# 1911 Encyclopædia Britannica/Thermodynamics

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​THERMODYNAMICS (from Gr. θέρμη, hot, δύναμις, force). 1. The name thermodynamics is given to that branch of the general science of Energetics which deals with the relations between thermal and mechanical energy, and the transformations of heat into work, and vice versa. Other transformations of heat are often included under the same title (see Energetics). An historical account of the development of thermodynamics is given in the article Heat. The object of the present article is to illustrate the practical application of the two general principles— (1) Joule’s law of the equivalence of heat and work, and (2) Carnot's principle, that the efficiency of a reversible engine depends only on the temperatures between which it works; these principles are commonly known as the first and second laws of thermodynamics. The application will necessarily be confined to simple cases such as are commonly met with in practice, or are required for reference in cognate subjects.
2. Application of the First Law.—The complete transformation of mechanical energy into heat by friction, or some analogous process of degradation, is always possible, and is made the basis of experiments for the determination of the mechanical equivalent of the heat unit (see Calorimetry). The converse process of the transformation of heat into mechanical work or other forms of energy is subject to limitations.
When a quantity of heat, H, is supplied to a body, part is expended in raising the temperature of the body, or in expanding the volume against molecular forces, and is represented by an increase in the total quantity of energy contained in the body, which is generally called its Intrinsic Energy, and will be denoted by the symbol E. The remainder is equivalent to the external work, W, done by the body in expanding or otherwise, which can be utilized for mechanical purposes, and ceases to exist as heat in the body. The application of the first law leads immediately to the equation,
| H=E−E$_0$+W, | (1) |
in which E$_0$ represents the quantity of energy originally present in the body, and all the quantities are supposed, as usual, to be expressed in mechanical units. This equation is generally true for any series of transformations, provided that we regard H and W as representing the algebraic sums of all the quantities of heat supplied to, and of work done by the body, heat taken from the body or work done on the body being reckoned negative in the summation. E−E$_0$, then, represents the total increase of the intrinsic energy of the body in its final state, which may be determined by measuring H and W. If after any series or cycle of transformations the body is restored to its initial state, we must have E=E$_0$, whence it follows that H=W. But this simple relation is only true of the net balances of heat and work in a complete cyclical process, which must be adopted for theoretical purposes if we wish to eliminate the unknown changes of intrinsic energy. The balance of work obtainable in such a cycle depends on the limits of temperature in a manner which forms the subject of the second law.
3. Indicator or p.v. Diagram.—The significance of relation (1) is best appreciated by considering the graphic representation of quantities of heat and energy on a work-diagram.
On the familiar indicator diagram the state of the working substance is represented by the position of a point called the “ state point, ” defined by the values of the pressure p and volume v of unit mass, as ordinate and abscissa respectively (fig. 1). Any line (“path” or “graph”) on the diagram, such as BCD, represents an “operation” or “process” i.e. a continuous series of states through which the substance may be made to pass in any transformation. It is tacitly assumed that the motion is relatively so slow that the pressure and temperature of the substance are practically uniform throughout its mass at any stage of the process. Otherwise the transformation could not be fully represented on the diagram, and would not be reversible. The area BCDdb under the path represents the external work done by the substance in ​ expanding from B to D, which is analytically represented by the integral of pdv taken along the given path. Any closed path or figure, such as ABCD, represents a complete cycle or series of operations, in the course of which the substance is restored to its original state with respect to temperature, intrinsic energy and other properties. The area DABbd under the return path (v diminishing) represents work done on the substance, or against the back-pressure, and is negative. The area of the cycle, viz., that enclosed by the path BCDA represents the balance of external work done by the substance in one cycle, and is positive if the cycle is described clockwise as indicated by the arrows. The simplest types of process or operation are:–(1), heating or cooling at constant volume, represented by vertical lines such as Bb, called Isometrics, in which the pressure varies, but no external work is done. (2) Heating or cooling at constant presrure, represented by horizontal lines such as NA, called Isopiestics, in which the external work done is the product of the pressure p and the expansion v″−v′. (3) Expansion or compression at constant temperature, represented by curves called Isothermals, such as BC, AD, the form of which depends on the nature of the working substance. The isothermal are approximately equilateral hyperbolas (pv=constant), with the axes of p and v for asymptotes, for a gas or unsaturated vapour, but coincide with the isopiestics for a saturated vapour in presence of its liquid. (4) Expansion or compression under the condition of heat-insulation, represented by curves called Adiabatics, such as BAZ or CDZ′, which are necessarily steeper than the isothermal.
* Fig. 1.
A cycle such as ABCD enclosed by parts of two isothermal, BC, AD, and two adiabatic, AB, CD, is the simplest form of cycle for theoretical purposes, since all the heat absorbed, H', is taken in during the process represented by one isothermal at the temperature θ′, and all the heat rejected, H″, is given out during the process represented by the other at the temperature θ″. This is the cycle employed by Carnot for the establishment of his fundamental principle of reversibility as the criterion of perfect efficiency in a heat engine. The area ABCD, representing the work, W, per cycle, is the difference (H′−H ) of the quantities of heat absorbed and rejected at the temperatures θ′ and θ″. As the temperature 0” is lowered, the area of the cycle increases, but since W can never exceed H', there must be a zero limit of temperature at which the pressure would vanish and the area of the cycle become equal to the whole heat absorbed at the higher temperature. Taking this ideal limit as a theoretical or absolute zero, the value of H may be represented on the diagram by the whole area included between the two adiabatic BAZ, CDZ′ down to the points where they intersect the isothermal of absolute zero, or the zero isopiestic OV asymptotically at infinity.
If the substance in any state such as B were allowed to expand adiabatically (dH=0) down to the absolute zero, at which point it contains no heat and exerts no ressure, the whole of its available heat energy might theoretically be recovered in the form of external work, represented on the diagram by the whole area BAZcb under the adiabatic through the state-point B, bounded by the isometric Bb and the zero isopiestic bV. The change of the intrinsic energy in passing from one state to another, as from B to C is represented by the addition of the heat-area H=BCZZ′, and the subtraction of the work-area W=BCcb. It follows from the first law that the intrinsic energy of a substance in a given state must always be the same, or that the change of E in any transformation must depend only on the initial and final states, and not on the path or process. It will be observed that the areas representing H and W both depend on the form of the path BC, but that the difference of the areas representing the change of intrinsic energy dE is independent of BC, which is a boundary common to both H and W. This is mathematically expressed by the statement that dE is an exact differential of a function of the co-ordinates defining the state of the body, which can be integrated between limits without reference to the relation representing the path along which the variations are taken.
