Heat capacity molar, definition

The partial molar properties are not measured directly per se, but are readily derivable from experimental measurements. For example, the volumes or heat capacities of definite quantities of solution of known composition are measured. These data are then expressed in terms of an intensive quantity—such as the specific volume or heat capacity, or the molar volume or heat capacity—as a function of some composition variable. The problem then arises of determining the partial molar quantity from these functions. The intensive quantity must first be converted to an extensive quantity, then the differentiation must be performed. Two general methods are possible (1) the composition variables may be expressed in terms of the mole numbers before the differentiation and reintroduced after the differentiation or (2) expressions for the partial molar quantities may be obtained in terms of the derivatives of the intensive quantity with respect to the composition variables. In the remainder of this section several examples are given with emphasis on the second method. Multicomponent systems are used throughout the section in order to obtain general relations. 

The specific and molar heat capacities of some common substances are given in Table 6.1. Note that, although the values of the specific heat capacities are listed in joules per degree Celsius per gram (J-(°C) 1 -g 1), they could equally well be reported in joules per kelvin per gram (J-K 1-g ) with the same numerical values, because the size of the Celsius degree and the kelvin are the same. We can calculate the heat capacity of a substance from its mass and its specific heat capacity by rearranging the definition Cs = dm into C = mCs. Then we can use... 

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 ... 

The heat capacities that have been discussed previously refer to closed, single-phase systems. In such cases the variables that define the state of the system are either the temperature and pressure or the temperature and volume, and we are concerned with the heat capacities at constant pressure or constant volume. In this section and Section 9.3 we are concerned with a more general concept of heat capacity, particularly the molar heat capacity of a phase that is in equilibrium with other phases and the heat capacity of a thermodynamic system as a whole. Equation (2.5), C = dQ/dT, is the basic equation for the definition of the heat capacity which, when combined with Equation (9.1) or (9.2), gives the relations by which the more general heat capacities can be calculated. Actually dQ/dT is a ratio of differentials and has no value until a path is defined. The general problem becomes the determination of the variables to be used in each case and of the restrictions that must be placed on these variables so that only the temperature is independent. 

For an ideal mixture, the enthalpy of mixing is zero and so a measured molar enthalpy of mixing is the excess value, HE. The literature concerning HE -values is more extensive than for GE-values because calorimetric measurements are more readily made. The dependence of HE on temperature yields the excess molar heat capacity, while combination of HE and GE values yields SE, the molar excess entropy of mixing. The dependences of GE, HE and T- SE on composition are conveniently summarized in the same diagram. The definition of an ideal mixture also requires that the molar volume is given by the sum, Xj V + x2 V2, so that the molar volume of a real mixture can be expressed in terms of an excess molar volume VE (Battino, 1971). 

These definitions accommodate both molar heat capacities and specific heat capacities (usually called specific heats), depending on whether U and H are molar or specific properties. 

This definition accommodates both the molar heat capacity and the specific heat capacity (usually called specific heat), depending on whether U is the molar or specific internal energy. Although this definition makes no reference to any process, it relates in an especially simple way to a constant-volume process in a closed system, for which Eq. (2.16) may be written ... 

Gas mixtures of constant composition may be treated in exactly the same way as pure gases. An ideal gas, by definition, is a gas whose molecules liave no influence on one another. Tills means that each gas in a mixture exists independent of the others its properties are unaffected by the presence of different molecules. Thus one calculates the ideal-gas heat capacity of a gas mixture as the mole-fraction-weightedsum of the heat capacities of the individual species. Consider 1 mol of gas mixture consisting of species A, B, and C, and let yA,jB, and yc represent the mole fractions of these species. The molar heat capacity of the mixture in the ideal-gas state is ... 

The specific heat of a substance is represented by Cp and is the energy as heat needed to raise the temperature of one gram of substance by one kelvin. Remember that molar heat capacity of a substance, C, has a similar definition except that molar heat capacity is related to moles of a substance not to the mass of a substance. Because the molar mass is the mass of 1 mol of a substance, the following equation is true. 

Both these considerations would be taken into accoimt if the activation process were assumed to occur at a constant pressure, p, such that the partial molar volume of the solvent is independent of the temperature, though this possibility does not appear to have been considered. A full discussion is beyond the scope of this chapter, but the resulting heat capacities of activation are unlikely to differ greatly from those determined at a constant pressme of, say, 1 atm. (see p. 137). Unfortunately, this approach requires the definition of rather clumsy standard states for solutes, e.g., hypothetically ideal, 1 molal, under a pressure such that a given mass of the pure solvent occupies a particular volume. 

U, H, and S as Functions of T and P or T and V At constant composition, molar thermodynamic properties can be considered functions of T and P (postulate 5). Alternatively, because V is related to T and P through an equation of state, V can serve rather than P as the second independent variable. The useful equations for the total differentials of U, H, and S that result are given in Table 4-1 by Eqs. (4-22) through (4-25). The obvious next step is substitution for the partial differential coefficients in favor of measurable quantities. This purpose is served by definition of two heat capacities, one at constant pressure and the other at constant volume ... 

IR. Relative Partial Molar Heat Contents.—The partial molar thermal properties, namely, heat content and heat capacity, are of particular interest, as well as of practical importance, as will be seen from some of the examples to be given below. In accordance with the general definition ( 26a), the partial molar heat content of any constituent of a solution is represented by... 

