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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=Re, 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 p. The adiabatic elasticity is equal to y p, where -y is the ratio S/s of the specific heats. The heat absorbed in isothermal expansion from vo to v at a temperature 0 is equal to the work done by equation (8) (since d0 =o, and 0(dp/d0)dv =pdv), and both are given by the expression RO log e (v/vo). The energy E and the total heat F are functions of the temperature only, by equations (9) and (I I), and their variations take the form dE = sdO, d F = Sd0. The specific heats are independent of the pressure or density by equations (to) 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 00 to 0 may be written E - Eo = s(0 - 0 0), F - Fo = S(0-00). In thiscase the ratio of the specific heats is constant as well as the difference, and the adiabatic equation takes the simple form, pv v = constant, which is at once obtained by integrating the equation for the adiabatic elasticity, - v(dp/dv) =yp.
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 0, s= so(I +aO), the adiabatic equation takes the form, s 0 log e OW +aso(0 - Oo) +R loge(v/vo) =o
(14) where (00,v), (e 0, vo) are any two points on the adiabatic. The corresponding expressions for the change of energy or total heat are obtained by adding the term 2as 0 (02-002) to those already given, thus: E - Eo = so (0-00) + 2 aso (02-002), F - Fo=S0(0-00) + zaso (02-002), where So= so+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 pv 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 sufficient, 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 pv = f (0), where f (0) is any arbitrary function of 0, 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=o at constant temperature. This gives by equation (9) the condition Odp/d0 =p, which is satisfied by any substance possessing the characteristic equation p/0=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.
to. Method of Joule and Thomson. - William Thomson (Lord Kelvin), who wars 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, p'v', done on it by the fluid behind and carries its energy, E'+ 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), Sd0 = (Odv/d0 - v) d p,. ... (15) where d0 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, d0 = o, and we have the condition Odv/d0 =v. The characteristic equation of the fluid must then be of the form v/0=f(p), where f(p) is any arbitrary function of p. If the fluid is a gas also obeying Boyle's law, pv = f (0), then it must be an ideal gas. As the result of their experiments on actual gases (air, hydrogen, and C02), Joule and Thomson (Phil. Trans., 1854, 1862) found that the cooling effect, do, 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 d0/dp was nearly constant for each gas over a range of pressure of five or six atmospheres. By experiments at different temperatures between o° and 00° 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 d0/dp=A/0 2 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/0 =f(p) - SA /30 3 , in which f(p) is an arbitrary function of p. Assuming that the gas should approximate indefinitely to the ideal state pv = R0 at high temperatures, they put f(p)=Rip, which gives a characteristic equation of the form v= Re/p - SA /30 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 do/dp at higher pressures, which is otherwise rendered probable by the form of the isothermals 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 C02, and a more rapid rate of variation with temperature.
I I. Modified Joule-Thomson Equation. - G. A. Hirn (Theorie Mec. de la Chaleur, ii. p. 211, Paris, 1869) proposed an equation of the form (p+po)(v - b) =RO, in which the effect of the size of the molecules is represented by subtracting a quantity b, the " covolume," from the volume occupied by the gas, and the effect of the mutual attractions of the molecules is represented by adding a quantity po, 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, po, 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=RO/p - co(Oo/O)=V - c,.. (17) in which c is a small quantity (expressing the defect from the ideal volume V =Re/p due to co-aggregation of the molecules) which varies inversely as the nth power of 0, but is independent of p to a first approximation at moderate pressures. The constant co is the value of c at some standard temperature oo. 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 JouleThomson equation. For steam between Ioo° 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 41/2 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 Re/p. Differentiating equation (17) at constant pressure to find dv/do, and observing that dcldO= - nc/O, we find by substitution in (is) the following simple expression for the cooling effect do/dp in terms of c and b, Sdo/dp= (n+I)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 (I) p=RO/v±f(v), which makes p a linear function of 0 at constant volume, as in van der Waal's equation; (2) v=RO/p+f(p), which makes v a linear function of 0 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.