4. Application of Carnot’s Principle.—Carnot adopted as the analytical expression of his principle the statement that the ethciency W/H, or the work obtainable per unit of heat by means of a perfect engine taking in heat at a temperature t° C. and rejecting heat at 0° C., must be some function F(t) of the temperature t, the lower limit 0° C. being supposed constant. He was unable to apply the principle directly in this form, as it would require an exact knowledge of the properties of substances through a wide range of temperature. He therefore employed the corresponding expression for a cycle of infinitesimal range dt at the temperature t in which the work dW obtainable from a quantity of heat H would be represented by the equation
dW=HF'(t)dt,
where F ′(t) is the derived function of F (t), or dF (t) /dt, and represents the work obtainable per unit of heat per degree fall of temperature at a temperature t. The principle in this form is readily applicable to all cases, and is independent of any view with regard to the nature of the heat. It simply asserts that the efficiency function F ′(t), which is known as Carnot’s function, is the same for all substances at the same temperature. Carnot verified this by calculating the values of F ′(t) at various temperatures from the known properties of vapours and gases, and showed that the efficiency function diminished with rise of temperature, as measured on the scale of the mercury or gas thermometer, from about 1.40 kilogram metres per kilo-calorie per degree C. at 0° C. to about 1.11 at 100° C., according to the imperfect data available in his time. Applying the above equation to a gas obeying the law pv=RT, for which the work done in isothermal expansion from a volume 1 to a volume r is W=RT log$_e$r, whence dW/=R log$_e$rdt, he deduced the expression for the heat absorbed by a gas in isothermal expansion
H=R log$_e$r/F ′(t).
He also showed that the difference of the specific heats at constant pressure and volume, S=s, must be the same for equal volumes of all gases at the same temperature and pressure, being represented by the expression R/TF ′(t). He remarks that “the law according to which the motive power of heat varies at different points of the thermometric scale is intimately connected with that of the variations of the specific heats of gases at different temperatures—a law which experiment has not yet made known to us with sufficient exactness." If he had ventured to assume the difference of the specific heats constant, it would have followed that F ′(t) must vary inversely as T. The same result follows if the work W=RT log$_e$r done by a gas in isothermal expansion is assumed to be equivalent or proportional to the heat absorbed, H=R log$_e$r/F ′(t). Mayer (1842) made this assumption in calculating the mechanical equivalent of heat. Joule (1845) was the first to prove it approximately by direct experiment, but did not see his way to reconcile Carnot’s principle, as stated by Clapeyron, with the mechanical theory. Holtzmann (1845) by the same assumption deduced the value J/T for the function F ′(t), but obtained erroneous results by combining this assumption with the caloric theory. Clausius (1850), apply in the same assumption, deduced the same value of F ′(t), and showed that it was consistent with the mechanical theory and Joule’s experiments, but required that a vapour like steam should deviate more considerably from the gaseous laws than was at that time generally admitted. The values of F ′(t) calculated previously by Sir W. Thomson (Lord Kelvin) from Regnault’s tables of the properties of steam, assuming the gaseous laws, did not vary exactly as J/T. Joule’s experiments on the equivalence of W and H were not sufficiently precise to decide the question. This most fundamental point was finally settled by a more delicate test, devised by Lord Kelvin, and carried out in conjunction with Joule (1854), which showed that the fundamental assumption W=H in isothermal expansion was very nearly true for permanent gases, and that F ′(t) must therefore vary very nearly as J/T. Kelvin had previously proposed to define an absolute scale of temperature independent of the properties of any particular substance in terms of Carnot’s function by making F ′(t) constant. He now proposed to define absolute temperature as proportional to the reciprocal of Carnot’s function, so as to agree as closely as possible with the scale of the gas thermometer. With this definition of temperature θ, if the heat H is measured in work units, the expression of Carnot’s principle for an infinitesimal cycle of range dθ reduces to the simple form dW/dθ=H/θ. Combining this with the first law, for a Carnot cycle of finite range, if H′ is the heat taken in at θ′, and H″ is the heat rejected at θ″, the work W done in the cycle is equal to the difference H′-H″, and we have the simple relations,
| W/(θ′−θ″)=H′/θ=H″θ″ | (2) |
5. Thermodynamical Relations.—The most important and most useful of the relations between the thermodynamical properties of a substance may be very simply deduced from a consideration of the indicator diagram by a geometrical method, which is in many respects more instructive than the analytical method generally employed. Referring to fig. 2, let BC be a small portion of any isothermal corresponding to the temperature θ′, and AD a neighbouring isothermal θ″. Let BE be an isometric through B meeting AD in E, and EC an isopiestic through E meeting BC in C. Let BA, CD be adiabatic through B and C meeting the isothermal θ″ in A and D. Then by relations (2) the heat, H, absorbed in the isothermal change BC, is to the work, W, done in the cycle ABCD in the ratio of θ′ to (θ′−θ″). If the difference ​of temperature (θ′−θ″) is small, the figure ABCD may be regarded as a parallelogram, and its area W as equal to the rectangle BE XEC. This is accurately true in the limit when (θ′−θ″) is infinitesimal, but in practice it is necessary to measure specific heats, &c., over finite ranges of temperature, and the error involved is generally negligible if the range does not exceed a few degrees. BE is the increase of pressure (p'−p”) produced by the rise of temperature (θ′−θ″) if the volume is kept constant. EC is (v”−v′) produced by the same rise of temperature is kept constant. Substituting these symbols in for the area, the relation becomes
| H=θ(p′−p″)(v′−v″)/(θ′−θ″) | (3) |
This relation may be interpreted in two ways, according as we require the heat absorbed in terms of the change of pressure or volume. (1) The heat, H, absorbed in isothermal expansion (latent heat of expansion) from p′ to p″ is equal to the diminution of pressure (p′−p″) multiplied by the absolute temperature and by the expansion per degree (v"−v')/(θ′−θ″) at constant pressure. (2) The heat, H, absorbed in isothermal expansion from v′ to v" is equal to the increase of volume (v"−v′) multiplied by the absolute temperature, and by the increase of pressure per degree (p′−p″)/(θ′−θ″), at constant volume. In the notation of the calculus the relations become
−dH/dp (θ const) =θdv/dθ (p const) (4) dH/dv (θ const) =θdp/dθ (v const)
* Fig. 2.
The negative sign is prefixed to dH/dp because absorption of heat +dH corresponds to diminution of pressure −dp. The utility of these relations results from the circumstance that the pressure and expansion coefficients are familiar and easily measured, whereas the latent heat of expansion is difficult to determine.
