Pure substance, heat capacity

For the material and energy balances, pure-component heat capacity and density data are needed. These are among the most widely measured data and are available on process simulators for more than a thousand substances. (See Chapter 13 for details of process simulators.) There are also reasonably accurate group-contribution techniques for use when no data are available [8]. The enthalpies of mixtures require an accurate equation of state for gases and nonionic liquids. The equations of state available on process simulators are accurate enough for these systems. However, additional heat of solution data are needed for electrolyte solutions, and these data may not be as readily available. For these systems, care should be taken to use accurate experimental data, because estimation techniques are not as well defined. 

Equation (4.2) can be used to determine the entropy of a substance. A pure crystalline sample is placed in a cryogenic calorimeter and cooled to low temperatures. Increments of heat, q, are added and the temperature change, AT, is measured, from which the heat capacity can be calculated from the relationship... 

E4.2 Solid phases a and j3 are in equilibrium for a pure solid substance at 12 K. Below 12 K, the heat capacities of a and 3 vary with temperature according to the equations... 

Point c is a critical point known as the upper critical end point (UCEP).y The temperature, Tc, where this occurs is known as the upper critical solution temperature (UCST) and the composition as the critical solution mole fraction, JC2,C- The phenomenon that occurs at the UCEP is in many ways similar to that which happens at the (liquid + vapor) critical point of a pure substance. For example, at a temperature just above Tc. critical opalescence occurs, and at point c, the coefficient of expansion, compressibility, and heat capacity become infinite. 

These expressions may be rearranged to calculate the specific or molar heat capacity from the measured temperature rise caused by a known quantity of heat. The specific heat capacity of a dilute solution is normally taken to be the same as that of the pure solvent (which is commonly water). Table 6.2 lists the specific and molar heat capacities of sume common substances. 

A total heat capacity has units of j/°C, whereas a molar heat capacity has units of J/mol °C. If we have n moles of a pure substance, its total heat capacity is = ft Cmotar. 

It is convenient that the temperature correction to the enthalpy of reaction 2.2 is rather small, because it suggests that the difference Ar//3io — Ar//298 for reaction 2.1 will also be negligible. In fact, we would be in some trouble to evaluate the temperature correction for the process under the experimental conditions, as some of the necessary data are not readily available. To calculate the solution enthalpies shown in figure 2.1 at 310 K (from the values at 298.15 K), both the (known) values of the heat capacities of the pure substances and the (unknown) values of these quantities in solution are required. 

A comprehensive collection of evaluated heat capacities of 1624 pure substances in the liquid state. This was updated in reference[19]. 

Useful quantitation of heat q as a quantity of energy can be traced to the studies of Joseph Black around 1803. Black recognized that different substances vary in their capacity to absorb heat, and he undertook systematic measurements of the heat capacity C (the ratio of heat absorbed to temperature increase) for many substances. He recognized that a fixed quantity of any pure substance (e.g., 1 g of water) has a unique value of C, which can be chosen as a calorimetric standard for defining quantity of heat in a convenient way. In this manner, he introduced the calorie as a unit of heat ... 

Heat capacities, C, are also reported for pure substances, not just for the complicated assembly of substances that makes up a typical calorimeter. For instance, we can report the heat capacity of water or of copper. More heat is needed to raise the temperature of a large sample of a substance by a given amount than is required to raise the temperature of a small sample by the same amount, so heat capacity is an extensive property the larger the sample, the greater its heat capacity (Fig. 6.15). It is therefore common to report either the specific heat capacity (often called just specific heat ), Cs, which is the heat capacity divided by the mass of the sample (Cs = dm), or the molar heat capacity, Cm, the heat capacity divided by the number of moles in the sample (Cm = dn). Specific heat capacities and molar heat capacities are intensive properties. 

If we want to calculate the entropy of a liquid, a gas, or a solid phase other than the most stable phase at T =0, we have to add in the entropy of all phase transitions between T = 0 and the temperature of interest (Fig. 7.11). Those entropies of transition are calculated from Eq. 5 or 6. For instance, if we wanted the entropy of water at 25°C, we would measure the heat capacity of ice from T = 0 (or as close to it as we can get), up to T = 273.15 K, determine the entropy of fusion at that temperature from the enthalpy of fusion, then measure the heat capacity of liquid water from T = 273.15 K up to T = 298.15 K. Table 7.3 gives selected values of the standard molar entropy, 5m°, the molar entropy of the pure substance at 1 bar. Note that all the values in the table refer to 298 K. They are all positive, which is consistent with all substances being more disordered at 298 K than at T = 0. 

