Heat Capacity and the Enthalpy Function

The first law of thermodynamics leads to a broad array of physical and chemical consequences. In the following Sections 3.6.1-3.6.8, we describe the formal theory of heat capacity and the enthalpy function, the measurements of heating effects that clarified the energy and enthalpy changes in real and ideal gases under isothermal or adiabatic conditions, and the general first-law principles that underlie the theory and practice of thermochemistry, the measurement of heat effects in chemical reactions. 

The provisional characterization (Section 3.4) of heat capacity C as the ratio of heat absorbed to the temperature increase corresponds to a differential ratio of the form (on a per-mole basis, unless otherwise specified) 

However, the definition (3.42) leaves heat capacity ill-determined, because the imperfect differential dq harbors a dependence on the (unspecified) path along which dq is measured ( path referring to how the remaining non-T degree of freedom is specified). 

Two common experimental paths for heat measurement are conditions of constant volume (qv) or constant pressure (qP). These conditions lead to the corresponding constant-volume (Cv) or constant-pressure (CP) heat capacities 

The definitions (3.43), although improving on (3.42), are still somewhat unsatisfactory owing to the presence of the nonstate property q. 

---

Use the MCPH function to calculate the mean heat capacity and the HRB function for the residual enthalpy. 

In spite of the wealth of information available on the preparative and structural aspects of the lanthanide chlorides (1-3), experimental thermodynamic, and, in particular, high-temperature vaporization data are singularly lacking. The comprehensive estimates of the enthalpies of fusion, vaporization, heat capacities and other thermal functions for the lanthanide chlorides by Brewer et ah (4, 5) appear internally consistent, but the relatively few experimental measurements (6-/2) do not permit confirmation of the estimates due to the narrow temperature ranges of study. Additionally, the absence of accurate molecular data for the gaseous species has hampered third-law treatment of the limited experimental vapor pressure data available. The one reported study (12) of the vaporization of EuC12 effected by a boiling-point method lacks accuracy for these reasons. 

A new type of rotational degrees of freedom parameter will be defined for the backbones and side groups of polymers, and correlations for the heat capacity and related thermodynamic functions (enthalpy, entropy and Gibbs free energy) will be developed utilizing both the connectivity indices and the rotational degrees of freedom, in Chapter 4. 

Literature values of Zrl4(g) pressures in equilibrium with solids of the zirconium-iodine system were used to calculate Zr and Zxf activities as functions of composition at 700 K. These activities, together with estimated heat capacities and Gibbs energy functions, were used to derive values of the activity of iodine (12(g) and 1(g)) and the enthalpies of formation of stoichiometric Zrl3(s), Zrl2(s) and Zrl(s). The thermodynamic functions were estimated according to the relationship ... 

Generally, therefore, these additional functions are connected with the departures from additivity shown by the volume F, the heat capacity and the chemical constant i and the enthalpy H on dilution of the solution. They find their tangible expression in volume contractions, heat effects and anomalous behavior of specific heats. Physically they should be attributed to an excess or deficiency in attraction between the molecules of solvent and solute over the cohesion of identical molecules. Hildebrand has termed solutions in which additional entropy terms such as 2, 3 and 4 are missing, regular solutions (see p. 222). In them the excess and deficiency attractions may be related quantitatively to the heat of dilution, since in the insertion of molecules of one component between those of the other, a heat effect other than zero results because the energy necessary for the separation of identical molecules differs from that obtained in bringing together dissimilar particles. 

The enthalpy as a function of time is readily available from, for example, drop calorimetry experiments or from adiabatic calorimeters with incremental temperature increases. Scanning calorimeters, however, furnish the heat capacity of the sample. In these cases, the phase transition shows as a peak and the enthalpy of transition is calculated by integration of the peak area. Traditionally, this is done after constructing a proper baseline under the peak between the start and the end of the peak. The definition of the start and the end of the peak and the shape of the baseline under the peak are somehow arbitrary, particularly when the phase transition is accompanied by a heat capacity change. The enthalpy change at the transition temperature trs can be calculated from the heat capacity curve by... 

In eqn [18], the first three terms represent baseline heat capacity and the fourth term represents the excess heat capacity due to fusion. If heat capacity and enthalpy functions are available, for example, from the ATHAS-DB, then eqns [16] and [17] can be solved. Because the partition between Wma and Wra is not known, eqns [18] and [19] cannot be solved without further assumptions. If at the lateral surfaces of each lamellae an RAF layer with the same thickness (often 2 nm) exists, the problem can be solved iteratively." " Righetti et developed a method based on an assumption regarding the availability of baseline heat capacity in certain temperature regions from TMDSC measurements. 

