Expressions for Heat Capacity

In Eqs. 5.5.6-5.5.9, the coefficient /x/ is the chemical potential of speeies i. The equations show that /Xj ean he equated to four different partial derivatives, similar to the equalities shown in Eq. 5.5.5 for a pure substance  

The partial derivative dG/dni)T,p,nj i called the partial molar Gibbs energy of species i, another name for the ehemieal potential as will be discussed in See. 9.2.6. 

As explained in See. 3.1.5, the heat eapaeity of a closed system is defined as the ratio of an infinitesimal quantity of heat transferred across the boundary under specified conditions and the resulting infinitesimal temperature change heat capacity = dq/dT. The heat capacities of isoehorie (eonstant volume) and isobarie (eonstant pressure) processes are of particular interest. 

The heat capacity at constant volume, Cy, is the ratio dq / dT for aproeess in a closed constant-volume system with no nonexpansion work—that is, no work at all. The first law shows that under these eonditions the internal energy change equals the heat dU = dq (Eq. 5.3.9). We can replace dq by dU and write Cy as a partial derivative  

If the closed system has more than two independent variables, additional conditions are needed to define Cy unambiguously. For instance, if the system is a gas mixture in which reaction can occur, we might specify that the system remains in reaction equilibrium as T changes at constant V. 

---

The expression for heat capacity brings out the fact that it is an indefinite quantity even when mass is specified, since 8q is so. This is no longer the case when certain conditions, particularly constant volume or constant pressure conditions, are specified. The heat capacity then becomes a definite quantity as a consequence of 8q becoming a definite quantity. 

Either arbitrary constants or coefficients of a polynomial expression for heat capacity, such as those listed in Appendix B.2. 

This equation is derived from an empirical expression for heat capacity, where a, b, c, d and B are constants which can be found in most of the handbooks of chemistry and physics (e.g. 8). For a dissolved species, however, very few experimental data are available for elevated temperatures and pressures. 

This chapter begins with a discussion of mathematical properties of the total differential of a dependent variable. Three extensive state functions with dimensions of energy are introduced enthalpy, Helmholtz energy, and Gibbs energy. These functions, together with internal energy, are called thermodynamic potentials. Some formal mathematical manipulations of the four thermodynamic potentials are described that lead to expressions for heat capacities, surface work, and criteria for spontaneity in closed systems. 

If the pressure varies in a BR (nonisobaric operations), the expressions for heat capacities, CpL and CpG, should be replaced with the heat capacities for a constant volume, Cypand CyG-... 

Fortunately, most expressions for heat capacity are simple power series in T, whose integrals are easy to evaluate on a term-by-term basis. 

Numerically determine AS for the isobaric change in temperature of 4.55 g of gallium metal as it is heated from 298 K to 600 K if its molar heat capacity is given by the expression Cp = 27.49 - 2.226 X 10 T+ 1.361 X IOVtI Assume standard units on the expression for heat capacity. 

The temperature dependency for heat capacity can usually be described by a polynomial expression, e.g.,... 

From classic thermodynamics alone, it is impossible to predict numeric values for heat capacities these quantities are determined experimentally from calorimetric measurements. With the aid of statistical thermodynamics, however, it is possible to calculate heat capacities from spectroscopic data instead of from direct calorimetric measurements. Even with spectroscopic information, however, it is convenient to replace the complex statistical thermodynamic equations that describe the dependence of heat capacity on temperature with empirical equations of simple form [15]. Many expressions have been used for the molar heat capacity, and they have been discussed in detail by Frenkel et al. [4]. We will use the expression... 

The SI unit for heat capacity is J-K k Molar heat capacities (Cm) are expressed as the ratio of heat supplied per unit amount of substance resulting in a change in temperature and have SI units of J-K -moC (at either constant volume or pressure). Specific heat capacities (Cy or Cp) are expressed as the ratio of heat supplied per unit mass resulting in a change in temperature (at constant volume or pressure, respectively) and have SI units of J-K -kg . Debye s theory of specific heat capacities applies quantum theory in the evaluation of certain heat capacities. 

Most equations for heat capacities of substances are empirical. Heat capacity at constant pressure is generally expressed in terms of temperature with a power,series type formula ... 

In Equation 2.21, the index i refers to all compounds of the reaction mass and to the reactor itself. However, in practice, for stirred tank reactors, the heat capacity of the reactor is often negligible compared to that of the reaction mass. In order to simplify the expressions, the heat capacity of the equipment can be ignored. This is justified by the following example. For a 10 m3 reactor, the heat capacity of the reaction mass is in the order of magnitude of 20000kJ K 1 whereas the metal mass in contact with the reaction medium is about 400 kg, representing a heat capacity of about 200 kj K"1, that is, ca. 1% of the overall heat capacity. Further, the error leads to a more critical assessment of the situation, which is a good practice... 