The most instructive example of the application of relations (1) and (2) is afforded by the change of state of a substance at constant temperature and pressure. Starting with unit mass of the substance in the first state (e.g. liquid) possessing volume v' at a temperature θ” and pressure p” represented by the point A in fig. 3, the heat absorbed in raising the temperature to θ′ and the pressure to p′ without change of state may be written s′ (θ′−θ″), where s′ is the specific heat of the substance in the first state at saturation pressure. If now the substance in the 'state B is entirely converted at constant temperature and pressure into the second state (e.g. saturated vapour), in which it occupies a volume v", the line BC represents the change of volume (v"−v'). The heat absorbed in this change is called the latent heat of change of state, and may be represented by the symbol L′. The substance is then cooled to the lower temperature θ” along the path CD, keeping it in the saturated state. The heat evolved in this process may be represented by s″(θ′−θ”), where S” is the specific heat of the substance in the second state at saturation pressure. Finally, the substance is reconverted into the first state at the tem erature θ″, completing the cycle by the abstraction of a quantity of heat L″. By the application of the first law, the difference of the quantities of heat absorbed and evolved in the cycle must be equal to the work represented by the area of the cycle, which is equal to (p'−p)(v"−v) in the limit when the difference of pressure is small. By the application of the second law, relations (2), the same work area is equal to (θ′−θ″)L′/θ′. Dividing by (θ′−θ″), and writing dp/dθ and dL/dθ for the limiting values of:the ratios (p′−p″)/(θ'−θ″) and (L′−L”)/(θ′−θ″), we obtain the important relations
s'−s"+dL/dθ=(v"−2/)dp/dθ=L/θ, . . (5)
in which dp/dθ is the rate of change of pressure with temperature when the two states are in equilibrium. It is not necessary in this example that AB, CD should be adiabatic, because the change of volume BC is finite. The same equations apply to the case of fusion of a solid, if L is the latest heat of fusion, and v', s', 11", 5” the specific volumes and specific heats of the solid and liquid respectively.
6. Ratio and Difference of Specific Heats.—If we take unit mass of the substance at B, fig. 2, and cool it at constant volume to E, through an interval of temperature (θ′−θ”), the amount of heat abstracted may be written h=s(θ′−θ”), where s is the specific heat at constant volume. If, starting from E, the same amount of heat h is restored at constant pressure, we should arrive at the point F on the adiabatic through B, since the substance has been transformed from B to F by a reversible path without loss or gain of heat on the whole. In order to restore the substance to its original temperature θ′ at constant pressure, it would be necessary to supply a further quantity of heat, H, represented by the area between the two adiabatics from FC down to the absolute zero; This quantity of heat is the same as that already found in equation (3), but for the small area BFC, which is negligibly small in the limit compared with H. The whole quantity of heat required to raise the temperature from θ″ at constant pressure the path EC is H+h, which is equal to S(θ′−θ″), where S is the specific heat at constant pressure. Since h=s(θ'-θ″), the difference S−s between the specific heats at constant pressure and volume is evidently H/(θ′−θ”). Substituting for H its value from (3), and employing the notation of the calculus, we obtain the relation
S−s =θ(dp/dθ) (dv/dθ), .... (6)
in which the partial differential coefficients have the same meaning as in (4).
* Fig. 3.
Since the amounts of heat supplied at constant pressure from E to F and from E to C are in the limit proportional to the expansions EF and EC which they produce, the ratio S/s is equal to the ratio EC/EF. EF is the change of volume corresponding to a change of pressure BE when no heat is allowed to escape and the path is the adiabatic BF. EC is the change of volume for the same change of pressure BE when the path is the isothermal BC. These changes of volume are directly as the compressibilities, or inversely as the elasticities. If we write K for the adiabatic elasticity, and k for the isothermal elasticity, we obtain
S/s=EC/EF=K/k .... (7)
The value of the specific heat S at constant pressure can always be determined by experiment, and in practice is one of the most important thermodynamical properties of a substance. The value of the specific heat s at constant volume can also be measured in a few cases, but it is generally necessary to deduoe it from that at constant pressure by means of relation (6). It is often impossible to observe the pressure-coefficient dp/dθ directly, but it may be deduced from the isothermal compressibility by means of the geometrically obvious relation, BE=(BE/EC)×EC. The ratio BE/EC of the diminution of pressure to the increase of volume at constant temperature, or −dp/dv, is readily observed.
The amount of heat absorbed in any small change of state, as from E to G in fig. 2, may be found by adding to the heat required for the change of temperature at constant volume, sdθ, or at constant pressure, Sdθ, the heat absorbed in isothermal expansion as given by relations (4). We thus obtain the expressions
dH=sdθ+θ(dp/dθ)dv =Sdθ−θ(dv/dθ)dp . . . (8)
The first is equivalent to measuring the heat along the path EBG, the second along the path ECG. The two differ by the area BEC, which can be neglected if the change is small. For a finite change it is necessary to represent the path by a series of small steps, which is the graphic equivalent of integration along the path represented by the given relation between v and θ, or p and θ. If we put dH=0 in equations (8), we obtain the relations between dv and dθ, or dp and dθ, under the condition of no heat-supply, Le. along the adiabatic, which can be integrated, giving the equations to the adiabatic, provided that the values of the specific heats and expansion-coefficients are known.
6. Intrinsic Energy.—The change of intrinsic energy E along any path is found by subtracting the work pdv from either of the expressions for dH. Since the change of energy is independent of the path, the finite change between any two given states may be found by integration along any convenient path. It is generally convenient to divide the path into two steps, isothermal and isometric, or isothermal and isopiestic, and to integrate along each separately. The change of energy at constant volume is simply sdθ, the change at constant temperature is (θdp/dθ−p)dv, which may be written
dE/dθ (v const) =s, dE/dv (θ const) =θdp/dθ−p . (9)
These must be expressed as functions of v and θ, which is theoretically possible if the values of s, p, and dp/dθ are known. Since the two expressions (9) are the partial differential-coefficients of a single function E of the independent variables v and θ, we shall obtain the same result, namely d$_2$E/dθdv, if we differentiate the first with respect to v and the second with respect to θ. We thus obtain the relation
ds/dv(θ const) =θd$_2$p/dθ$_2$ (v const), . (10) ​which is useful for calculating the variation of the specific heat s with variation of density at constant temperature. A similar expression for the variation of the specific heat S at constant pressure is obtained from the second expression in (8), by taking p and θ as independent variables; but it follows more directly from a consideration of the variation of the function (E+pv).
7. Total Heat.—The function (E+pv), like E itself, has a value depending only on the state of the body. It may conveniently be called the Total Heat, by a slight extension of the meaning of a term which has been for a long time in use as applied to vapours (see Vaporization). Since $dE=dH-pdv,$ we have evidently for the variation of the total heat from the second expression (8),
| $dF=d(E+pv)=dH+vdp=Sd\theta -(\theta dv/d\theta -v)dp$ | (11) |
This expression shows that the rate of variation of the total heat with temperature at constant pressure is equal to the specific heat at constant pressure. To find the total heat of a substance in any given state defined by the values of p and $\theta$, starting from any convenient zero of temperature, it is sufficient to measure the total heat required to raise the substance to the final temperature under a constant pressure equal to p. For instance, in the boiler of a steam engine the feed water is pumped into the boiler against the final pressure of the steam, and is heated under this constant pressure up to the temperature of the steam. The total heat with which we are actually concerned in the working of a steam engine is the total heat as here defined, and not the total heat as defined by Regnault, which, however, differs from $(E+pv)$ only by a quantity which is inappreciable in ordinary practice.