dv/dO (p const) = (R/p) (I +nc/V).. | (19) |
d 2 v/d0 2 „ _ - n(n+ I)c/e 2. . | 20) |
dp/do (v const) = (R/V) (I +nc/V). | 21) |
d 2 p/d0 e „ =Rnc(I - n+2ne/V)/0V 2. | (22) |
d(pv)/dp(o const) = b - c | . (23) |
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: - Substituting these values in equations already given, we find, from (6) S - s =R(I +nc/V)2 (24) „ (9) dE/dv (o const) =ncp/V . (25) „ (11) dF/dp „ = n+i)c - b . (26) „ (10) ds/dv (I - n+2nc/V)Rnc/V2 (27) „ (12) dS/dp „ =n(n+I)c/e. ..) 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 =Re/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 So, so, these constant limiting values at p=o, we may obtain the values at any pressure by integrating the expressions (27) and (28) from co to v and from o to p respectively. We thus obtain S=So±n(n+I)pc/O. s= so+ (n - I - nc/V)ncp/o. 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 poOo to po is readily found by substituting these values of the specific heats in equations (II) or (13), and integrating between the given limits. We thus obtain E - g o= s 0 (B-o)) - n(pc - poco) (31) We have similarly for the total heat F = E + pv, F - Fo=So(O - oo) - (n+1)(cp - copo)+b(p - p.).
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 diatomic 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-= so/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 0 from one adiabatic 4' (Fig. I) to any other adiabatic 0", the quotient H/o of the heat absorbed by the temperature at which it is absorbed is the same for the same two adiabatics whatever the temperature of the isothermal path. This quotient is called the change of entropy, and may be denoted by (4,"-0'). In passing along an adiabatic there is no change of entropy, since no heat is absorbed. The adiabatics 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, 0, at which it is acquired, and taking the sum or integral of the quotients, dH/o, so obtained.
(29) (30) The expression for the change of entropy between any two states is found by dividing either of the expressions for dH in (8) by 0 and integrating between the given limits, since dH/B 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 0-00 =s log e0/00. If the substance at the temperature 0 undergoes a change of state, absorbing latent heat, L, we have merely to add the term Lie to the above expression. In the case of an ideal gas, dp/d9 at constant volume =R/v, and dvld6 at constant pressure =R/p; thus we obtain the expressions for the change of entropy 0-4)0 from the state poeovo to the state pev, log e e/eo+R logev/vo =S log e 9/00-R (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 4-¢o = Sologe0/00 - R logep/po-n(cp/B-copo/Bo)
(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, 4, diagram represent quantities of heat in the same way as areas on the indicator diagram represent quantities of work. The 0, 4) 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 gas into a vacuum or against a pressure less than its own. In this case the work of expansion, pdv, is expended in the first instance in producing kinetic energy 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 isentropic, 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 diagram 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 to. 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 isentropic path, dE= -pdv. As the section of the tube varies, the change of kinetic energy of flow, dU, is represented by 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) = o; 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, d0, must be equal to dH/9. If the change is not reversible, but the final state is the same, the change of entropy, do, is the same, but it is no longer equal to dII/B. By Carnot's principle, in all irreversible processes, dH/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=o, 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 any 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 Od4 cannot be less than dH, the difference (61d4-dE) cannot be less than dW. This inequality holds in all cases, but cannot in general be applied to an irreversible change, because Od4 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 transformation is conducted in an isothermal enclosure, a common condition easily realized in practice, the temperature at the end of the transformation is reduced to its initial value throughout the substance. The value of Od4 is then the same as d(64), which is a perfect differential, so that the condition may be written d(46-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=o, the value of the function (00-E) cannot diminish, or (E--94,) 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 94, may be represented at any point such as D in Fig. I by the whole area B"DZ'VO under the isothermal 9"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 90-E is represented by the area 9"DdO to the left of the isometric Dd under the isothermal B"D. The increment of this area (or the decrement of the negative area E--04) 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 done in adiabatic expansion. The function J = E-94,, has been called the " free energy " of the substance by Helmholtz, and 90 the " bound energy." These functions do not, however, represent energy existing in the substance, like the intrinsic energy; but the increment of 90 represents heat supplied to, and the decrement of (E-04) 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 condition of minimum potential energy as the criterion of stable equilibrium in statics.
