Molar and Specific Heat Capacities

The molar heat capacity of a gas denoted by the capital letter C and expressed in J-mol K , represents the heat stored by a mole of the gas when heated from a temperature T, to a temperature Tj. At constant pressure (i.e., isobaric transformation), the heat change corresponds to the variation of the enthalpy of the gas as follows  

The respective isobaric and isochoric molar heat capacities for ideal gases can be calculated from the kinetic theory and are given in Table 19.6. 

Type of gas molecule Molar heat capacities Isentropic exponent 

The difference between the molar heat capacities at constant pressure and constant volume is given by the general Mayer s relation  

For ideal gases, the above equation becomes the Mayer s equation for ideal gases  

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The molar and specific heat capacities are related by the equation ... 

Usually, the molar and specific heat capacities or pure substances and compounds vary with temperature, and the molar heat capacity is usually provided by the polynomial equation ... 

The molar and specific heat capacities of mixtures can be approached quite accurately by the weighted average of the molar or specific heat capacities of each component ... 

Cp, Cp molar and specific heat capacities respectively, at constant pressure... 

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. 

Note that Eq. 5.2.31 converts the specific mass-based heat capacity to molar-based specific heat capacity. Also recall from Section 2.7 that for liquid-phase reactions, (Atot)o includes all the reactants and products, but not a solvent (inert species). Hence, the numerical value of is usually large since it also accounts for the heat capacity of the solvent. 

For gas-phase reactions, the heat capacity of the reacting fluid depends on both composition and temperature. Also, the specific heat capacity of the injection stream is usually different from that of the inlet stream. To accoimt for the effect of composition and temperature, molar-based specific heat capacities are used. The molar-based and the mass-based specific heat capacities are related by... 

This relation can be used to define various other isochoric heat quantities such as integral and differential, molar and specific heats of reaction and the corresponding heat capacities. The most well known of these quantities is the (global or integral) heat capacity at constant volume or isochoric heat capacity, which we got to know briefly in Sect. 9.1 ... 

Heat capacity is an extensive property the larger the sample, the more heat is required to raise its temperature by a given amount and so the greater is its heat capacity (Fig. 6.10). 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 amount (in moles) of the sample (Cm = C/n). For example, the specific heat capacity of liquid water at room temperature is 4.18 J-(°C) -g, or 4.18 J-K 1-g and its molar heat capacity is 75 J-K -mol1. 

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. 

More values are available in Appendices 2A and 2D values assume constant pressure. Specific heat capacities commonly use Celsius degrees in their units, whereas molar heat capacities commonly use kelvins. All values except that for ice are for 25°C. 

The high-temperature contribution of vibrational modes to the molar heat capacity of a solid at constant volume is R for each mode of vibrational motion. Hence, for an atomic solid, the molar heat capacity at constant volume is approximately 3/. (a) The specific heat capacity of a certain atomic solid is 0.392 J-K 1 -g. The chloride of this element (XC12) is 52.7% chlorine by mass. Identify the element, (b) This element crystallizes in a face-centered cubic unit cell and its atomic radius is 128 pm. What is the density of this atomic solid ... 

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. 

If a substance is heated without a change of state, the amount of heat required to change the temperature of 1 gram by 1° C is called the specific heat capacity of the substance. Similarly, the molar heat capacity is the amount of heat needed to raise the temperature of 1 mole of a substance by 1° C. Table 7-2 shows the heat capacities of several elements and compounds. 

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... 

The populations of other intermediate states are very small and can be neglected. For larger more complex proteins made up of multiple subunits, and in many fibrous proteins, this conclusion cannot be supported. Complex globular proteins appear to melt cooperatively in domains in which the smaller units melt independently, and the melting in fibrous proteins is even more complex. While the molar quantities for the heat capacity are dependent upon the size of the protein, the partial specific heat capacities of many proteins are very nearly the same. 

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. 

Ordinarily E and H are extensive, i.e., they are proportional to the amount (extent) of the system. If we use a molar heat capacity or a specific heat capacity, however, the heat transferred and the change in E or H refers to 1 mol or 1 g of the substance, and we must multiply by n or m to correct AE or AH to the actual amount of substance. 

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... 

Thermochemical measurements are based on the relationships between heat and temperature. The measurement that relates to the two is heat capacity, defined as the amount of heat that is required to raise the temperature of a substance 1°C. (The amount of substance is sometimes expressed in moles or in grams.) The heat capacity of a mole of a substance is known as the molar heat capacity, while the heat capacity for gram values of a substance are known as specific heat capacities. The specific heat of a substance is the amount of heat required to raise 1 gram of the substance 1°C. The formula that is used to calculate specific heat is Equation 17.4 ... 

The specific heat capacity is the heat that must be added per kg of a substance to raise the temperature by one Kelvin or one degree Celsius. The molar heat capacity is the specific heat multiplied by the molar mass (the molar mass of a structural unit in the case of polymers). Specific and molar heat capacity may be defined at constant volume or at constant pressure. The heat added causes a change in the internal energy (It) and in the enthalpy (heat content, H) of the substance. The following notations can be formulated ... 

It is assumed that the semi-crystalline polymer consists of an amorphous fraction with heat capacity Cp and a crystalline fraction with heat capacity of Cp. For a polymer with 30% crystallinity the estimated molar heat capacity is Cp(298) = 0.3 x 71.9 + 0.7 x 88.3 = 83.4 J moF1 K-1. The specific heat capacity is Cp/M= 1985 J kg-1 K 1... 

The complete course of the specific heat capacity as a function of temperature has been published for a limited number of polymers only. As an example, Fig. 5.1 shows some experimental data for polypropylene, according to Dainton et al. (1962) and Passaglia and Kevorkian (1963). Later measurements by Gee and Melia (1970) allowed extrapolation to purely amorphous and purely crystalline material, leading to the schematic course of molar heat capacity as a function of temperature shown in Fig. 5.2. 

In general a polymer sample is neither completely crystalline nor completely amorphous. Therefore, in the temperature region between Tg and Tm the molar heat capacity follows some course between the curves for solid and liquid (as shown in Fig. 5.1 for 65% crystalline polypropylene). This means that published single data for the specific heat capacity of polymers should be regarded with some suspicion. Reliable values can only be derived from the course of the specific heat capacity as a function of temperature for a number of samples. Outstanding work in this field was done by Wunderlich and his co-workers. Especially his reviews of 1970 and 1989 have to be mentioned here. 

So far only cp and Cp, the specific and the molar heat capacities at constant pressure, have been discussed. Obviously, these quantities are always dealt with in normal measurements. For the calculation of the specific heat capacity at constant volume, cv some relationships are available. An exact thermodynamic derivation leads to the equation ... 

Despite a wide range of specific heat capacities, the diatomic gases have molar heat capacities of about 29 J/moF°C, and the molar heat capacities of all the metallic elements are close to 26 J/moF°C. The latter generalization is known as the law of Dulong and Petit. 

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 ... 

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