Observing that $F$ is a function of the co-ordinates expressing the state of the substance, we obtain for the variation of S with pressure at constant temperature,
| $dS/dp(\theta$ const$)=d^{2}/d\theta dp=-\theta d^{2}v/d\theta ^{2}(p$ const$)$ | (12) |
If the heat supplied to a substance which is expanding reversibly and doing external work, $pdv$, is equal to the external work done, the intrinsic energy, $E$, remains constant. The lines of constant energy on the diagram are called Isenergics. The equation to these lines in terms of $v$ and $\theta$ is obtained by integrating
| $dE=sd\theta +(\theta d/d\theta -p)dv=0$ | (13) |
If, on the other hand, the heat supplied is equal to $-vdp$, we see from (II) that $F$ remains constant. The equation to the lines of constant total heat is found in terms of p and $\theta$ by putting $dF=0$ and integrating (11).
8. Ideal Gases.—An ideal gas is a substance possessing very simple thermodynamic properties to which actual gases and vapours appear to approximate indefinitely at low pressures and high temperatures. It has the characteristic equation $pv=R\theta$, and obeys Boyle's law at all temperatures. The coefficient of expansion at constant pressure is equal to the coefficient of increase of pressure at constant volume. The difference of the specific heats by equation (6) is constant and equal to $R$. The isothermal elasticity $-v(dp/dv)$ is equal to the pressure <m. The adiabatic elastic it is equal to $\gamma p$, where $\gamma$ is the ratio $S/s$ of the specific heats. The heat absorbed in isothermal expansion from $v_{0}$ to $v$ at a temperature 0 is equal to the work done by equation (8) (since $d0=0$, and $0(dp/d\theta )dv=pdv)$, and both are given by the expression $R\theta \log _{e}(v/v_{0})$. The energy $E$ and the total heat $F$ are functions of the temperature only, by equations (9) and (11), and their variations take the form $dE=sd\theta$, $dF=Sd\theta$. The specific heats are independent of the pressure or density by equations (10) and (12). If we also assume that they are constant with respect to temperature (which does not necessarily follow from the characteristic equation, but is generally assumed, and appears from Regnault's experiments to be approximately the case for simple gases), the expressions for the change of energy or total heat from $\theta _{0}$ to $\theta$ may be written $E-E_{0}=s(\theta -\theta _{0}),F-F_{0}=S(0-\theta _{0})$. In this case the ratio of the specific heats is constant as well as the difference, and the adiabatic equation takes the simple form, $pv^{\gamma }=$ constant, which is at once obtained by integrating the equation for the adiabatic elasticity, $-v(dp/dv)=-\gamma p$.
The specific heats may be any function of the temperature consistently with the characteristic equation provided that their difference is constant. If we assume that s is a linear function of $\theta$, $s=s_{0}(1-a\theta )$, the adiabatic equation takes the form,
| $s_{0}\log ^{e}(\theta /\theta _{0})+as_{0}(\theta -\theta _{0})+R\log _{e}(v/ve)=0$ | (14) |
where $(\theta _{0},v),<math>(\theta _{0},v_{0})$ are any two points on the adiabatic. The corresponding expressions for the change of energy or total heat are obtained by adding the term éas0(02-002) to those already given, thus:
* $E-E_{0}=s_{0}(\theta -\theta _{0})+{\frac {1}{2}}as_{0}(\theta ^{2}-{\theta _{0}}^{2}),$
* $F-F_{0}=_{0}(\theta -\theta _{0})+{\frac {1}{2}}as_{0}(\theta ^{2}-{\theta _{0}}^{2}),$
where $S_{0}=s_{0}+R$.
9. Deviations of Actual Gases from the Ideal State.—Since no gas is ideally perfect, it is most important for practical purposes to discuss the deviations of actual gases from the ideal state, and to consider how their properties may be thermodynamically explained and defined. The most natural method of procedure is to observe the deviations from Boyle's law by measuring the changes of $pv$ at various constant temperatures. It is found by experiment that the change of pw with pressure at moderate pressures is nearly proportional to the change of p, in other words that the coefficient d(pv)/dp is to a first approximation a function of the temperature only. This coefficient is sometimes called the “angular coefficient, " and may be regarded as a measure of the deviations from Boyle's law, which may be most simply expressed at moderate pressures by formulating the variation of the angular coefficient with temperature. But this procedure in itself is not SLll:flCi€I1l1, because, although it would be highly probable that a gas obeying Boyle's law at all temperatures was practically an. ideal gas, it is evident that Boyle's law would be satisfied by any substance having the characteristic equation p/v=f(θ), where f(θ) is any arbitrary function of θ, and that the scale of temperatures given by such a substance would not necessarily coincide with the absolute scale. A sufficient test, in addition to Boyle's law, is the condition dE/dv=0 at constant temperature. This gives by equation (9) the condition θdp/dθ=p, which is satisfied by any substance possessing the characteristic equation p/θ =f(v), where f(v) is any arbitrary function of v. This test was applied by Joule in the well-known experiment in which he allowed a gas to expand from one vessel to another in a calorimeter without doing external work. Under this condition the increase of intrinsic energy would be equal to the heat absorbed, and would be indicated by fall of temperature of the calorimeter. Joule failed to observe any change of temperature in his apparatus, and was therefore justified in assuming that the increase of intrinsic energy of a gas in isothermal expansion was very small, and that the absorption of heat observed in a similar experiment in which the gas was allowed to do external work by expanding against the atmospheric pressure was equivalent to the external work done. But owing to the large thermal capacity of his calorimeter, the test, though sufficient for his immediate purpose, was not delicate enough to detect and measure the small deviations which actually exist.
10. Method of Joule and Thomson.—William Thomson (Lord Kelvin), who was the first to realize the importance of the absolute scale in thermodynamics, and the inadequacy of the test afforded by Boyle's law or by experiments on the constancy of the specific heat of gases, devised a more delicate and practical test, which he carried out successfully in conjunction with Joule. A continuous stream of gas, supplied at a constant pressure and temperature, is forced through a porous plug, from which it issues at a lower pressure through an orifice carefully surrounded with non-conducting material, where its temperature is measured. If we consider any short length of the stream bounded by two imaginary cross-sections A and B on either side of the plug, unit mass of the fluid in passing A has work, i>"1/, done on it by the fluid behind and carries its energy, E'~l-U, with it into the space AB, where U' is the kinetic energy of flow. In passing B it does work, p”v", on the fluid in front, and carries its energy, E”+ U", with it out of the space AB. If there is no external loss or gain of heat through the walls of the pipe, and if the flow is steady, so that energy is not accumulating in the space AB, we must evidently have the condition $E'+U'+p'v'=E''+U''+p''v''$ at any two cross-sections of the stream. It is easy to arrange the experiment so that U is small and nearly constant. In this case the condition of flow is simply that of constant total heat, or in symbols, $d(E+pv)=0.$ We have therefore, by equation, (11),
| $Sd\theta =(\theta dv/d\theta -v)dp,$ | (15) |
where $d\theta$ is the fall of temperature of the fluid corresponding to a diminution of pressure dp. If there is no fall of temperature in passing the plug, $d\theta =0$, and we have the condition $\theta dv/d\theta =v.$ The characteristic equation of the fluid must then be of the form v/θ =f(p), where f(p) is any arbitrary function of p. If the fluid is a gas also obeying Boyle's law, $pv=f(\theta ),$ then it must be an ideal gas. As the result of their experiments on actual gases (air, hydrogen, and CO$_2$), Joule and Thomson (Phil. Trans., 1854, 1862) found that the cooling effect, $d\theta$, was of the same order of magnitude as the deviations from Boyle's law in each case, and that it was proportional to the difference of pressure, dp, so that 110/dp was nearly constant for each gas over a range of pressure of five or six atmospheres. By experiments at different temperatures between 0° and 100° C., they found that the cooling effect per atmosphere of pressure varied inversely as the square of the absolute temperature for air and CO$_2$. Putting dt?/dp=A/02 in equation (15), and integrating on the assumption that the small variations of S could be neglected over the range of the experiment, they found a solution of the type, v/θ =f(p) — SA/3θ$^3$, in which f(p) is an arbitrary function of p. Assuming that the gas should approximate indefinitely to the ideal state pv=Rθ at high temperatures, they put f(p)=R/p, which gives a characteristic equation of the form
| $v=R\theta /p=SA/3\theta ^{2}$ | (16) |
An equation of a similar form had previously been employed by Rankine (Trans. Roy. Soc. Ed., 1854) to represent Regnault's experiments on the deviations of CO$_2$ from Boyle's law. This equation is practically identical for moderate pressures with that ​ devised by Clausius (Phil. Mag., 1880) to represent the behaviour of CO$_2$ up to the critical point. Experiments by Natanson on CO$_2$ at 17° C. confirm those of Joule and Thomson, but show a slight increase of the ratio dθ/dp at higher pressures, which is otherwise rendered probable by the form of the isothermal as determined by Andrews and Amagat. More recent experiments by J. H. Grindley (Proc. Roy. Soc., 1900, 66, p. 79) and Callendar (Proc. Roy. Soc., 1900) on steam confirm this type of equation, but give much larger values of the cooling effect than for CO$_2$, and a more rapid rate of variation with temperature.