Hea.t Ca.pa.cities. The heat capacities of real gases are functions of temperature and pressure, and this functionaHty must be known to calculate other thermodynamic properties such as internal energy and enthalpy. The heat capacity in the ideal-gas state is different for each gas. Constant pressure heat capacities, (U, for the ideal-gas state are independent of pressure and depend only on temperature. An accurate temperature correlation is often an empirical equation of the form ... 

The heat flux and energy calibrations are usually performed using electrically generated heat or reference substances with well-established heat capacities (in the case of k ) or enthalpies of phase transition (in the case of kg). Because kd, and kg are complex and generally unknown functions of various parameters, such as the heating rate, the calibration experiment should be as similar as possible to the main experiment. Very detailed recommendations for a correct calibration of differential scanning calorimeters in terms of heat flow and energy have been published in the literature [254,258-260,269]. 

Table III gives values of the changes in Gibbs energy, enthalpy, entropy, and heat capacity of the solution process as calculated from the equations of Table I. Figure 1 shows the recommended noble gas mole fraction solubilities at unit gas partial pressure (atm) as a function of temperature. The temperature of minimum solubility is marked.

We introduced the enthalpy function particularly because of its usefulness as a measure of the heat that accompanies chemical reactions at constant pressure. We will find it convenient also to have a function to describe the temperature dependence of the enthalpy at constant pressure and the temperature dependence of the energy at constant volume. Eor this purpose, we will consider a new quantity, the heat capacity. (Historically, heat capacity was defined and measured much earlier than were enthalpy and energy.)... 

Because of this relationship between (TT — and p-j x.. the former quantity frequently is referred to as the Joule-Thomson enthalpy. The pressure coefficient of this Joule-Thomson enthalpy change can be calculated from the known values of the Joule-Thomson coefficient and the heat capacity of the gas. Similarly, as (H — is a derived function of the fugacity, knowledge of the temperature dependence of the latter can be used to calculate the Joule-Thomson coefficient. As the fugacity and the Joule-Thomson coefficient are both measures of the deviation of a gas from ideahty, it is not surprising that they are related. 

With most properties (enthalpies, volumes, heat capacities, etc.) the standard state is infinite dilution. It is sometimes possible to obtain directly the function near infinite dilution. For example, enthalpies of solution can be measured in solution where the final concentration is of the order of 10-3 molar. With properties such as volumes and heat capacities this is more difficult, and, to get standard values, it is usually necessary to measure apparent molal quantities 0y at various concentrations and extrapolate to infinite dilution (y° = Y°). Fortunately, it turns out that, at least with volumes and heat capacities, the transfer functions AYe (W — W + N) do not vary significantly with the electrolyte concentration as long as this concentration is relatively low (3). With most of the systems investigated, the transfer functions were calculated from apparent molal quantities at 0.1m and assumed to be equivalent to the standard values. 

In equation 33, the superscript I refers to the use of method I, a T) is the activity of component i in the stoichiometric liquid (si) at the temperature of interest, AHj is the molar enthalpy of fusion of the compound ij, and ACp[ij] is the difference between the molar heat capacities of the stoichiometric liquid and the compound ij. This representation requires values of the Gibbs energy of mixing and heat capacity for the stoichiometric liquid mixture as a function of temperature in a range for which the mixture is not stable and thus generally not observable. When equation 33 is combined with equations 23 and 24 in the limit of the AC binary system, it is termed the fusion equation for the liquidus (107-111). 

The currently preferred method for the study of gas forming reactions as function of temp is DSC. Here the specimen and the ref sample are heated at programmed heating rates (in controlled atms, if desired) while the differential energy input to the specimens is recorded. Hence the pen movement is directly proportional to the heat capacity while the area under the curve represents the enthalpy change. New equipment is now on the market which can operate up to 1200° (Ref 79) and which is therefore adequate for expl and propint studies. Limitations on the use of DSC for kinetics studies of expls will be discussed under the entry Thermochemistry in this Vol... 

Similar arguments and definitions can be applied to the other partial molar thermodynamic functions and properties of the components in solution. By differentiation of Equation (8.71), the following expressions for the partial molar entropy, enthalpy, volume, and heat capacity of the kth component are obtained ... 

---