions where \(p < 0\), if there exists any practically attainable temperature \(t'\) at which the substance has the properties of a perfect gas, except for the case when its volume is less than some quantity \(v'\). Indeed, the equations of the isotherm on the thermodynamic surface for a perfect gas have the form [see equations \((B)\) and \((E)\) in the article “Graphical Methods in the Thermodynamics of Gases”]

\[ \varepsilon = C, \]

\[ \eta = a \lg v + C'. \]

The isotherm \(t'\) on the thermodynamic surface of the substance under consideration must therefore have the same equations in that part in which its volume is greater than some constant quantity \(v'\). But if, for some point of this surface, \(p < 0\) and \(t > 0\), then the equation of the tangent plane at this point will have the form

\[ \varepsilon = m\eta + nv + C'', \]

where \(m\) denotes the temperature, and \(n\) the pressure at the point of tangency, so that \(m\) and \(n\) are positive. But it is obvious that the value of \(v\) in the equations of the isotherm can be made so large that the corresponding point lies below the tangent plane. Consequently, the tangent plane intersects the primary surface, and the point on the thermodynamic surface for which \(p < 0\) cannot belong to the surfaces from which, according to the above, one continuous surface is formed.

plane touches the primitive surface at these three points, it can begin to roll over the surface as a plane of double tangency, not only leaving the surface at one of these points, but also turning in the opposite direction. In the latter case, however, the lines formed on the primitive surface by the points of tangency, although they are continuations of the curves described above, are not any part of the boundary of absolute stability. Likewise, the portions of the enveloping surfaces of the rolling plane between these lines, although they are continuations of the developable surfaces described above and represent states of the body from which at least some can be realized, are of secondary interest, since they do not form any part of the surface of dissipated energy and, on the other hand, do not have the theoretical significance possessed by the primitive surface.

PROBLEMS RELATING TO THE SURFACE OF DISSIPATED ENERGY

The surface of dissipated energy has an important range of application to a certain class of problems concerning the results theoretically possible with a given body or system of bodies in a specified initial state.

Suppose, for example, that it is required to find the greatest amount of mechanical work that can be obtained from a given quantity of a certain substance in a given initial state, without increasing its total volume and with no giving up by it, or access to it, of heat from surrounding bodies except those which at the end of the processes remain in their original state. This quantity was called the available energy of the body. The initial state of the body is here assumed to be such that the body can be brought from it into states of dissipated energy by means of reversible processes.

When a body is in a state represented by some point on the surface of dissipated energy, under the prescribed conditions no work can, of course, be obtained from it. But even when the body is in a state of thermodynamic equilibrium, i.e. in a state represented by a point on the thermodynamic surface, if this point does not lie on the surface of dissipated energy, some quantity of energy will be available (available), provided the conditions necessary for the performance of work are present, since the equilibrium of the body is unstable with respect to discontinuous changes. Or if the body is in a solid state, then even when it is entirely homogeneous in state, the magnitude of the pressure (or stress) may be different in it in different directions, and because of this the body may possess a certain amount of useful energy. Or if different parts of the body are in different states, then this fact, generally speaking, must be a source of useful energy. Finally, we must not exclude

that case in which the body has appreciable motions and its vis viva constitutes useful energy. In each case we must find the initial volume, entropy, and energy of the body, which will be equal to the sums of the initial volumes, entropies, and energies of its parts (the concept “energy” here includes the vis viva of appreciable motions). These values \(v\), \(\eta\), and \(\varepsilon\) will determine the position of a certain point, which we shall call the point representing the initial state.

But the condition that no heat be given up to the surrounding bodies requires that the final entropy of the body be not less than the initial entropy, since the entropy of the body can decrease only as a result of violating this condition. The problem, therefore, may be reduced to the following: to find that quantity of energy by which the energy of the body can be decreased without increasing its volume or decreasing its entropy. This quantity will be represented geometrically by the distance of the point representing the initial state from the surface of dissipated energy, measured parallel to the \(\varepsilon\)-axis.