11. Modified Joule-Thomson Equation.—G. A. Hirn (Théorie Mec. de la Chaleur, ii. p. 211, Paris, 1869) proposed an equation of the form (p+p$_0$)(v−b) Rθ in which the effect of the size of the molecules is represented by subtracting a quantity b, the “co-volume,” from the volume occupied by the gas, and the effect of the mutual attractions of the molecules is represented by adding a quantity p$_0$, the internal pressure, to the external pressure, p. This type of equation, was more fully worked out by van der Waals, who identified the internal pressure, p$_0$, with the capillary pressure of Laplace, and assumed that it varied directly as the square of the density, and could be written a/v$^2$ This assumption represents qualitatively the theoretical isothermal of James Thomson (see Vaporization) and the phenomena of the critical state (see Condensation of Gases; but the numerical results to which it leads differ so widely from experiment that it is necessary to suppose the constant, a, to be a function of the temperature. Many complicated expressions have been suggested by subsequent writers in the attempt to represent the continuity of the gaseous and liquid states in a single formula, but these are of a highly empirical nature, and beyond the scope of the present inquiry. The simplest assumption which suffices to express the small deviations of gases and vapours from the ideal state at moderate pressures is that the coefficient a in the expression for the capillary pressure varies inversely as some power of the absolute temperature. Neglecting small terms of the second order, the equation may then be written in the form
| v − b＝Rθ/p − c$_0$(θ$_0$/θ)$^n$＝V − c | (17) |
which c is a small quantity (expressing the defect from the ideal volume V＝Rθ/p due to co-aggregation of the molecules) which varies inversely as the nth power of θ, but is independent of p to a first approximation at moderate pressures. The constant to is the value of c at some standard temperature θ$_0$. The value of the index, n, appears to be different for different types of molecule. For CO$_2$ at ordinary temperatures n＝2, as in the Joule-Thomson equation. For steam between 100° and 150° C. it approaches the value 3·5. It is probably less than 2 for air and the more perfect gases. The introduction of the covolume, b, into the equation is required in order to enable it to represent the behaviour of hydrogen and other gases at high temperatures and pressures according to the experiments of Amagat. It is generally taken as constant, but its value at moderate pressures is difficult to determine. According to van der Waals, assuming spherical molecules, it should be four times; according to O. E. Meyer, on slightly different assumptions, it should be 4√2 times, the actual volume of the molecules. It appears to be a quantity of the same order as the volume of the liquid, or as the limiting volume of the gas at very high pressures. The value of the co-aggregation volume, c, at any temperature, assuming equation (17), may be found by observing the deviations from Boyle's law and by experiments on the Joule-Thomson effect. The value of the angular coefficient d(pv)/dp is evidently (b−c), which expresses the defect of the actual volume v from the ideal volume Rθ/p. Differentiating equation (17) at constant pressure to find dv/dθ, and observing that dc/dθ＝−nc/0, we find by substitution in (15) the following simple expression for the cooling effect dθ)/dp in terms of c and b,
| Sdθ/dp＝(n + 1)c − b | (18) |
Experiments at two temperatures suffice to determine both c and n if we assume that b is equal to the volume of the liquid. But it is better to apply the Boyle's law test in addition, provided that errors due to surface condensation can be avoided. The advantage of this type of equation is that c is a function of the temperature only. Other favourite types of equation for approximate work are (1) p＝Rθ/v+ f(v), which makes p a linear function of 6 at constant volume, as in van der Waal's equation; (2) v＝Rθ/p+f(p), which makes v a linear function of θ at constant pressure. These have often been employed as empirical formulae (e.g. Zeuner’s formula for steam), but they cannot be made to represent with sufficient approximation the deviations from the ideal state at moderate pressures and generally lead to erroneous results. In the modified Joule-Thomson equation (17), both c and n have simple theoretical interpretations, and it is possible to express the thermodynamical properties of the substance in terms of them by means of reasonably simple formulae.
12. Application of the Modified Equation.—We may take equation (17) as a practical example of the thermodynamical principles already given. The values of the partial differential coefficients in terms of n and c are as follows:-
| dv/dθ (p const)＝(R/p) (1 + nc/V) | (19) |
| d$^2$v/dθ$^2$ ,, ＝−n(n+1)c/θ$^2$ | (20) |
| dp/dθ (v const)＝(R/V)(1+nc/V) | (21) |
| d$^2$p/dθ$^2$ ,, ＝Rnc(1−n+2nc/V)/θV$^2$ | (22) |
| d(pv)/dp(θ const)＝b − c | (23) |
Substituting these values in equations already given, we find,
| from (6) S−s | ＝R(1+nc/V)$^2$ | (24) |
| ,,  (9) dE/dv (θ const) | ＝ncp/V | (25) |
| ,, (11) dF/dp ,, | ＝(n+1)c−b | (26) |
| ,, (10) ds/dv ,, | ＝(1−n+2nc/V)Rnc/V$^2$ | (27) |
| ,, (12) dS/dp ,, | ＝n(n+1)c/θ | (28) |
In order to deduce the complete variation of the specific heats from these equations, it is necessary to make some assumption with regard to the variation of the specific heats with temperature. The assumption usually made is that the total kinetic energy of the molecules, including possible energy of rotation or vibration if the molecules consist of more than one atom, is proportional to the energy of translation in the case of an ideal gas. In the case of imperfect gases, all the available experimental evidence shows that the specific volume tends towards its ideal value, V＝Rθ/p, in the limit, when the pressure is indefinitely reduced and the molecules are widely separated so as to eliminate the effects of their mutual actions. We may therefore reasonably assume that the limiting values of the specific heats at zero pressure do not vary with the temperature, provided that the molecule is stable and there is no dissociation. Denoting by S$_0$, s$_0$, these constant limiting values at p＝0, we may obtain the values at any pressure by integrating the expressions (27) and (28) from ∞ to v and from 0 to p respectively. We thus obtain
| S＝S$_0$+n(n+1)pc/θ | (29) |
| s＝s$_0$+(n−1−nc/V)ncp/θ | (30) |
In working to a first approximation, the small term nc/V may be omitted in the expression for s.
