THE ENTHALPY-POROSITY METHOD WITH A RECTANGULAR SPECIFIC HEAT CURVE

2018 ◽  
Author(s):  
Wim Beyne ◽  
Bernd Ameel ◽  
Alvaro de Gracia ◽  
Gabriel Zsembinszki ◽  
Luisa F Cabeza ◽  
...  
Keyword(s):  
1969 ◽  
Vol 47 (14) ◽  
pp. 1485-1491 ◽  
Author(s):  
Neil Waterhouse

The specific heat of copper heated in hydrogen at 1040 °C has been measured over the temperature range 0.4 to 3.0 °K and found to be anomalous. The anomaly occurs in the same temperature range as the solid hydrogen λ anomaly which, in conjunction with evidence of ortho to para conversion of hydrogen in the sample, suggests the presence of molecular hydrogen in the copper. The anomaly reported by Martin for "as-received" American Smelting and Refining Company (ASARCO) 99.999+ % pure copper has been briefly compared with the present results. The form of the anomaly produced by the copper-hydrogen specimen has been compared with Schottky curves using the simplest possible model, that for two level splitting of the degenerate J = 1 rotational state of the ortho-hydrogen molecule.Maintenance of the copper-hydrogen sample at ~20 °K for approximately 1 week removed the "hump" in the specific heat curve. An equation of the form Cp = γT + (464.34/(θ0c)3)T3 was found to fit these experimental results and produced a value for γ which had increased over that for vacuumannealed pure copper by ~2%.


The exact solution of the three-dimensional Ising model of a ferromagnetic presents diffi­culties of a very fundamental nature. It therefore seems that the most reliable information on the behaviour of the model is provided by exact series expansions of the partition func­tion at low and high temperatures. However, the usual low -temperature expansion fails to converge in the neighbourhood of the critical point. By rearranging the terms of the series on the basis of physical considerations, it is possible to obtain a systematic set of successive approximations, each approximation taking exact account of clusters of a given size or less (metastable approximations). By extrapolation accurate estimates can be derived of the Curie point and critical values of the energy and entropy. It is found that there is a marked difference in behaviour between two- and three-dimensional lattices, a far larger proportion of the entropy change taking place in the temperature region below the Curie point in the latter case. The corre­sponding specific heat curves are therefore much closer to those observed experimentally. Finally, a brief discussion is given of the dependence of the specific heat curve on lattice structure.


In a recent article F. Hund has treated the problem of the specific heat of the hydrogen molecule on the basis of the wave mechanics. The total number of rotational states are divided due to the homopolar character of the molecule into two groups, to the one of which belong wave functions symmetrical in the two nuclei, and to the other wave functions which are antisymmetrical in the nuclei. Hund has suggested that the presence of both groups in hydrogen may be accounted for by assuming that the nuclei possess a spin, in which case transitions between symmetrical or between antisymmetrical states will have their usual intensity but transitions between symmetrical and antisymmetrical states will be very weak, of the order of the coupling of the nuclear spins. He then writes the following expression for the rotational specific heat, C r /R = σ 2 d 2 / d σ 2 log Q, Q = β [1 + 5 e -6σ + 9 e -20σ + ...] + 3 e -2σ + 7 e -12σ + 11 e -30σ +...., (1) where σ = h 2 /8π 2 I k T and β is the ratio of the weights of the symmetrical group of states to the antisymmetrical group. Hund has found that he obtains a close agreement between (1) and the observed specific heat curve only when β has about the value 2, that is when the symmetrical states have twice the weight of the antisymmetrical. He further obtains for this case I = 1·54 × 10 -41 gm. cm. 2 , the moment of inertia of the H 2 molecule.


1928 ◽  
Vol 24 (2) ◽  
pp. 290-290
Author(s):  
W. H. McCrea

In a recent paper in these Proceedings the writer suggested the possibility of a transition from one molecular form to another in CO2. The suggestion is embodied in the equation (10) and the resulting specific heats for low temperatures given. He greatly regrets that it was not till after those results were published that he found they gave a high and altogether impossible maximum in the specific heat curve for higher temperatures before it returns to the neighbourhood of the unmodified curve Cv′.