Let us consider another problem. Suppose, as before, that a certain initial state of the body is given. No work is allowed either by the surrounding bodies or upon them. Heat may be given to the surrounding bodies and received from them only on the condition that the algebraic sum of all transferred quantities of heat be equal to zero. From both of these conditions those bodies may be exempted which, by the end of the processes, remain in the initial state. Moreover, an increase in the volume of the body is also not allowed. It is required to find the greatest magnitude by which, under such conditions, the entropy of the surrounding system of bodies can be decreased. This magnitude will, evidently, be equal to that by which the entropy of the body can be increased without changing the energy of the body or increasing its volume, and which is represented geometrically by the distance of the point representing the initial state from the surface of dissipated energy, measured parallel to the \(\eta\)-axis. It may be called the capacity for entropy of the body in the given state\(^1\).

\(^1\) It is not without use to draw attention to the analogy and the difference between the two statements of the problem. In the first case the question is in fact reduced to this: what weight can the state of the given body allow us to raise to a given height, so that no residual changes occur in the surrounding bodies? In the second case the question is reduced to this: what quantity of heat can, by using the state of the body under consideration, be taken from some body with a definite temperature and transferred to another body with a higher definite temperature? In order that the numerical values of useful energy and of the capacity for entropy should be identical with the answers to these questions, it is necessary, in the first case, if the weight is measured in units of force, that the given distance, measured vertically, be equal to unity, and in the second case that the difference of the reciprocals of the two given temperatures be equal to unity. If we choose, as these given temperatures, the freezing point and the boiling point of water, and since

\[ \frac{1}{273}-\frac{1}{373}=0.00098, \]

then the capacity for entropy of a body in any given state will be equal to \(0.00098\), multiplied by

Thirdly, let a definite initial state of the body again be given. No work is allowed to be done either by external bodies or upon them, and likewise heat passes neither to them nor from them. Under these conditions, as before, one may set free the bodies in which no residual changes are produced. It is required to find the quantity—

the amount of heat which, with its aid, we can transfer from the freezing point to the boiling point (i.e. remove from a body whose temperature remains all the time the same as at the freezing point, and impart to a body whose temperature remains all the time the same as at the boiling point).

The relations between these quantities and their connection with the surface of dissipated energy are illustrated in Fig. 3, where there is represented the plane perpendicular to the axes \(\upsilon\) and \(u\) and passing through the point \(A\), which depicts the initial state of the body. The line \(MN\) is a section of the surface of dissipated energy. The straight lines \(Q\varepsilon\) and \(Q\eta\) are sections of the planes \(\eta=0\) and \(\varepsilon=0\), and therefore are parallel respectively to the axes \(\varepsilon\) and \(\eta\). The segments \(AD\) and \(AE\) depict the energy and entropy of the body in its initial state, while \(AB\) and \(AC\) depict its available energy and its capacity for entropy. It is easy to see that when either the available energy or the capacity for entropy is equal to zero, the other also becomes zero. With the exception of this case, these quantities may vary independently of one another. Indeed, owing to the curvature of the surface of dissipated energy it is evidently possible to change the position of the point representing the initial state of the body in such a way that the change in the distance of the point from the surface, measured parallel to one coordinate axis, is not accompanied by a change in the distance measured parallel to the other axis.

Since the term “entropy” is used by different authors in different senses, which may give rise to misunderstandings, it will not be superfluous to add a few words concerning the terminology used in the present question. If Professor Clausius had defined the quantity of entropy by means of the equation

\[ dS=-\frac{dQ}{T} \]

instead of the equation introduced by him (Mechanische Wärmetheorie, ch. IX, § 14; Pogg. Ann., July 1865)

\[ dS=\frac{dQ}{T}, \]

where \(S\) denotes entropy, \(T\) the temperature of the body, and \(dQ\) the elementary amount of heat communicated to the latter, then the quantity which we have called above the capacity for entropy would naturally be called available (available) entropy, a term also more convenient because of its analogy with the term available energy. Such a change in the definition of the concept of entropy would entail no changes either in the form of the thermodynamic surface or in any of its geometrical constructions, if only it were supposed that the values of entropy are measured in the opposite direction. We should have only to replace \(-\eta\) by \(\eta\) in our equations and to make the corresponding changes in the verbal formulation of the propositions. Prof. Tait proposed using the word “entropy” in the sense opposite to that in which Clausius used it (Thermodynamics, § 48, see also § 178), evidently meaning by this the definition of entropy by means of the first of the equations given above. Subsequently, however, he applies this term to denote available energy (§ 182). Prof. Maxwell uses the term “entropy” as a synonym of available energy, erroneously asserting that Clausius uses-

to which the volume of the body can be reduced by applying for this purpose, in accordance with the conditions, only the force obtained from the body itself. These conditions require that the energy of the body not change and that its entropy not decrease. Consequently, the desired quantity is represented by the distance of the point representing the initial state of the body from the surface of dissipated energy, measured parallel to the axis of volumes.