The expression for the change of intrinsic energy E between any given limits p$_0$θ$_0$ to pθ is readily found by substituting these values of the specific heats in equations (11) or (13), and integrating between the given limits. We thus obtain
| E−E$_0$＝s$_0$(θ−θ$_0$)−n(pc−p$_0$c$_0$) | (31) |
We have similarly for the total heat F＝E+pv,
F−F$_0$＝S$_0$(θ−θ$_0$)−(n+1)(cp−c$_0$p$_0$)+b(p−p$_0$)
The energy is less than that of an ideal gas by the term npc. If we imagine that the defect of volume c is due to the formation of molecular aggregates consisting of two or more single molecules, and if the kinetic energy of translation of any one of these aggregates is equal to that of one of the single molecules, it is clear that some energy must be lost in co-aggregating, but that the proportion lost will be different for different types of molecules and also for different types of co-aggregation. If two monatomic molecules, having energy of translation only, equivalent to 3 degrees of freedom, combined to form a diatomic molecule with 5 degrees of freedom, the energy lost would be pc/2 for co-aggregation, c, per unit mass. In this case n＝1/2. If two diatonic molecules, having each 5 degrees of freedom, combine to form a molecule with 6 degrees of freedom, we should have n＝2, or the energy lost would be 2pc per unit mass. If the molecules and molecular aggregates were more complicated, and the number of degrees of freedom of the aggregates were limited to 6, or were the same as for single molecules, we should have n＝s$_0$/R. The loss of energy could not be greater than this on the simple kinetic theory, unless there were some evolution of latent heat of co-aggregation, due to the work done by the mutual attractions of the co-aggregating molecules.
It is not necessary to suppose that the co-aggregated molecules are permanently associated. They are continually changing partners, the ratio c/V representing approximately the ratio of the time during which any one molecule is paired to the time during which it is free. At higher densities it is probable that more complex aggregates would be formed, so that as the effect of the collisions became more important c would cease to be a function of the temperature only; experiment, indeed, shows this to be the case.
13. Entropy.—It follows from the definition of the absolute scale of temperature, as given in relations (2), that in passing at constant temperature θ from one adiabatic φ′ (Fig. 1) to any other adiabatic φ″, the quotient H/θ of the heat absorbed by the temperature at which it is absorbed is the same for the same two adiabatic whatever the temperature of the isothermal path. This quotient is called the change of entropy and may be denoted by (φ″−φ′). In passing along an adiabatic there is no change of entropy, since no heat is absorbed. The adiabatic are lines of constant entropy, and are also called Isentropics. In virtue of relations (2), the change of entropy of a substance between any two states depends only on the initial and final states, and may be reckoned along any reversible path, not necessarily isothermal, by dividing each small increment of heat, dH, by the temperature, θ, at which it is acquired, and taking the sum or integral of the quotients, dH/θ so obtained.
The expression for the change of entropy between any two states is found by dividing either of the expressions for dH in (8) by θ and integrating between the given limits, since dH/θ is a perfect differential. In the case of a solid or a liquid, the latent heat of isothermal expansion may often be neglected, and if the specific heat, s, be also taken as constant, we have simply φ−φ$_0$＝s log$_e$θ/θ$_0$. If the substance at the temperature θ undergoes a change of state, absorbing latent heat, L, we have mercl to add the term L/θ to the above expression. In the case of an ideal gas, dp/110 at constant volume＝R/v, and dv/dθ at constant pressure＝R/p; thus we obtain the expressions for the change of entropy φ−φ$_0$ from the state p$_0$θ$_0$v$_0$ to the state pθv,
φ−φ$_0$＝s log$_e$θ/θ$_0$+R log$_e$v/v$_0$
| ＝S log$_e$θ/θ$_0$−R log$_e$ p/p$_0$ | (32) |
In the case of an imperfect gas or vapour, the above expressions are frequently employed, but a more accurate result may be obtained by employing equation (17) with the value of the .specific heat, S, from (29), which gives the expression
| φ−φ$_0$＝S$_0$log$_e$θ/θ$_0$−R log$_e$p/pP$_0$−n(cp/θ−c$_0$p$_0$/θ$_0$) | (33) |
The state of a substance may be defined by means of the temperature and entropy as co-ordinates, instead of employing the pressure and volume as in the indicator diagram. This method of representation is applicable to certain kinds of problems, and has been developed by Macfarlane Gray and other writers in its application to the steam engine. (See STEAM ENGiNE.) Areas on the temperature-entropy or 0, ¢ diagram represent quantities of heat in the same way as areas on the indicator diagram represent quantities of work. The 0, ¢ diagram is useful in the study of heat waste and condensation, but from other points of view the utility of the conception of entropy as a “ factor of heat ” is limited by the fact that it does not correspond to any directly measurable physical property, but is merely a mathematical function arising from the form of the definition of absolute temperature. Changes of entropy must be calculated in terms of quantities of heat, and must be interpreted in a similar manner. The majority of thermodynamical problems may be treated without any reference to entropy, but it affords a convenient method of expression in abstract thermodynamics, especially in the consideration of irreversible processes and in reference to the conditions of equilibrium of heterogeneous systems.
