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Rh in which the suffix (c) indicates the values of T, p, and V, at the critical point. Taking the values, ^ = 72-9 atmos. = 1071 Ib./sq.in., at T e = 304-6 we find,

b = 0-0156 cub.ft./lb. =0-974 cc./gm., V = 2-92 c.c./gm. c c = 0-05265" "=3-287

C, = 0-0655 ' " =4-090

With these values of the constants, equation (34) represents the observations of Jenkin and Pye on H, S, and C, very satisfactorily, but the theoretical expressions, applying to any equation of the Van der Waals' type, are somewhat complicated and inconvenient for practical use, namely,

H=S m <+B- 3 cRT/V+&RT/(V-&),. . . (36) SC = a[ 3 c-6VV(V-6) 2 ]/[V 2 /(V-6) 2 -2 C /V]. . . (37)

. . . (38).

It will be observed that the value of the critical volume V f is too large to reconcile with the observed value 2-15 c.c./gm. given by Amagat. The value of b is also larger than the observed volume of the liquid at 5OC., and the equation does not represent the latent heat or the saturation pressures at all satisfactorily.

Equation of Saturation Pressure. Maxwell was the first to show how the saturation pressure could be calculated at any tempera- ture from the continuous isothermal of James Thomson (see 27.898), as represented by Van der Waals' equation. According to Carnot's principle, that no work can be obtained from heat at constant temperature, the integral of PdV along the continuous isothermal represented by the equation (34), must be equal to the external work of vaporization p(Vv) between the limits V and v; and the latent heat of vaporization must be equal to the integral of aT(dP/d'T) v dV, between the same limits, at constant T. Applying these conditions to Van der Waals' equation, in which c varies as I/T, we obtain,

= log.(V-&)-Iog.(- b)+c/V-c/v. . . (39) = log e (V-6)- log. (-&), . . . (40).

These give the increase of p between o and 3OC. only half the observed value, and the calculated value of p at 5OC. is more than twice too large. The calculated value of L at oC. is less than half the observed value, showing that Van der Waals' theory re- quired serious modification.

The equation of Clausius (Phil. Mag., 13, p. 132, 1882) for CO 2 is still most commonly quoted. He reverted to Rankine's assump- tion for the variation of c, but introduced an additional empirical constant b" in the denominator of the term representing the effect of coaggregation on the density,

aP/RT = i/(V-6')-<:/(V+6") 2, . . . (41).

This has the effect of reducing the value of V for any given values of P and T by the constant quantity b", but makes no difference to any of the other properties in terms of P and T. The values of c and b as deduced from P c and T c remain unaltered, but b' = b b". Clausius selected 6" to make the volume of the liquid agree with observation at 2OC., but the slope of the curve is unaltered, and the calculated value of by (41) is 26 % too small at 5OC., whereas by (34) it is 40% too large. The calculated values of p and L at the same point by either equation are 34 % too small for p, and 37 % too large for L, if Maxwell's theorem is employed. But it is unjusti- fiable to apply Maxwell's theorem to an equation which represents the properties of the liquid so badly, and it may be doubted whether the theorem is strictly applicable to an unstable transformation, such as that required by the James Thomson isothermal. It is always possible to choose the variation of c to fit the saturation pressures, but this is purely empirical, and fails in other respects.

Since the application of Maxwell's theorem is doubtful and diffi- cult in any case, it seems preferable for practical purposes to calcu- late the saturation pressures, as in the case of steam (see 27.903), by combining an equation of the type (32) for the liquid with a suitable expression for V. This method, as applied by Callendar (Properties of Steam, p. 186), seemscapable of giving very accurate values of p, without upsetting the agreement with H and V, or introducing intolerable complications in the theoretical expressions, such as have frequently been proposed by mathematicians. It may fairly be regarded as confirming the correctness of the principles applied in the case of steam, and the exact definition of formula (32) for the total heat of the liquid, on which the result mainly depends.

Critical Relations. -The critical point is most conveniently defined, especially in the case of transcendental equations, by the conditions, (dP/ = o, and ( 2 ) ( = o. . . (42)

which imply that the isothermal elasticity becomes zero of the second order, vanishing without change of sign at the critical point. Apply- ing these conditions to the equation of Dieterici (Ann. Phys., 5, p. 51, 1901)

oP(V-&) = RTe-'/,. . . (43) we obtain, showing that it gives a value of the ratio of the critical volume to the ideal volume agreeing better with experiment than that found from Van der Waals' equation in (35). There are, however, many other conditions to be satisfied which limit the possible choice of equations. The general expressions for S and SC, which are as follows :

SC = (dH/dP) t =-(dH/dV),/dP/dV),, . . . (44) S = (dH/dT) p = (dH/dT),+SC(dP/dT)v, . . . (45)

show that SC and S become infinite of the second order at the critical point, but that C remains finite and becomes equal to the reciprocal of the pressure coefficient (dP/dT)* to the second order of small quantities.