Fourthly, as before, a certain initial state of the body is given. An increase of its volume is not allowed. The performance of work either by external bodies or upon them is not allowed, nor is the transfer to them or from them of heat, with the exception of some body at the given constant temperature \(t'\). From the latter conditions one may, as before, exempt bodies in which no residual changes are produced. It is required to determine the greatest quantity of heat that can be communicated to the body with constant temperature, and also the greatest quantity of heat that can be taken from it under the stated conditions. If through the point of the initial state a straight line is drawn in the plane perpendicular to the axis \(v\), so that the tangent of the angle of its inclination to the direction of the axis \(\eta\) is equal to the given temperature \(t'\), then it is easy to show that the vertical projections of two segments of this straight line between the point of the initial state and the surface of dissipated energy will respectively represent the two desired quantities\(^{1}\).

Fig. 3.

These problems may be modified so that they approximate the practical problems that usually arise, if it is assumed that the body is surrounded by a medium with constant pressure and temperature, and the body considered in the preceding

use the word “entropy” to designate that part of the energy which is not useful (Theory of Heat, pp. 186 and 188). However, the term “entropy,” in the sense in which Clausius uses it, does not denote a quantity of the same kind (i.e., one that can be measured in the same units) as energy, as is evident from the derivation of his equation, where \(Q\) (heat) denotes a quantity measured in units of energy, and since the units in which \(T\) (temperatures) are measured are arbitrary, it is obvious that \(S\) and \(Q\) are measured in different units. It may also be added that entropy, as defined by Clausius, is analogous (synonymous) to the thermodynamic function defined by Rankine.

\(^{1}\) Thus, if in Fig. 3 the straight line \(MAN\) is drawn so that \(NAC = t'\), then the segment \(MR\) will be equal to the greatest quantity of heat that can be communicated to the body with constant temperature, and the segment \(NS\) to the greatest quantity of heat that can be taken from this body.

problems, the body and the medium being taken together. Then we obtain the following results.

If we assume that the plane representing constant pressure and temperature of the medium is tangent to the surface of dissipated energy for the given body, then the distance of the point representing the initial state of the body from this plane, measured parallel to the axis $\varepsilon$, will represent the available energy of the body and the medium; the distance measured parallel to the axis $\eta$ will represent the capacity for entropy of the body and the medium; and the distance measured parallel to the axis $v$ will represent the amount of greatest rarefaction that can be produced in the body or in the medium (if all the applied force is exerted by the body and the medium). If a line is drawn through the above-mentioned point in the plane perpendicular to the axis $v$, then the vertical projection of the segment of this line, bounded by the point and the tangent plane, will represent the greatest quantity of heat that can be imparted to or withdrawn from some other body with a constant temperature equal to the tangent of the angle of inclination of the straight line to the horizontal plane (this segment represents the greatest quantity of heat that can be imparted to a body with constant temperature if the latter is higher than the temperature of the medium; in the opposite case it represents the greatest quantity of heat that can be withdrawn from this body). In all these cases the point of contact of the plane with the surface of dissipated energy represents the final state of the given body.

If, through a point representing any given initial state of the body, one draws the plane representing the pressure and temperature of the medium, then the part of this plane lying inside the surface of dissipated energy will represent all states, in respect of volume, entropy, and energy, into which the given body can be brought by reversible processes without producing residual changes in external bodies (with the exception of the medium), and the three-dimensional region enclosed between this plane figure and the surface of dissipated energy will represent all states into which the body can be brought by any processes whatever that do not cause residual changes in external bodies (with the exception of the medium).1