14. Irreversible Processes.—In order that a process may be strictly reversible, it is necessary that the state of the working substance should be one of equilibrium at uniform pressure and temperature throughout. If heat passes “of itself” from a higher to a lower temperature by conduction, convection or radiation, the transfer cannot be reversed without an expenditure of work. If mechanical work or kinetic energy is directly converted into heat by friction, reversal of the motion does not restore the energy so converted. In all such cases there is necessarily, by Carnot's principle, a loss of efficiency or available energy, accompanied by an increase of entropy, which serves as a convenient measure or criterion of the loss. A common illustration of an irreversible process is the expansion of a as into a vacuum or against a pressure less than its own. In tiiiis case the work of expansion, pdv, is expended in the first instance in roducing kinetic ener of motion of parts of the gas. If this could) be co-ordinated and utilized without dissipation, the gas might conceivably be restored to its initial state; but in practice violent local differences of pressure and temperature are produced, the kinetic energy is rapidly converted into heat by viscous eddy friction, and residual differences of temperature are equalized by diffusion throughout the mass. Even if the expansion is adiabatic, in the sense that it takes place inside a non-conducting enclosure and no heat is supplied from external sources, it will not be is entropic, since the heat supplied by internal friction must be included in reckoning the change of entropy. Assuming that no heat is supplied from external sources and no external work is done, the intrinsic energy remains constant by the first law. The final state of the substance, when equilibrium has been restored, may be deduced from this condition, if the energy can be expressed in terms of the co-ordinates. But the line of constant energy on the dia ram does not represent the path of the transformation, unless it be supposed to be effected in a series of infinitesimal steps between each of which the substance is restored to an equilibrium state. An irreversible process which permits a more complete experimental investigation is the steady flow of a fluid in a tube already referred to in section IO. If the tube is a perfect non-conductor, and if there are no eddies or frictional dissipation, - the state of the substance at any point of the tube as to E, p, and v, is represented by the adiabatic, or is entropic path, dE＝-pdv. As the section of the tube varies, the change of kinetic energy of flow, dU, is represented by −vdp. The flow in this case is reversible, and the state of the fluid is the same at points where the section of the tube is the same. In practice, however, there is always some frictional. dissipation, accompanied by an increase of entropy and by a fall of pressure. In the limiting case of a long fine tube, the bore of which varies in such a manner that U is constant, the state of the substance along a line of flow may be represented by the line of constant total heat d(E+pv)＝0; but in the case of a porous plug or small throttling aperture, the steps of the process cannot be followed. though the final state is the same.
In any small reversible change in which the substance absorbs heat, dH, from external sources, the increase of entropy, d¢, must be equal to dH/θ. If the change is not reversible, but the final state is the same, the change of entropy, dφ, is the same, but it is no longer equal to dH/θ. By Carnot's principle, in all irreversible processes, dH/θ=0 must be algebraically less than do, otherwise it would be possible to devise a cycle more efficient than a reversible cycle. This affords a useful criterion (see Energetics) between transformations which are impossible and those which are possible but irreversible. In the special Case of a substance isolated from external heat supply, dH＝0, the change of entropy is zero in a reversible process, but must be positive if the process is not reversible. The entropy cannot diminish. Any change involving decrease of entropy is impossible. The entropy tends to a maximum, and the state-is one of stable equilibrium when the value of the entropy is the maximum value consistent with the conditions of the problem.
15. Heterogeneous Equilibrium.—In a system, as distinguished from a homogeneous substance, consisting of two or more states or phases, a similar condition of equilibrium applies. In an spontaneous irreversible change, if the system is heat-isolated, there must be an increase of entropy. The total entropy of the system is found by multiplying the entropy per unit mass of the substance in each state by the mass existing in that state, and adding the products so obtained. The simplest case to consider is that of equilibrium between solid and liquid, or liquid and vapour. The more general case is discussed in the article Energetics, and in the original memoirs of Willard Gibbs and others. Since the condition of heat-isolation is impracticable, the condition of maximum entropy cannot, as a rule, be directly applied, and it is necessary to find a more convenient method of expression. If dW is the external work done, dH the heat absorbed from external sources, and dE the increase of intrinsic energy, we have in all cases by the first law, dH-dE＝dW. Since 0d¢ cannot be less than dH, the difference (θdφ−dE) cannot be less than dW. This inequality holds in all cases, but cannot in general be applied to an irreversible change, because θdφ is not a perfect differential, and cannot be evaluated without a knowledge of the path or process of transformation. In the special case, however, in which the transformational is conduérted in an isothermal enclosure, a common condition easily realized in practice, the temperature at the end of the transformation is reduced to its initial li)/alue throughout the substance. The value of θdφ is then the same as d(θφ), which is a perfect differential, so that the condition may be written d(θφ−E)＝dW. The condition in this form can be readily applied provided that the external work dW can be measured. There are two special cases of importance:—(a) If the volume is constant, or dW＝0, the value of the function (θφ−E) cannot diminish, or (E-041) cannot increase, if the temperature is kept constant. This function may be represented, for each state or phase of the system considered, by an area on the indicator diagram similar to that representing the intrinsic energy, E. The product θφ may be represented at any point such as D in Fig. 1 by the whole area θ″DZ′VO under the isothermal θ″D and the adiabatic DZ′, bounded by the axes of pressure and volume. The intrinsic energy, E, is similarly represented by the area DZ′Vd under the adiabatic to the right of the isometric Dd. The difference θφ−E is represented by the area 0"DdO to the left of the isometric Dd under the isothermal θ″D. The increment of this area (or the decrement of the negative area E−θφ) at constant temperature represents the external work obtainable from the substance in isothermal expansion, in the same way that the decrement of the intrinsic energy represents the work one in adiabatic expansion. The function J＝E−θφ, has been called the “ free energy ” of the substance by Helmholtz, and 04> the “bound energy.” These functions do not, however, represent energy existing in the substance, like the intrinsic energy; but the increment of θφ represents heat su plied to, and the decrement of (E−θφ) represents work obtainable from, the substance when the temperature is kept constant. The condition of stable equilibrium of a system at constant temperature and volume is that the total J should be a minimum. This function is also called the “thermodynamic potential at constant volume ” from the analogy with the cinrédition of minimum potential energy as the criterion of stable equilibrium in statics.
As an example, qwe may apply this condition to the case of change of state. If J′, J″ represent the values of the function for unit mass of the substance of specific volumes v′ and v″ in the two states at temperature 0 and pressure p, and if a mass tn is in the state v′, and 1−m in the state v", the value of J for unit mass of the mixture is m]'+(1-m)]”. This must be a minimum in the state of equilibrium at constant temperature. Since the volume is constant, we have the condition m°v'+(1-m)v"=constant. Since d]=-4>d6-pdv, we have also the relations dJ′/dv=−p=dJ″/dv″, at constant temperature. Putting dJ/dm=0 at constant volume, we obtain as the condition of equilibrium of the two states J′+p′v′=J″+p″v″. This may be interpreted as ​the equation of the border curve giving the relation between p and 0, but is more easily obtained by considering the equilibrium at constant pressure instead of constant volume.