As an example, we 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 and if a mass m is in the state v', and 1-m in the. state v", the value of J for unit mass of the mixture is mJ' + (1m) This must be a minimum in the state of equilibrium at constant temperature. Since the volume is constant, we have the condition mv'--l-(I-m)v"=constant. Since dJ=-4d9-pdv, we have also the relations dJ'/dv' = - p = dJ"/dv", at constant temperature. Putting dJ /dm =o 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=pdv=d(pv), a perfect differential, so that the external work done is known from the initial and final states. In any possible transformation d(D4 - E) cannot be less than d(pv), or the function (E - D4)+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 potential at constant pressure." The product pv for any state such as D in fig. r is represented by the rectangle MDdO, bounded by the isopiestic and the isometric through D. The function G is represented by the negative area D"DM under the isothermal, bounded by the isopiestic DM and the axis of pressure. The increment of 00 is always greater than that of the total heat F=E+pv, 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 0', 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 pressure 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(4)"-4)'), and this by the first law is equal to the change of intrinsic energy, E" - E', plus the external work done, p(v" - v'), i.e. 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 0 corresponding to equilibrium between the two states, which is the solution of the relation (v" - v')dp/dO=L/D, (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 (3r) and (33) for the energy and entropy with indeterminate constants A and B, instead of taking them between limits, we obtain the following expressions for the thermodynamic functions in the case of the vapour: " =Solog e 0 - R log e p - ncp/D+A". E"=s:0 - ncp+B".. F" =S 0 0 - (n+1)cp+bp+B„ dG"/dO (p const) =0" = dJ"/dB (v const).. (39) dG"/dp (D const) =v, dJ"/dv (0 const) = p. (40) And all the properties of the substance may be expressed in terms of G or J and their partial differential coefficients. The values of the corresponding functions for the liquid or solid cannot be accurately expressed, as the theoretical variation of the specific heat is unknown, but if we take the specific heat at constant pressure s to be approximately constant, and observe the small residual variation dh of the total heat, we may write F'=s'D+dh+B'. 4)' =s'loge0+d4)+A'.. G' =s'e(i - log e 0) +(dh - Dd4) - A'D+B'. where do is the corresponding residual variation of 0', and is easily calculated from a table of values of h. To find the border curve of equilibrium between the two states, giving the saturation pressure as a function of the temperature, we have merely to equate the values of G and G". Rearranging the terms, and dividing throughout by 0, we obtain an equation of the form R log ep= A - B/D - (s' - So)loge0+(c - b)p/D+(dh/D - d4)) (44) in which B=B" - B', and A = A'+s' - So. The value of A is determined by observing the value of Do at some known pressure po, e.g. at the boiling-point. The value of B is determined by observing the latent heat, Lo = F"o - F'0, which gives B =B" - B' =L0+(s' - So)00+(n+r)copo - bpo+dho (45) This constant may be called the absolute latent heat, as it expresses the thermal value of the change of state in a manner independent of temperature.
The term (dh/0 - d4)) depending on the variation of the specific heat of the liquid may be made very small in the case of water by 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 Vaporization, § 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 (c - b) p/D. 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 cp, E, and F. Although the value of G in any case cannot be found without that of 0, 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 the criterion 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 (London), Parker's Thermodynamics (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.
0, Entropy. Section 13.
b, Covolume of molecules of gas. Equation (17).
c, co, Co-aggregation volume per unit mass. Equation (17). e, Base of Napierian logarithms.
E, Intrinsic energy per unit mass. Section 2.
F= E+pv, Total heat. Section 7.
G, J, Thermodynamic potential functions. Section 15.
H, Quantity of heat (in mechanical units). Section 2.
K, 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 c. Equation (17).
p, Pressure of fluid. po, Initial pressure.
so, Constant in gas-equation (17).
S, Specific heat of gas at constant pressure.
So, Limiting value of S when p=o. 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 To.
it, Mean velocity of gaseous molecules. Section 8.
V =RD/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.)
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