Heat-capacity rule

The melting point is defined as the temperature at which the liquid is in equilibrium with the solid so that the difference in Gibbs free energy between the two phases is zero. The entropy of fusion can then be expressed as. 

According to the rule, if the polymorph with the higher melting point has the lower entropy of fusion, the two modifications are enantiofropically related. If the lower-melting form has the lower entropy of fusion then the two forms are monotropically related (Burger 1982a,fe). 

For a pair of polymorphs, if the modification wifh fhe higher melting poinf also has a higher heat capacity at a given temperature than the second polymorph, then there exists an enantiotropic relationship between them. Otherwise, the system is 

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Another paper of Einstein s showed that quantization of energy also predicted that Dulong and Petit s heat capacity rule would only be valid at high temperatures. Assume that the only allowed energies are E = 0, hv, 2hv,... nhv, where n can be arbitrarily large. The average energy is... 

A relatively simple example of a group contribution technique is the method for estimating Hquid and soHd heat capacities (159). This method is a modification of Kopp s rule (160,161) which was originally proposed in 1864. Kopp s rule states that, at room temperature, the heat capacity of a soHd compound is approximately equal to a stoichiometric summation of the heat capacities of its atoms (elements). The Hurst-Harrison modified equation is as follows ... 

There are no reliable prediction methods for solid heat capacity as a function of temperature. However, the atomic element contribution method of Hurst and Harrison,which is a modification of Kopp s Rule, provides estimations at 298.15 K and is easy to use ... 

The heat capacity, the heat required to raise 1 g-mole drrough one kelvin, can be calculated at temperatures generally above 300 K by two simple empirical rules. The first of these, Dulong and Petit s rule, was discovered in the course of calorimeU ic smdies of the heat capacities of the elements and shows drat the heat capacity has a value,... 

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elemental solid, which reflected the vibrational energy of a tliree-dimensional solid, should be equal to 3f JK moH The anomaly that the free electron dreory of metals described a metal as having a tliree-dimensional sUmcture of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add anodier (3/2)7 to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas die quanmtii theory of free electrons shows that diese quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

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... 

If the cold and hot fluids heat capacity rates are equal, then R = 1. Equation (9.22) gives an indefinite value, and this equation cannot be used directly. Using I Hopital s rule as R —> 1 gives d... 

The deviation of the heat capacity of a solid solution from the heat capacity calculated by the additivity hypothesis (Kopp-Neumann rule), a quantity of importance for the evaluation of the... 

Its validity at normal temperatures was shown for more than 60 materials, ranging from pure metals to glassy polymers. Obviously, the polymers of the present study are good examples for Barker s rule. The product ot2E is linked to the difference of the two heat capacities c0 and cE, measured under constant stress and under constant strain, respectively [58], Also, a2E is linked to the difference of two Young s moduli Es, and ET measured adiabatically and isothermally [59]. 

To design the network for maximum energy recovery start at the pinch and match streams following the rules on stream heat capacities for matches adjacent to the pinch. Where a match is made transfer the maximum amount of heat. 

CrCl2 molecules have been isolated in solid inert-gas matrices and their i.r. spectrum has been recorded 33—1000 cm The isotopic shifts and i.r. selection rules indicate a linear 10°) structure. The d-d spectrum of gaseous CrClj has been discussed in terms of ligand field theory.The heat capacity of anhydrous CrCl3 in the temperature interval 2—20 K has been determined and the sublimation and decomposition pressures of the compound have been recorded. ... 

The specific heat capacity of a mixture can be estimated from the specific heat capacities of the different compounds by a mixing rule ... 

Figure 11.7 Calculation of TD24 as a function of the temperature at which the detection limit of lOWkg-1 is reached with different values of the activation energy of 50 kj mol-1 (solid line) 75, 100, and 150kjmof (dashed lines) and a specific heat capacity of 1 kjkg 1 K 1. The dotted line represents the 100 K-rule. ...

Most of the existing HEN synthesis methods rely on either heuristic rules (for example, pinch analysis method [2]) or mathematical programming (for example, simultaneous optimization approach [3-6]). And further, to some typical objectives considered in the HEN synthesis such as utility consumption, total number of matches, and total exchanger area, the flexibility of the HENs for feasible operation under possible variation of source-stream temperatures and/or heat-capacity flow rates has been emphasized in some recent articles [6-10]. For HEN synthesis, the analysis of this flexibility, defined as the size of the region of feasible operation in the space of desired or undesired deviations of pa-... 

EuRu4Sbn is metallic and becomes ferromagnetic for temperatures below 3.3 K (Takeda and Ishikawa, 2000b). The low temperature saturation moment is about 6.2/xb, 89% of the Eu+2 value. Low temperature heat capacity measurements indicate that the magnetic entropy removed due to magnetic order is also only about 90% of its expected value (Rln8). It is likely that the lanthanide site is not completely filled in this compound although mixed valence behavior can not be ruled out with the available data. 

Luria, M. Benson, S.W., "Heat Capacities of Liquid Hydrocarbons, Estimation of Heat Capacities at Constant Pressure as a Temperature Function Using Additivity Rules" J. Chem. Eng. Data 1977, 22, 90. 

It should be noted that experimental measurements of heat capacity at T > T2, as a rule, give the values decreased in a gradual manner up to zero (Fig. 1), but not nonzero heat capacity, that is caused by the existence of short-range order in crystal or by correlation in substitution of lattice sites by < > i. [Pg.224]

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