(b) The second case, which is of greater practical utility, is that in which the external pressure, p, is kept constant. In this case dW=pd1/=d({>v), a perfect differential, so that the external work done is known from the initial and final states. In any possible transformation d(0¢-E) cannot be less than d(pv), or the function (E-6¢+pv) =G cannot increase. The condition of stable equilibrium is that G should be a minimum, for which reason it has been called the “thermodynamic otential at constant pressure.” The product po for any state such as D in Hg. 1 is represented by the rectangle MDdO, bounded by the isopiestic and the isometric through D. The function G is represented by the negative area 6”DM under the isothermal, bounded by the isopiestic DM and the axis of pressure. The increment of 0¢ is always greater than that of the total heat F=E+{>v, except in the special case of an equilibrium change at constant temperature and pressure, in which case both are equal to the heat absorbed in the change, and the function G remains constant. This is geometrically obvious from the form of the area representing the function on the indicator diagram, and also follows directly from the first law. The simplest application of the thermodynamic potential is to questions of change of state. If ¢', E', v'; and 4>", E", v", refer to unit mass of the substance in the first and second states respectively in equilibrium at a temperature 0 and ressure p, the heat absorbed, L, per unit mass in a change from the first to the second state is, by definition of the entropy, equal to 0(¢”-¢'), and this by the first law is equal to the change of intrinsic energy, E”—E', plus the external work done, p(v'-v'), Le. to the change of total heat, F'-F. If G' and G” are the values of the function G for the two states in equilibrium at the same pressure and temperature, we must have G'=G". Assuming the function G to be expressed in terms of p and 0, this condition represents the relation between p and 6 corresponding to equilibrium between the two states, which is the solution of the relation (v"—'v')dp/d0=L/0, (5). The direct integration of this equation requires that L and v”-'v' should be known as functions of p and 0, and cannot generally be performed. As an example of one of the few cases where a complete solution is possible, we may take the comparatively simple case equation (17), already considered, which is approximately true for the majority of vapours at moderate pressures.
Writing formulae (31) and (33) for the energy and entro y with indeterminate constants A and B, instead of taking them fietween limits, we obtain the following expressions for the thermodynamic functions in the case of the vapour:-
    ¢>” =S0log,0-R log.p-nop/0-1-A” .... (34)
    E”=s¢,0-nog:-l-B' ....... 35
    F; =S00-(n-l-1)cp+bp+B” ..... (36)
    G =S00(I-log,0)+R0log, p-(c-b)p-A"6-l-B" . (37)
    J”=s0¢9-S00log,0-|-R0log, p-A"0-I-B" . . . (38)
The function J” may be expressed in terms of 0 and 'U by writing for p its value, namely, R0/(v-i-c-b). We have also in any case the relations
    dG”/d9 (F const) =¢" =dJ”/d0 (v const) . . (39)
    dG”/dp 0 const) =z1, dJ”/dv (6 const) =p . . (40)
And all the properties of the substance may be expressed in terms of G or J and their artial differential coefficients. The values of the correspond in f)unctions for the liquid or solid cannot be accurately expresseti as the theoretical variation of the specific heat is unknown, but if we take the specific heat at constant pressure 5' to be approximately constant, and observe the small residual variation dh of the total heat, we may write
    F' =s'0+dh-}-B' ..... (41)
    41' = s”l0g,0 -l-dda +A ' .... . (42)
    G' = S'0(I -loged) -l-(dh -0d4>) - A '0-l-B' . . (43)
where dd> is the corresponding residual variation of ¢', and is easily calculated from a table of values of h.
To find the border curve of equilibrium between the two states, iving' the saturation pressure as a function of the temperature, we have merely to equate the values of G' and G". Rearranging the germs, and dividing throughout by 0, we obtain an equation of the form
    R l0g.P=/1-B/9-(S'-5o)10E¢0+(0-bl?/0+(dh/9-d¢) - (44)
in which B=B"-B', and A =A”-A'-l-5'-Sq. The value of A is determined by observing the value of 00 at some known pressure po, e.g. at the boiling-point. The value of B is determined by observing the latent heat, L0= F "0-F '0, which gives
    B =B”'B' =Lo'l'(-Y'"So)90'l'(7l+1)CoPo"bPo'l'dh0 ' » (45)
This constant may be called the absolute latent heat, asitexpresses the thermal value of the change of state in a manner independent of temperature.
The term (dh/0-d¢) depending on the variation of the specific heat of the liquid may be made very small in the case of waterby a proper choice of the constant s'. It is of the same order as the probable errors of observation, and may be neglected in practice. (See VAPoR1zATION, ,§ 16.) The expression for R logp for an imperfect gas of this type differs .from that for a perfect gas only by the addition of the term (0-b)p/0. This simple result is generally true, and the corresponding expressions for G" and J” are valid, provided that c-b in formula (17) is a function of the temperature only. It is not necessary to suppose that c varies inversely as the nth power of the' temperature, and that b is constant, as assumed in deducing the expressions for 4>, E, and F.
Although the value of G in any case cannot be found without that of 4>, and although the consideration of the properties of the thermodynamic potential cannot in any case lead to results which are not directly deducible from the two fundamental laws, it affords a convenient method of formal expression in abstract thermodynamics for the condition of equilibrium between different phases, or thetcriterion of the possibility of a transformation. For such purely abstract purposes, the possibility of numerical evaluation of the function is of secondary importance, and it is often possible to make qualitative deductions with regard to the general nature of a transformation without any knowledge of the actual form of the function. A more common method of procedure, however, is to infer the general relations of the thermodynamic potential from a consideration of the phenomena of equilibrium.
As it would be impossible within the limits of this article to illustrate or explain adequately the applications which have been made of the principles of thermodynamics, it has been necessary to select such illustrations only as are required for other reasons, or could not be found elsewhere. For fuller details and explanations of the elements of the subject, the reader must be referred to general treatises such as Baynes's Thermodynamics (Oxford), Tait's Thermodynamics (Edinburgh), Maxwell's Theory of Heat Xliondon), Parker's Therrnodynarmks (Cambridge), Clausius's mechanical Theory of Heat (translated by Browne, London), and Preston's Theory of Heat (London). One or two chapters on the subject are also generally included in treatises on the steam engine, or other heat engines, -such as those of Rankine, Perry or Ewing. Of greater interest, particularly from a historical point of view, are the original papers of joule, Thomson and Rankine, some of which have been reprinted in a collected form. A more complete and more elaborate treatment of the subject will be found in foreign treatises, such as those of Clausius, Zeuner, Duhem, Bertrand, Planck and others.
Alphabetical Index of Symbols Employed.
    0, Thermodynamic 'or absolute temperature.
    ¢, Entropy. Section 13.
    b, Covolume of molecules of gas. Equation (17).
    c, co, Co-a gregation, volume per unit mass. Equation (17).
    e, Base of blapierian logarithms.
    E, Intrinsic energy per unit mass. Section 2.
    F =E -l-gil, Total heat. Section 7.
    G, J, T ermodynamic potential functions. Section 15.
    H, Quantity of heat (in mechanical units). Section 2.
    K, ,, Adiabatic and 'isothermal elasticities. Equation (7).
    L, Latent heat of fusion or vaporization. Equation (5).
    M, Molecular weight. Section 8.
    m, Mass of substance or molecule.
    n, Index in expression for:L Equation (17).
    p, Pressure of fluid. po, Initial pressure.
    R=S0-st., Constant ingas-equation (17).
    S, Specific heat of gas at constant pressure.
    So, Limiting value of S when p=0. Section 12.
    S, Specific -heat of"gas at constant volume.
    so, Limiting value of s when p=o. Section 12.
    s', s”, Specific heat under other conditions. Equation (5).
    U, Kinetic energy of flow of fluid. Section Io.
    u, Mean velocity of gaseous molecules. Section 8.
    V=R0/p, Ideal volume of gas per unit mass. Equation (17).
    v, Specific volume of fluid, reciprocal of density.
    W, External work done by fluid.
(H. L. C.)