On a heating curve for a pure / V I substance the temperature remains constant during a phase change. Within a given state, temperature rises with a slope that depends on the heat capacity of the substance in that state. 

Despite its age, Lyman s Handbook is still a valuable source of basic information on chemical properties, and many of the estimation methods contained therein remain valid. It also contains a suite of chapters on pure substance properties that normally are not of direct interest to environmental scientists e.g., flash point and heat capacity. 

The heat capacity of a body is the amount of heat required to raise the temperature of that body 1K (1°C). For pure substances, it is most convenient to refer to quantities of molar heat capacity (heat capacity per mole) and, as discussed above, the specific heat capacity or, more commonly, the specific heat (heat capacity per unit of mass). As an example, the average specific heat of water is... 

It is clear that temperature oscillations during heating-cooling cycles depend on the fixed-bed heat capacity. Figure 1.10 shows a simplified picture of the effect of phase change on the effective heat capacity of pure substances. Considerable amounts of... 

In words, aP can be described as the fractional volume increase (dV/V) with respect to a temperature increase (dT) under isobaric conditions, while /3y is the corresponding fractional volume decrease (—dV/V) with respect to a pressure increase (dP) under isothermal conditions. Of course, both aP = ah P, T) and = fir(P, T) vary with P, T, as do other thermodynamic properties. Numerical values of aP, fiT (e.g., for 1 atm, 25°C) are often tabulated with other material properties, such as density, boiling point, or heat capacity, as unique fingerprints of a pure substance. [Throughout this book, experimental values are commonly drawn from standard sources, such as J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954) or any recent edition of the CRC Handbook of Chemistry and Physics (CRC Press, Boca Raton, FL).]... 

Thermochemical data on the separate phases in equUibrium are needed to constmct accurate phase diagrams. The Gibbs energy of formation for a pure substance as a function of temperature must be calculated from experimentally determined temperature-dependent thermodynamic properties such as enthalpy, entropy, heat capacity, and equihbrium constants. By a pure substance, one generally means a stoichiometric compound in which the atomic constituents ate present in an exact, simple reproducible ratio. 

The infinite heat capacity at the critical point looks unusual, but this is true of all pure substances. This has been observed experimentally and can be demonstrated using the principles of classical thermodynamics. However, even though the heat capacity is infinite, the enthalpy at the critical point is finite. 

As mentioned in Sections 1.1 and 2.9, the third law of thermodynamics makes it possible to obtain the standard Gibbs energy of formation of species in aqueous solution from measurements of the heat capacity of the crystalline reactant down to about 10 K, its solubility in water and heat of solution, the heat of combustion, and the enthalpy of solution. According to the third law, the standard molar entropy of a pure crystalline substance at zero Kelvin is equal to zero. Therefore, the standard molar entropy of the crystalline substance at temperature T is given by... 

Gibbs-Duhem Relationship The partial molar properties of a multicomponent phase cannot be varied independently (the mole fractions, jc, = ,/E of the components total unity). For example, for the chemical potentials, /i, the Gibbs-Duhem relationship is En, dni = 0 (for details, see e.g., Atkirs, 1990 Blandamer, 1992 Denbigh, 1971). Similar constraints apply to the partial molar volumes, enthalpies, entropies, and heat capacities. For pure substances, the partial molar property is equal to the molar property. For example, the chemical potential of a pure solid or liquid is its energy per mole. For gaseous, liquid, or solid solutions, X, = X,(ny), that is, the chemical potentials and partial molar volumes of the species depend on the mole fractions. 

The molar heat capacity of a pure substance is the energy as heat needed to increase the temperature of 1 mol of the substance by 1 K. Molar heat capacity has the symbol C and the unit J/K mol. Molar heat capacity is accurately measured only if no other process, such as a chemical reaction, occurs. 

In particular, if the system consists of one mole of a single component in a single physical state, its heat capacity at constant volume is called the molar heat capacity of the pure substance at constant volume, and is denoted by Cy. 

Figure 4.5 illustrates the behavior of the heat capacity of a pure substance over a wide range of absolute temperatures. Observe that at zero degrees absolute the heat... 

Figure 4.5 Heat capacity as a function of temperature for a pure substance.

Experimental evidence indicates that the heat capacity of a substance is not constant with temperature, although at times we may assume that it is constant in order to get approximate results. For the ideal monoatomic gas, of course, the heat capacity at constant pressure is constant even though the temperature varies (see Table 4.1). For typical real gases, see Fig. 4.7 the heat capacities shown are for pure components ... 

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