EDITOR’S NOTE

The translation of the article by J. W. Gibbs, published in Transactions of the Connecticut Academy, pp. 382–404 (1873), is the second article published by Gibbs; it, like his first article, “Graphical Methods in the Thermodynamics of Fluids” [Transactions of the Connecticut Academy, 309–342 (1873)], is devoted to geometrical methods of thermodynamic investigation. Gibbs’s work is appearing in Russian print for the first time. At the same time, just as Gibbs’s name is well known to chemists and physicists, since his thermodynamic investigations laid the foundation for modern thermodynamics and, chiefly, for its numerous applications to problems of heterogeneous equilibrium, surface phenomena, chemical equilibrium, etc., the original works of Gibbs are little known not only to the broad circle of physical chemists, but even to specialist thermodynamicists. Therefore one must welcome the initiative of the editors of Uspekhi Fizicheskikh Nauk, who are publishing in the pages of their journal a translation of one of Gibbs’s thermodynamic papers. This work played a major role in the history of thermodynamics, since it opened up broad possibilities for the geometrical method of investigating various physicochemical phenomena, whose significance goes beyond the simple geometrical interpretation of phenomena and which, in Gibbs, becomes a completely independent form of investigation. In this article Gibbs solves, by general methods of geometrical investigation, problems concerning the surface formed, for a chemically homogeneous substance, by the variables entropy \(-\eta\), internal energy \(\varepsilon\), and volume \(v\), for questions connected with the investigation of the coexistence of different aggregate states of a substance; he derives the general conditions of equilibrium, shows, by way of illustration, the derivation of the Clausius–Clapeyron equation, outlines the general solution of certain practical questions—for example, finding the work that can be obtained under specified conditions from a body in a certain state, etc. The article identifies a number of quantities (for example, the thermodynamic potential) which in Gibbs’s later work “On the equilibrium of Heterogeneous substances” received broad application and analytic form. It is interesting to note that the paths of geometrical investigation of thermodynamic surfaces outlined in this article subsequently made it possible for van der Waals to construct a coherent thermodynamic theory of binary mixtures. Like Gibbs’s other works, this article is written in an extremely concise form. The rigorous presentation of the material attracts one by the coherent sequence of the ideas developed by the author. A term introduced in the article serves not so much as the name of a quantity as, for a visible degree, as its interpretation. Such are “available energy, dissipated energy,” etc. These points made the translation of the article into Russian very difficult. The present translation was made from the English complete collected works of Gibbs; moreover, the editor tried to make the translation as close to the original as possible, sometimes even at the expense of the harmonious construction of Russian phrases, and only in those places where a literal translation no longer fit well with the form of a Russian phrase did the editor allow himself to depart somewhat from the original, while still striving to convey as accurately as possible the meaning of the text. In order to render the content of Gibbs’s thought more correctly, the editor checked his translation against the German translation, published, as is known, by W. Ostwald in 1892 and reviewed by Gibbs. In the text, only in two places did the editor allow himself to correct formulas, since obvious

will be represented by the inclination of the surface of dissipated energy according to the rule set forth on p. 400. In application to problems similar to those considered above, this surface will have, with respect to the system represented by it, properties completely analogous to the properties of the surface of dissipated energy for a homogeneous body.

misprint, namely \(\left(\dfrac{d^{2}\varepsilon}{dv^{2}}\right)_{\eta}\), the editor corrected to \(\left(\dfrac{d^{2}\varepsilon}{dv^{2}}\right)_{\eta}\), and in the formula
\(\varepsilon''' - \varepsilon' = t'(\eta''-\eta') - p'(v''-v')\) the difference \(\varepsilon''-\varepsilon'\) was replaced by \(\varepsilon'''-\varepsilon'\), as required by the meaning of the formula. As for the translations of terms, the editor sought to convey their meaning, comparing with how this was done in the German translation, namely:

1) Surface of dissipated energy — surface of scattered energy (German: Fläche Zerstreuter Energie).
2) available energy — useful energy (German: nutzbare Energie).

In one instance the editor introduced a Russian term which, it seems to him, is better suited in meaning to the concepts being denoted. In his work Gibbs often speaks of “states, in prolonged contact,” by which he means various aggregate states of a substance existing jointly and satisfying the condition of stable equilibrium; the editor termed such states “coexisting,” without, of course, adding the condition of the prolonged presence of these aggregate states in contact. In order that the translation should be closer to the original text, the editor did not replace the concept “body” with the word substance, although in meaning it would more closely correspond to the essence of the matter; therefore in the translation there are passages which express the circumstance that the body under consideration, consisting of a liquid body and a solid body, may not quite correspond to the understanding of the concept “body” generally accepted in Russian literature. In one of the notes the author speaks of a “weight” that must be “lifted to a given height”; here the editor replaced the word “lift” by the word “overcome,” on the grounds that, since weight is a force, it cannot be lifted, but can be overcome.