A. Example CD8-1 AHitj(T) for Heat Capacities Expressed as Quadratic... 

For this purpose it is convenient to express the heat capacities of gases in a power series of the form... 

Most of the equations for the heat capacities of solids, liquids, and gases are empirical. We usually express the heat capacity at constant pressure Cp as a function of temperature in a power series, with constants a, b, c, and so on for example. 

This is the Maier-Kelley equation for heat capacities (Maier and Kelley, 1932). Several other equations are also commonly used to fit the heat capacity (see for example Tables 7.1, 7.2 and 7.3). Rather than try to present the rest of the equations in this chapter for all current versions of this equation, we continue to use the Maier-Kelley equation as an example. Commonly used equations derived using other heat capacity expressions are presented in Appendix B. 

Ihbles 3a, 3b, and 4 also show that the values of several different quantities are expressed in the same SI unit. For example, the joule per kelvin (J/K) is the SI unit for heat capacity as well as for entropy. Thus the name of the unit is not sufficient to define the quantity measured. 

By a reduction of oscillations per frequency in the low-frequency domain from 3 to 2, method A shortens the time span for one TIS measurement by 25% to 18.4 h. Heat capacity and heat exchange with enviromnent still have similar values. For heat capacity, the standard deviation from several evaluation runs increases slightly but stays below 1%. Regarding thermal conductivity, the minor changes in mean value and standard deviation of 0.1 W/mK he clearly below the uncertainties of the reference measurement expressed by standard deviation. Consequently, thermal parameters confirm that the reduced number of oscillations has practically no influence on accuracy. 

Both (6.16) and (6.17) are therefore approximations, to be used only over a small temperature interval, or in cases where the result need only be approximate. More accurate formulae involve the heat capacity, but as there are a variety of equations expressing the heat capacity as a function of T, there are a variety of more accurate expressions for A,.G°. We looked at two of these in Chapter 5, Equations (5.29) and (5.38). 

In this section, we show that the heat capacity (here, at constant volume) is not expressible in terms of the pair distribution function. In fact, we shall see that molecular distribution functions of up to order four are required for this purpose. In Chapter 5, we discuss a different possibility of expressing the heat capacity in terms of generalized molecular distribution functions. 

Of the derived units, it may be noted that the unit for force is the newton (N), which, expressed in terms of SI base units, is mkgs the unit for density or mass density is kilogram per cubic metre (kgm ) the unit for concentration (of amount of substance) is mole per cubic metre (mol m ) the unit for specific volume is cubic metre per kilogram (m kg ). Some other derived units have special names for example, the name for the unit of energy, work, or quantity of heat is the joule (J), equal to the newton metre (N m). The imit of power is the watt (W), equal to J s. The unit for heat capacity and entropy is the joule per kelvin (JK ) the unit for molar energy is joule per mole (J mol" ) and for molar entropy and molar heat capacity is joule per kelvin mole (J mol" ). 

The heat capacity calculated above is for a sample weighing 1058.44 g. We can express the heat capacity per gram of solution (in other words, as a specific heat capacity) by dividing the heat capacity above by 1058.44 g. We obtain... 

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... 

Residue Hea.tup. Equations 27—30 can be used to estimate the time for residue heatup, by replacing the Hquid properties, such as density and heat capacity, with residue properties, and considering the now smaller particle in evaluating the expressions for ( ), and T. In the denominator of T, 0is replaced by and is replaced by T the ignition temperature of the residue. 

Evaluation of the integrals requires an empirical expression for the temperature dependence of the ideal gas heat capacity, (3p (8). The residual Gibbs energy is related to and by equation 138 ... 

The standard Gibbs-energy change of reaction AG° is used in the calculation of equilibrium compositions. The standard heat of reaclion AH° is used in the calculation of the heat effects of chemical reaction, and the standard heat-capacity change of reaction is used for extrapolating AH° and AG° with T. Numerical values for AH° and AG° are computed from tabulated formation data, and AC° is determined from empirical expressions for the T dependence of the C° (see, e.g., Eq. [4-142]). 

Exponential cost correlations have been developed for individual items of equipment. Care must be taken in determining whether the cost of the eqmpment has been expressed as free on Board (FOB), delivered (DEL), or installed (INST), as this is not always clearly stated. In many cases the cost must be correlated in terms of parameters related to capacity such as surface area for heat exchangers or power for grinding equipment. There are four main sources of error in such cost correlations ... 

These heat capacity approximations take no account of the quantal nature of atomic vibrations as discussed by Einstein and Debye. The Debye equation proposed a relationship for the heat capacity, the temperature dependence of which is related to a characteristic temperature, Oy, by a universal expression by making a simplified approximation to the vibrational spectimii of die... 

---