Although the heat capacity of iron at different temperatures has been the subject of a number of investigations in the past, it is only recently that iron of purity greater than 99.9 % has been available. Furthermore, in most previous determinations the property actually measured has been the total heat over a relatively large temperature range. Specific heats deduced from such measurements are liable to appreciable error, since if the total heat curve is smoothed, small fluctuations in the specific heat will be concealed, whereas if the actual observations are retained without smoothing, fluctuations which have no physical existence may appear in the result. Thus, suppose that the total heat is measured from 50 to 145 and from 50 to 155° C, the former being in error by 1 part in 1000 in excess and the latter the same amount in defect, the error in the specific heat over the range 145-155° C will be almost 2%. Evidently a real variation of 1 or 2% would be liable to pass unnoticed if any smoothing is undertaken, and conversely, fluctuations of this order may be introduced spuriously if the observations are used without smoothing. In general, calorimetry from high temperatures cannot be carried out to an accuracy of 1 part in 1000, and in any case, even this accuracy is insufficient at temperatures of the order of 1000° C.


The work described in this paper started from the researches upon the properties of aqueous solutions, which have occupied one of us for some years past. In the course of this work it had been found that the measurement of various physical properties of solutions, including density, conductivity, and viscosity, at various temperatures and various concentrations, threw considerable light, not only on the constitution of solutions, hut upon that of water itself and upon the amounts of water combined with a solute at various temperatures and concentrations. It seemed probable that similar series of observations upon the specific heat of solutions over considerable ranges of concentration and temperature would throw further light upon these matters, and the apparatus described in this paper was therefore primarily designed for the observation of the specific heat of solutions with the desired degree of accuracy, and with the facility and ease of manipulation which are essential when it is required to amass a large body of data in a reasonable time. At an early stage it became apparent that the temperature-specific heat curve of water was entirely altered in character by the introduction of a small amount of solute. With a half-normal solution of KCI the more or less parabolic curve for water becomes nearly a straight line, and even with fairly dilute solutions the water curve is greatly modified. The appreciation of this modification necessarily involved as a starting-point the consideration of the curve for pure water, as to the form of which different observers have come to widely different conclusions. A reference to fig. 10 (Section 14 post ), where the curves given by different observers are plotted, shows that the latest form of the curve, which is the result of the researches of Callendar and Barnes, differs widely from the curves given by Regnault and by Lüdin. At 80° C. the values of the specific heat of water in terms of the 15° calorie are.


The present paper is concerned with the detailed calculation of the partition functions for a two-dimensional quadratic lattice from the matrix derived previously. A brief discussion is given of the general methods of calculation available. The method here employed is the expansion of the eigenvector and eigenvalue as power series about zero temperature. This is readily applicable to finite matrices of quite high order, and, when a suitable notation has been introduced, to the infinite case. It is found more convenient to deal with the general problem of an unsymmetric net with different interactions in the two directions; for a ferromagnetic in the absence of a magnetic field a symmetry relation observed empirically enables terms to be derived successively, and it is assumed that this solution is equivalent to Onsager’s (1944). The method is applied in the presence of a magnetic field, and several terms of a generalized series are deduced. Several terms are hence obtained of a series for the spontaneous magnetization. An inspection of the generalized series leads one to conjecture that the specific heat curve becomes continuous in the presence of a magnetic field. A rearrangement of the terms of the generalized series enables several terms of the high-temperature expansion to be deduced. Finally, the results are applied to the theory of binary solid solutions; the solubility curve for the two substances is formally closely related to the spontaneous magnetization. The separation into two phases is established, and the corresponding specific heat singularities are analyzed.


This paper is a continuation of the previous investigation (Part II) on the vibrational spectrum of a crystal. The influence of the maxima of the density of the vibrations on the form of the θ D — T diagram is discussed in some detail. The main result is the discovery that more than one region of constant θ D value is possible—which is equivalent to the possibility of pseudo-T 3 regions in the specific heat curve. A further result is an explanation of the discrepancies hitherto found between the θ D values derived from thermal and from elastic data at low temperatures. 1—We shall start with an examination of the one-dimensional case which is important because it provides a striking example of the influence of the lattice structure on the specific heat curve.


1882 ◽  
Vol 14 (342supp) ◽  
pp. 5451-5452
Author(s):  
J. MacFarlane Gray
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