The equality occurring in the text

\[ d\varepsilon = t\,d\eta - p\,dv, \]

for the derivation of which the author refers to his preceding work, is, as is easy to see, a generalization for equilibrium processes of the first and second laws of thermodynamics, since for such processes, by the first law of thermodynamics, \(dQ = d\varepsilon + p\,dv\), where \(dQ\) is the quantity of heat communicated to the system under consideration in the given process, and, by the second law, \(d\eta = \dfrac{dQ}{t}\). By combining these two relations one obtains the equality given above. The relations \(p = -\left(\dfrac{d\varepsilon}{dv}\right)_{\eta}\) and \(t = \left(\dfrac{d\varepsilon}{d\eta}\right)_{v}\) immediately follow from the fact that \(d\varepsilon\) is the total differential of a function of the state of the system; therefore, taking \(\varepsilon = \varepsilon(\eta,v)\) and taking into account that \(\eta\) and \(v\) may be regarded as independent variables, we conclude that \(d\varepsilon = \left(\dfrac{d\varepsilon}{dv}\right)_{\eta}\cdot dv + \left(\dfrac{d\varepsilon}{d\eta}\right)_{v}\cdot d\eta\), and, comparing with the expression \(d\varepsilon = t\,d\eta - p\,dv\), we find the two indicated relations.

The equality given in the text

\[ \varepsilon' - t'\eta' + p'v' = \varepsilon'' - t''\eta'' + p''v'' \]

gives the condition that the tangent planes to the surface at the points \(\varepsilon', v', \eta'\) and \(\varepsilon'', v'', \eta''\) coincide. Indeed, from the equation of the plane in intercepts

\[ \frac{\varepsilon}{a}+\frac{\eta}{b}+\frac{v}{c}=1, \]

where \(a, b, c\) are the intercepts on the axes \(\varepsilon, \eta, v\) cut off by the plane, it follows that for the two planes

\[ \frac{\varepsilon'}{a_1}+\frac{\eta'}{b_1}+\frac{v'}{c_1}=1 \quad\text{and}\quad \frac{\varepsilon''}{a_2}+\frac{\eta''}{b_2}+\frac{v''}{c_2}=1, \]

and since, by the assumption made in the text, \(a_1=a_2\), then

\[ \varepsilon' + \eta'\frac{a_1}{b_1}+v'\frac{a_1}{c_1} = \varepsilon'' + \eta''\frac{a_2}{b_2}+v''\frac{a_2}{c_2}; \]

but it is easy to see that from the fact that the surface is tangent to the planes it follows that

\[ \frac{a_1}{b_1}=-\frac{\partial \varepsilon'}{\partial \eta'}=-t'; \qquad \frac{a_1}{c_1}=-\frac{\partial \varepsilon'}{\partial v'}=p' \]

and

\[ \frac{a_2}{b_2}=-\frac{\partial \varepsilon''}{\partial \eta''}=-t''; \qquad \frac{a_2}{c_2}=-\frac{\partial \varepsilon''}{\partial v''}=p''. \]

Consequently

\[ \varepsilon' - t'\eta' + p'v' = \varepsilon'' - t''\eta'' + p''v''. \]

1. Throughout this entire article the body under consideration has been assumed homogeneous in composition. But if we imagine any material system whatever and suppose that each possible state of this system is determined by a point whose coordinates are equal to the total volume, entropy, and energy of the system, then these points evidently form a three-dimensional region bounded, in some directions of the surface, by a surface representing the state of dissipated energy. In these states the temperature is necessarily one and the same in all parts of the system; the pressure may vary (as occurs in the case of a body of very large mass, like a planet), but it will always be possible to preserve the equilibrium of the system (in the state of dissipated energy) by applying at its surface one and the same normal pressure. This pressure and the everywhere equal temperature of the system ↩↩↩↩↩↩↩↩↩