The heat capacity

For systems in which no change in composition (chemical reaction) occurs, things are even simpler to a very good approximation, the enthalpy depends only on the temperature. This means that the temperature of such a system can serve as a direct measure of its enthalpy. The functional relation between the internal energy and the temperature is given by the heat capacity measured at constant pressure  

The difference between Cp and Cy is of importance only when the volume of the system changes significantly— that is, when different numbers of moles of gases appear on either side of the chemical equation. For reactions involving only liquids and solids, Cp and Cv are for all practical purposes identical. 

Heat capacity can be expressed in joules or calories per mole per degree (molar heat capacity), or in joules or calories per gram per degree the latter is called the specific heat capacity or just the specific heat. 

The greater the heat capacity of a substance, the smaller will be the effect of a given absorption or loss of heat on its temperature. 

In other words, the potential energy of a molecule depends on the time-averaged relative locations of its constituent electrons and nuclei. This dependence is expressed by the familiar potential energy curve which serves as an important description of the chemical bond between two atoms. 

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The heat capacity of an ideal vapor is a monotonic function of temperature in this work it is expressed by the empirical relation... 

Consider the simple flowsheet shown in Fig. 6.2. Flow rates, temperatures, and heat duties for each stream are shown. Two of the streams in Fig. 6.2 are sources of heat (hot streams) and two are sinks for heat (cold streams). Assuming that heat capacities are constant, the hot and cold streams can be extracted as given in Table 6.2. Note that the heat capacities CP are total heat capacities and... 

Taking the heat capacity of water to be 4.3 kJ kg K , heat duty on boiler feedwater preheating is... 

Heat carriers. If adiabatic operation produces an unacceptable rise or fall in temperature, then the option discussed in Chap. 2 is to introduce a heat carrier. The operation is still adiabatic, but an inert material is introduced with the reactor feed as a heat carrier. The heat integration characteristics are as before. The reactor feed is a cold stream and the reactor efiluent a hot stream. The heat carrier serves to increase the heat capacity fiow rate of both streams. 

Another important accomplislnnent of the free electron model concerns tire heat capacity of a metal. At low temperatures, the heat capacity of a metal goes linearly with the temperature and vanishes at absolute zero. This behaviour is in contrast with classical statistical mechanics. According to classical theories, the equipartition theory predicts that a free particle should have a heat capacity of where is the Boltzmann constant. An ideal gas has a heat capacity consistent with tliis value. The electrical conductivity of a metal suggests that the conduction electrons behave like free particles and might also have a heat capacity of 3/fg,... 

The value of at zero temperature can be estimated from the electron density ( equation Al.3.26). Typical values of the Femii energy range from about 1.6 eV for Cs to 14.1 eV for Be. In temis of temperature (Jp = p//r), the range is approxunately 2000-16,000 K. As a consequence, the Femii energy is a very weak ftuiction of temperature under ambient conditions. The electronic contribution to the heat capacity, C, can be detemiined from... 

The integral can be approximated by noting that the derivative of the Femii function is highly localized around E. To a very good approximation, the heat capacity is... 

As one raises the temperature of the system along a particular path, one may define a heat capacity C = D p th/dT. (The tenn heat capacity is almost as unfortunate a name as the obsolescent heat content for// alas, no alternative exists.) However several such paths define state functions, e.g. equation (A2.1.28) and equation (A2.1.29). Thus we can define the heat capacity at constant volume Cy and the heat capacity at constant pressure as... 

However, the possibility that might not go to zero could not be excluded before the development of the quantum theory of the heat capacity of solids. When Debye (1912) showed that, at sufficiently low... 

Each hamionic temi in the Hamiltonian contributes k T to the average energy of the system, which is the theorem of the equipartition of energy. Since this is also tire internal energy U of the system, one can compute the heat capacity... 

One can trivially obtain the other thennodynamic potentials U, H and G from the above. It is also interesting to note that the internal energy U and the heat capacity Cy can be obtained directly from the partition fiinction. Since V) = 11 exp(-p , ), one has... 

Fluctuations in energy are related to the heat capacity Cy and can be obtained by twice differentiating log Q with respect to p, and using equation (A2.2.69) ... 

If //is 00 (very large) or T is zero, tire system is in the lowest possible and a non-degenerate energy state and U = -N xH. If eitiier // or (3 is zero, then U= 0, corresponding to an equal number of spins up and down. There is a synnnetry between the positive and negative values of Pp//, but negative p values do not correspond to thennodynamic equilibrium states. The heat capacity is... 

This behaviour is characteristic of any two-state system, and the maximum in the heat capacity is called a Schottky anomaly. 

Once the partition function is evaluated, the contributions of the internal motion to thennodynamics can be evaluated. depends only on T, and has no effect on the pressure. Its effect on the heat capacity can be... 

The first mtegral is the energy needed to move electrons from fp to orbitals with energy t> tp, and the second integral is the energy needed to bring electrons to p from orbitals below p. The heat capacity of the electron gas can be found by differentiating AU with respect to T. The only J-dependent quantity is/(e). So one obtains... 

In typical metals, both electrons and phonons contribute to the heat capacity at constant volume. The temperaPire-dependent expression... 

Figure A2.2.6. Electronic contribution to the heat capacity Cy of copper at low temperatures between 1 and 4 K. (From Corak et al [2]).

This is of course an excess heat capacity, an amount in addition to the contributions of the heat capacities ( y... 

Although the previous paragraphs hint at the serious failure of the van der Waals equation to fit the shape of the coexistence curve or the heat capacity, failures to be discussed explicitly in later sections, it is important to recognize that many of tlie other predictions of analytic theories are reasonably accurate. For example, analytic equations of state, even ones as approximate as that of van der Waals, yield reasonable values (or at least ball park estmiates ) of the critical constants p, T, and V. Moreover, in two-component systems... 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

In 1953 Scott [H] pointed out that, if the coexistence curve exponent was 1/3, the usual conclusion that the corresponding heat capacity remamed finite was invalid. As a result the heat capacity might diverge and he suggested an exponent a= 1/3. Although it is now known that the heat capacity does diverge, this suggestion attracted little attention at the time. 

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

The brackets symbolize fiinction of, not multiplication.) Smce there are only two parameters, and a, in this expression, the homogeneity assumption means that all four exponents a, p, y and S must be fiinctions of these two hence the inequalities in section A2.5.4.5(e) must be equalities. Equations for the various other thennodynamic quantities, in particular the singidar part of the heat capacity Cy and the isothemial compressibility Kp may be derived from this equation for p. The behaviour of these quantities as tire critical point is approached can be satisfied only if... 

Similar equations apply to the extended scaling of the heat capacity and the coexistence curve for the determination of a and p. 

Sengers and coworkers (1999) have made calculations for the coexistence curve and the heat capacity of the real fluid SF and the real mixture 3-methylpentane + nitroethane and the agreement with experiment is excellent their comparison for the mixture [28] is shown in figure A2.5.28. 

Figure A2.5.28. The coexistence curve and the heat capacity of the binary mixture 3-methylpentane + nitroethane. The circles are the experimental points, and the lines are calculated from the two-tenn crossover model. Reproduced from [28], 2000 Supercritical Fluids—Fundamentals and Applications ed E Kiran, P G Debenedetti and C J Peters (Dordrecht Kluwer) Anisimov M A and Sengers J V Critical and crossover phenomena in fluids and fluid mixtures, p 16, figure 3, by kind pemiission from Kluwer Academic Publishers.

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