The definition of the specific molar heat capacity of the reference state, Cp, and the determination of the correction factor, CF(Zm, 6), are discussed next. The specific molar heat capacity of the reference state, Cp, is defined by... 

Definition of the specific molar heat capacity of the reference state, Cpg, for gas-phase and liquid-phase reactions... 

Figure A2.5.4 shows for this two-component system the same thermodynamic functions as in figure A2.5.2, the molar Gibbs free energy G= XjPj + X2P25 the molar enthalpy "w and the molar heat capacity C , again all at constant pressure, but now also at constant composition, x = 1/2. Now the enthalpy is continuous because the vaporization extends over an appreciable temperature range. Moreover, the heat capacity, while discontinuous at the beginning and at the end of the transition, is not a delta function. Indeed the graph appears to satisfy the definition of a second-order transition (or rather two, since there are two discontinuities).

The first partial derivative is the definition of the heat capacity, Cp, Notice this heat capacity is the extensive heat capacity of the reactor contents. Normally we express this quantity as an intensive heat capacity times the amount of material in the reactor. We can express the intensive heat capacity on either a molar or mass basis. We choose to use the heat capacity on a mass basis, so the total heat capacity can be expressed as... 

Because we are given the partial molar heat capacities (see Equation 3.51 for the definition of partial molar heat capacity), it is convenient to evaluate the total heat capacity as... 

The fact that we have not addressed all the different types of heat capacities became evident at the end of the subsection on "Neat content" where a certain difficulty became apparent in our two prototypical example systems. Along with the integral quantities dealt with above, we need various specific (related to the mass) and molar (related to the amount of substance) quantities derived from them. We can omit them here because their definitions and applications follow known patterns. 

Clearly, the relaxation term in (2.3.64) is always positive, independent of the definition of L and H. Of course, the magnitude of the relaxation term depends on the particular choice of the classification procedure. In order to explain the high value of the heat capacity of liquid water, one assumes that the frozen-in term, i.e. the first term on the right-hand side of (2.3.63), has a normal value. The excess heat capacity is then attributed to the relaxation term. In order for the latter to be large, we must have two components which differ appreciably in their partial molar enthalpy [otherwise (Hi — Hh) cannot be large], and none of the mole fractions xl and xh can be too small. 

Heat capacities. In the demonstration of the ideal gas equation, the variations of the capacitive (internal) energy 11 and the enthalpy J-f have been merged into a single equation describing the variation of the PVproduct. If these differential equations had been kept separate and the same procedure was followed for defining the gas constant as the constant common value of two second derivatives, one would have found the definitions of the molar heat capacities (also called specific heats). Depending on the energy form, there are two heat capacities, at constant volume and at constant pressure ... 

From the definition in Equation K2.12 of the chemical potential at constant temperature as a difference between the two partial chemical potentials used in the previous definitions of the molar heat capacities, their relationship with the gas constant ensues... 

The exact form of the function [/(T, V, Nk) for a particular system is obtained empirically. One way of obtaining the temperature dependence of U is to measure the molar heat capacity, Cy, at constant volume. (Box 2.1 gives basic definitions of heat capacity.) At constant volume, since no work is performed, dU = dQ. Hence... 

The random motion of molecules causes all thermodynamic quantities such as temperature, concentration and partial molar volume to fluctuate. In addition, due to its interaction with the exterior, the state of a system is subject to constant perturbations. The state of equilibrium must remain stable in the face of all fluctuations and perturbations. In this chapter we shall develop a theory of stability for isolated systems in which the total energy U, volume V and mole numbers Nk are constant. The stability of the equilibrium state leads us to conclude that certain physical quantities, such as heat capacities, have a definite sign. This will be an introduction to the theory of stability as was developed by Gibbs. Chapter 13 contains some elementary applications of this stability theory. In Chapter 14, we shall present a more general theory of stability and fluctuations based on the entropy production associated with a fluctuation. The more general theory is applicable to a wide range of systems, including nonequilibrium systems. 

The original definition of molar heat eapacities was defined as the quantity of heat needed to increase the temperature of one mole of substance by 1°C (without aity ehange in phase). In fact, as this heating can be carried out in different ways, reversible or non-reversible, usually the quantity of heat used, which is not a state function, depends on the heating method it is preferred to define the heat capacities from the entropy function which is a state function according to the second principle. 

Thus, the molar heat capacities are linked to two thermodynamic coefficients with temperature as the definition variable and entropy as the definition function. 

By nsing the definition of molar heat capacity at constant pressure ... 

It is worth mentioning the approximations in the background of the energy balance equations presented in this section. In fact, the term mCp(dT/df) represents the storage of internal energy in the system (dU/df). The derivative, dU/dt, can be written as dl7/df = (dU/dr)(dr/df). As a definition, we can write that the derivative dU/dT = c m. Here the heat capacity of the system at a constant volume is Cy. For liquid-phase systems, the difference between Cy and Cp is usually negligible, but for gas-phase systems, the molar heat capacities of a gas are related to each other by the following expression, provided that the ideal gas law is valid ... 

In Equation 6.65, it is assumed that the molar heat capacities at constant pressure and temperature have approximately the same values (CpL Cvl> CpG Cvg)- The catalyst mass meat and the energy flux can be expressed by Equations 6.7 and 6.58, respectively. By inserting these definitions, the energy balance for a BR becomes... 

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