The Specific Heat of Solids

The following table gives the values of fd and y which have been found for a number of sohds. The highest temperature for which the equation is apphcable is given in the last column.  

Calculating from this table, we find the true specific heat of silver at 100° to be 0 0566. The mean specific heat between 

The specific heat varies most rapidly with the temperature in the case of non-metals of low atomic weight. Particularly is this true of carbon, above all in the non-conducting crystalline form of diamond. The specific heat of diamond increases from 01128 to 046, that is to say, about four-fold in the interval between 0° and 1000°. 

The smaller the atomic weight of the metal, the more rapidly the specific heat decreases with the temperature. The variation of the specific heat at very low temperatures will be discussed later on. 

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The specific heats of solids at low temperatures are appreciably less than at higher temperatures. A maximum specific heat has been observed in the case of iron at 740° and nickel at 320° (Lecher, 1908). Since these are the temperatures at which recalescence and loss of magnetic properties occur, the close relation of specific heat to molecular structure is evident. 

The problem has been largely worked at from both sides from the theoretical side the point of view has been almost exclusively that of the kinetic gas theory. It must be kept in mind, however, that it is possible that a purely mechanical theory may not be sufficient to cover the phenomena, as has recently appeared in the case of the specific heats of solids. 

This remarkable result has been verified by experimental measurements of specific heats at very low temperatures, viz., in liquid air and liquid hydrogen (cf. references in Chap. I.). It was formerly believed that the specific heats of solids approached small positive limiting values at the absolute zero, but the form of the curve at very low temperatures alters appreciably, and it may be inferred that the specific heat is vanishingly small at... 

Pollitzer (ZeiUchr. Elcktrochem., 17, 5, 1911 Berechmuig Chan. Affinitiiten nach Xernstschen Warmetheoi em, 1912) has carried out calculations with the new formula for the specific heats of solids ( 204), but this introduces nothing new in principle. 

The coffee-cup calorimeter can be used to measure the heat changes in reactions that are open to the atmosphere, qp, constant pressure reactions. We use this type of calorimeter to measure the specific heats of solids. We heat a known mass of a substance to a certain temperature and then add it to the calorimeter containing a known mass of water at a known temperature. The final temperature is then measured. We know that the heat lost by the added substance (the system) is equal to the heat gained by the surroundings (the water and calorimeter, although for simple coffee-cup calorimetry the heat gained by the calorimeter is small and often ignored) ... 

The specific heat of solid anhydrous Mg(N03)2 may be calculated from the equation ... 

If the specific heat of the metal is known, an approximate atomic weight can be determined. This can be done using the relationship between the specific heat of solid metallic objects and their atomic weights observed by Pierre Dulong and Alexis Petit in 1819 it is known as the Law of Dulong and Petit. 

Equation (6.4) is the general one for vapor pressure, and it shows that the undetermined constant in In P, in Eq. (5.9), is just the chemical constant that we have already determined in Eq. (3.16) of Chap. VIII. The simplest experimental method of finding the chemical constants is based on Eq. (6.4) one measures the vapor pressure as a function of the temperature, finds the specific heats of solid and gas, so that one can calculate the term in the specific heats, and computes the quantity... 

Yet another cause for deviation from Eotvos s law may be found in Bom and Courant s theory1 of the motions of the molecules of liquids, which follows the lines of Debye s theory of the specific heat of solids. Assuming three degrees of freedom for the motions of the molecules, they obtained good agreement with experiment for several liquids for which the constant is about 2-1. If the number of degrees of freedom is n, the constant is altered on their theory in the ratio (n/3). ... 

The specific heat of solid iodine, according to Nemst and Lindemann is given by... 

We shall show in the next paragraph that the vapour pressure constants play an important part in the calculation of chemical equilibria in gases. The first problem which Nernst had to solve after the discovery of his theorem was therefore the calculation of at least the approximate value of C for as many simple substances as possible. For this purpose he made use of the theorem of corresponding states, and assumed further that the specific heat of solid and liquid bodies diminishes to a small but finite value (viz. nx 1 5, where n is the number of atoms in the molecule) as the temperature is lowered. On the evidence of the measurements published up to that time he also assumed that the molecular specific heat of gases and vapours is a linear function of the temperature which approaches the value 3-5-l-7ixl-5 at very low temperatures. In this way he arrived at the vapour pressure formula... 

The plan of the following sections is as follows. First, we undertake an analysis of the energetics of a solid when the atoms are presumed to make small excursions about their equilibrium positions. This is followed by a classical analysis of the normal modes of vibration that attend such small excursions. We will see in this case that an arbitrary state of motion may be specified and its subsequent evolution traced. The quantum theory of the same problem is then undertaken with the result that we can consider the problem of the specific heat of solids. This analysis will lead to an explicit calculation of thermodynamic quantities as weighted integrals over the frequencies of vibration. 

The solution arrived at in our linear elastic model may be contrasted with those determined earlier in the lattice treatment of the same problem. In fig. 5.13 the dispersion relation along an arbitrary direction in g-space is shown for our elastic model of vibrations. Note that as a result of the presumed isotropy of the medium, no g-directions are singled out and the dispersion relation is the same in every direction in g-space. Though our elastic model of the vibrations of solids is of more far reaching significance, at present our main interest in it is as the basis for a deeper analysis of the specific heats of solids. From the standpoint of the contribution of the thermal vibrations to the specific heat, we now need to determine the density of states associated with this dispersion relation. 

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