Neumann and Kopp

The law of Dulong atid Petit was extended to molecular substances by Neumann and Kopp, who suggested drat die heat capacity of a compound containing n atoms per molecule would be given by... 

As a further approximation, it may be assumed drat dre elements conform to Neumann and Kopp s rule that dre heat capacities of compounds and elements are about 25Jgatom Consequently the htral temperature, 7j, which is achieved at the reaction front is given by... 

For the crystal range 298.15 - 478.9 K, these estimated C values vary from 34.63 to 36.69 cal K fair agreement with the adopted values and were estimated by comparison with ZrCl (cr) and HfCl.(cr) estimated the heat capacity of NbCl based on Neumann and Koppe s rule, C 26.71 + 35.2 x lO T cal K... 

A useful approximation to estimate the possibility of a particular reaction which depends on internal heat generation to produce the products in the proper state for separation is to ignore the heat losses from die reactor, and assumes an average heat capacity calculated from die Neumann-Kopp law... 

The researches of F. Neumann (1831), Regnault (1840), and H. Kopp (1864), indicated that solid elements preserve unchanged their atomic heats when they unite to form solid compounds. Thus, the product molecular weight) X s )ecific heat) = (molecular heat) is composed additively of the atomic heats MC = niaiCi + h2 2c2 + n t s + (9)... 

If the elements are liquid in the standard state specified by the formation reaction, the Neumann-Kopp rule (115) suggests that ACp = 0. For the standard formation reaction in which Kn = 0 at the compound melting temperature, the standard entropy of formation can be obtained from the temperature derivative of AGf°[ij] or, often more accurately, from a combination of values of AGf°[ij, Tmij] and AHf°[ij, Tmij]. When Kn = 1 at TmiJ , the value of the natural standard entropy of formation, A Sf°[ij, Tmij], must be corrected as follows ... 

As it follows from Table 4.1, the estimated heats of fusion of Na3pS04 and K3FSO4 are lower than the measured values and the error in the estimation lies within the range 7-13%. This difference is probably due to the limited validity of the Neumann-Kopp s rule, which in most cases, yield higher values of heat capacities of crystalline AB compounds in Eq. (4.19). 

Now however complicated the law of force between the atoms f may be, the force driving the atom back to its equilibrium position must, for sufficiently small displacements from rest, be proportional to its distance from that point all theories of elasticity, for example, operate with this assumption. Thence it follows that at sufficiently low temperatures the Law of Dulong and Petit, or its generalization by Kopp and Neumann, must hold not only for elements but also for compounds. 

In addition, the atoms within the coniines of the molecule will oscillate to and from one another at temperatures sufficiently high to exclude the quantum theory, each atom has an energy of vibration 3RT (law of Kopp-Neumann) at lower temperatures the mean energy of vibration per... 

Estimation of the Values of v,—The experimental and theoretical investigations described in this and the previous chapter have apparently led to a complete explanation of the law of Dulong and Petit, and of its generalization by Kopp and Neumann. A knowledge only of the one value of fiv, for very simply constructed crystals, or of the various... 

The curve given above (Fig. 18), which is taken from Fischer s paper, shows the course of the two energy curves up to about io° abs. there is practically perfect contact, and at about 20° the difference is only a little more than 1 pro mille. At first, as required by equations (41), U departs from the dotted common prolongation of the line of contact more rapidly than does A. But then, as a result of the law of Kopp and Neumann coming more or less into operation, U shows a tendency to become independent of temperature,... 

We must point out that the standard entropies of formation of 81)2X03 and Bi2T03 calculated from our data are very small. This shows that the two tellurides obey the Kopp-Neumann law. 

The relation between Neumann s and Petit and Dulong s laws was pointed out by Joule (1844) and independently by Alphonse Cornill Woestyn, who studied in the ficole Polytechnique and became director of a sugar factory in Russia." They showed that the molecular heat of a solid compound Aa-B C. .. is the sum of the atomic heats of its elements or if =sp. ht., a= at. wt., it is ZxUiCi. This was confirmed in extensive investigations by Kopp and Carl Pape. ... 

The entropy of fusion L/T) of completely disordered intermetallic phases can generally be calculated by fractional addition from those of the components. For the completely ordered state the term — 19.146 (Afi log Afi-F A 2 log A 2) is to be added to the calculated entropy of fusion [1.216-218,221,222]. The molar heat capacity of the homogeneous alloy phases and intermetallic compounds, as calculated approximately from the atomic heat capacities of the components using Neumann-Kopps rule, is obeyed to within 3% in the temperature range 0-500 °C in the Ag—Au, Ag—Al, Ag—Al, and Ag—Mg alloy systems. The heat capacities of heterogeneous alloys may he calculated hy fractional addition from those of the components hy the empirical relation Cp = 4.1816(a-F lO- i r-F 105cr-2)J/(Kmol) to satisfactory accuracy. 

Neumann - Kopp Rule. The heat capacity of a compound is equal to the sum of those of its components in the same standard state at the same temperature. This implies that there is no change in heat capacity on formation of the compound, and so the entropy of formation is zero - an assumption more likely to be valid for non-ionic compounds. 

The absence of data on the heat capacity of lanthanide dichlorides in the liquid state makes us use approximate estimation methods. We applied the Neumann-Kopp rule of summation, according to which the heat capacity Cp of a compoimd can be calculated from the weighted component contributions constituting it. Considering the formation of RCI2 from R and RCI3, we find that... 

Y-lron is stable between 910 and I400 C. It has the face-centered cubic structure which is the basis of the austenitic steels. Since pure Y >ron is not stable at low temperatures the above values were calculated by application of the Kopp-Neumann rule to experimental data on two austenitic Fe-Mn alloys and of uncertain accuracy. 

The absolute values and temperature dependence of the calculated heat capacity (Cp ) agree well with the actual heat capacity of LaCrOs, which indicates that the heat capacity of those oxides can be estimated by Neumann-Kopp s theory. The heat capacity data of LaMnOs (Kobayashi et al., 1993), LaCo03 (Horinouchi et al., 1981), and Lai j Srj Co03 (Barkhatova, et al., 1990) measured by a laser flash method are shown in fig. 14. The heat capacities of LaMnOs and Lai-jcSr cCoOs show peaks due to the phase trairsition from orthorhombic to rhombohedral stracture, however, the heat capacity of LaCoOs shows complicated temperature dependence due to the magnetic phase transitions. 

---