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# Andrews Fig 2-A James Thomson Isothermal B Isothermals of Co

## saturation, equation, critical, pressure, temperature, pressures, liquid, values, calculated and line

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FIG. 2.-(A) JAMES THOMSON ISOTHERMAL; (B) ISOTHERMALS OF CO, (ANDREWS) zontal tangent as shown in Andrews' diagram, fig. 'B. The con dition for three equal roots at this point gives the critical rela tions, as given by van der Waals, omitting the reduction factor a. These relations give the values of V, P and T, at the critical point in terms of the constants A, b and R, or vice versa. It will be seen that the results depend mainly on the value assigned to b, which is very small, and could not be determined satisfactorily from Regnault's observations at comparatively low pressures. The value of the constant A was taken from Regnault's observations, following Rankine's procedure, but that of b was selected to fit the critical temperature, T=304°, which was known with a fair degree of accuracy. The calculated values of the critical pressure and volume did not agree very well with Andrews' observations, but these were somewhat uncertain. The qualitative agreement was incredibly perfect in all essential respects with the James Thomson theory, and the equation of van der Waals was immedi ately accepted as the true and final solution of the problem.

## Maxwell's Theorem.

The continuous isothermals repre sented by equation (7) could easily be plotted on the PV diagram as shown in fig. 2, but an exact quantitative comparison was diffi cult, because van der Waals' theory, as originally presented, gave no indication of the manner in which the line BCD, representing the saturation pressure, should be drawn in relation to the con tinuous isothermal. This question was first solved by J. C. Max well (1875), who showed by a simple application of Carnot's principle, that the work of vaporization must be the same along either path, or that the line BCD at each temperature must cut off positive and negative loops of equal area from the continuous isothermal. The work of vaporization along BCD is simply p(V-v); that along the curve is the integral of pdV from (7) between the same limits. The integration is easy, and gives the condition, Fig. 3 shows the saturation lines correctly drawn in accordance with Maxwell's rule in relation to the isothermal curves given by (7) for the temperatures o°, 1o°, 20° and 3o° C. Below o° down to —50° C, the saturation pressures given by (7) are marked on the saturation line aC, for the liquid. The dotted curve AC shows the saturation line for the liquid and the satura tion pressures as actually observed. Comparing the two curves, we see at once that the values of v calculated from (9) are all nearly i00% too large, and that the actual increase of saturation pressure between io° and 3o° C is nearly double that given by (7). This was not noticed at the time, because the saturation pressures were difficult to calculate from (9), and the volumes of the liquid were considered to be of little importance. Equation (7) gives a very simple expression for the internal latent heat regarded as being equivalent to the work done against the internal pressure when the volume is increased from v to V at constant temperature. We find immediately, A/v —A/V in work units. The work-equivalent of the latent heat L as ordi narily measured is obtained by adding to L1 the external work p(V-v). L could also be calculated directly from Clapeyron's first relation (see article on THERMODYNAMICS, equation 29) by inte grating T(dp/dt), from v to V, along the continuous isothermal given by (7). We thus obtain, in work units, a quite different

expression, namely, L= but it is easy to see that the two expressions for L are equal and exactly consis tent with Maxwell's condition (9). These expressions show that any error in v will entail a corresponding error in L as calculated from (7), as is found to be the case on comparison with actual measurements of L which have since become available.

Clausius' Equations for Carbonic Acid and Steam.—To deduce the equation of saturation pressure from (9) in the form of a relation between p and T, as in (4), it would be necessary to eliminate V and v with the aid of (7), which is satisfied by both. This proved to be extremely difficult, as is often the case with transcendental equations. Clausius (Phil. Meg. 1882) first succeeded in finding a practical solution in the form of a table of corresponding states, giving the values of p, V and v, as frac tions of the critical pressure and volume, in terms of the tempera ture expressed as a fraction of the critical temperature. The table in this form could be applied to any equation of the type (7) provided that A were a function of the temperature only. If A were constant, as assumed by van der Waals, the saturation pres sures given by the table could not be made to agree at all with observation. But by choosing a suitable expression for A as a function of the temperature, the saturation pressures could always the vapour, V; the middle root C corresponding to the unstable intermediate state imagined by James Thomson. The isothermal curve shown in the figure IA, corresponds roughly with that given by equation (7) at 21.5° C, or T= 294.6. As the temperature is raised towards the critical point at 31-1° C, the line BCD becomes rapidly shorter, and the three roots finally coalesce at the critical point, where the isothermal has a point of inflection with a hori be fitted as closely as desired, if the function were sufficiently complicated. In the case of in order to make the saturation pressures as given by (7) agree with those observed by Andrews over the very limited range of his experiments, it sufficed to replace Rankine's T in the expression for the internal pressure, making it as in (5). In order to make the equation represent the volumes of the liquid more closely, Clausius also found it neces sary to replace in this term by the empirical con stant b" being chosen to make the calculated value of v agree with observation at 20° C. The equation thus modified and FIG. 4 often required at the present day in the use of for refrig erating plant), as it would make the saturation pressure about 5o per cent too small, and the volume of the liquid about 35 per cent too small at —5o° C. Clausius could undoubtedly have made a much better approximation to the saturation pressures, if accu rate observations of p had been available over a wider range, but the volumes of the liquid, and the values of the latent heat (which are most important for practical purposes) would still have been most unsatisfactory.

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