Similarly, if we take the Joule-Thomson equation,

SC=aT(dV/dT) p -aV. . . (46)

and divide by S = aT( p (<2P/ we obtain,

C a =(dT!dp-)t-aV/S. . . (47)

which shows that the cooling-effect CH at constant H becomes equal to the cooling-effect C(j> in adiabatic expansion when aV/S becomes zero of the second order at the critical point. The three cooling-effects, CH, C,,, and Ccf. which are the reciprocals of the pressure coefficients, and are most easily measured for any sub- stance, or deduced from any assumed type of characteristic equa- tion, remain finite and become equal, to the second order of small quantities, at the critical point.

Again, if we take the general expression for the latent heat accord- ing to Maxwell's theorem, we see that the latent heat is equal to the product of v) by the mean value of (dP/aT), between V and v. Comparing this with Clapeyron's equation, L,=aT(dp/dt) (Vv), we observe that (dPjdT), must become equal to dp/dt, to the first order of small quantities, at the critical point, which affords a useful test of any type of equation, because the coefficient dp/dt is readily observed.

These simple conditions, which seem to have been overlooked, are fatal to most of the equations which have been proposed. For instance, the equation of Clausius for COi requires (dP/dT), to be equal to 7P/T at the critical point. But the observations of Amagat show that dp/dt = 6-$p/T at this point, so that we should expect to find some difficulty in reconciling the equation of Clausius with the saturation pressures, as already indicated. The equation of Dieterici (43) gives a very satisfactory representation of the cooling-effect, to which it has often been applied, provided that the quantity c is assumed to vary as i/T " 2 . But this gives a value only 4?/T for the coefficient (dP/OT),, at the critical point in place of 6-sP/T, so that it would be quite impossible to represent the saturation pres- sures consistently. Most of the equations which have been pro- posed are modifications of the cubic type of Van der Waals, but are too complicated and empirical to serve as a satisfactory basis for the physical interpretation of the phenomena of the critical state. There is an almost infinite variety of possible types if transcen- dental functions are introduced. Many of these will be difficult to manipulate, but, in spite of the complexity of the conditions to be satisfied, we need not despair of arriving ultimately, by a process of elimination, at some form which is in reasonable agreement with experiment and at the same time sufficiently simple to be intelligible.

REFERENCES. In addition to works cited in the earlier articles, the following may be recommended. On the practical side, Sir J. A. Ewing's Mechanical Production of Cold and Thermodynamics for Engineers (1920); on the theoretical side, H. S. Carslaw, Fourier's Scries and Integrals and J. H. Jeans, Dynamical Theory of Gases. For experimental details it is always necessary to refer to the original papers, but Physical and Chemical Constants by G. W. C. Kaye and T. H. Laby (1921) gives a very handy and up-to-date summary of numerical results. (H. L. C.) BEATON, SIR JOHN HENNIKER, 1ST BART. (1848-1914), English postal reformer, was born at Rochester, in Kent, May 18 1848, the son of Lt.-Col. Heaton. fie was educated at Kent House grammar school and King's College, London. In 1864 he went to Australia and became a landowner and newspaper proprietor in New South Wales. He returned to England and entered the House of Commons as M.P. for Canterbury in 1885, retaining the seat until 1910. All his energies were devoted to postal reform. He advocated penny postage throughout the British Empire, and lived to see it achieved and extended to the United States. He also promoted cheaper oceanic telegraphy, and many other postal reforms. He died at Geneva Sept. 8 1914.

See Life and Letters of Sir John Henniker Heaton, Bart., by his daughter, Mrs. Adrian Porter (1916). HEDIN, SVEN ANDERS (1865- ), Swedish geographer and explorer, was born at Stockholm Feb. 19 1865. He was educated at Stockholm and Upsala universities, and afterwards studied in Germany at Berlin and Halle. In 1885-6 he made a year's journey through Persia and Mesopotamia, and in 1890 was attached to the special embassy sent by King Oscar of Sweden to the Shah of Persia. The same year he visited Khorasan and Turkestan. Sven Hedin is, however, best known for his explorations in Tibet, which place him in the first rank of modern