Ideal gas heat capacities

The specific heat capacity of an ideal gas is the basic quantity for the enthalpy calculation, as it is independent from molecular interactions. It is also possible to define a real gas heat capacity, but for process calculations it is more convenient to account for the real gas effects with the enthalpy description of the equation of state used (see Section 6.2). In process calculations, the specific heat capacity of ideal gases mainly determines the duty of gas heat exchangers, and it has an influence on the heat transfer coefficient as well. 

The specific heat capacity of an ideal gas is defined as the heat per amount of substance in the ideal gas state necessary to obtain a certain temperature change. It must be distinguished between the specific isobaric heat capacity c p (at constant pressure) and the specific isochoric heat capacity c[f (at constant volume). For ideal gases, both quantities are related [52] via 

The specific isochoric heat capacity cjf consists of four parts, which refer to the kinetic energy of the molecules, the rotation energy of the molecules, the energy of the valence and deformation vibrations in the molecules, and the anharmonicity correction  

The term degree of freedom is important to understand how depends on temperature. Each fully activated degree of freedom contributes /iR to The kinetic energy part is pretty simple. The three translation coordinates can be regarded as the degrees of freedom. They are fully activated just above OK, therefore, the contribution to d is 

Considering two-atomic molecules, the rotation parts can be activated. There are three rotation degrees of freedom according to the three coordinates for the orientation of the rotation axis. They are fully activated when the absolute temperature exceeds a few K. However, the third coordinate, that is, the connection line between the two atoms, can again not be activated due to quantum mechanics because of the small moment of inertia. Thus, the rotation contribution of the rotation energy to d is 

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Appendix C-3 gives constants for the ideal-gas, heat-capacity equation... 

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 ... 

Values for the free energy and enthalpy of formation, entropy, and ideal gas heat capacity of carbon monoxide as a function of temperature are listed in Table 2 (1). Thermodynamic properties have been reported from 70—300 K at pressures from 0.1—30 MPa (1—300 atm) (8,9) and from 0.1—120 MPa (1—1200 atm) (10). 

Heat Capacity. The multiple property estimation methods for constant pressure ideal-gas heat capacities cover a broad range of organic compounds (188,216,217). Joback s method (188) is the easiest to use however, usage of all these methods has been recommended only over the range 280—1100 K (7). An accurate method for ideal-gas heat capacities (constant pressure), limited to hydrocarbons, has been presented (218) that involves a fit of seven variables, and includes steric, ring, branching, alkene, and even allene corrections. 

Heat Capacity, C° Heat capacity is defined as the amount of energy required to change the temperature of a unit mass or mole one degree typical units are J/kg-K or J/kmol-K. There are many sources of ideal gas heat capacities in the hterature e.g., Daubert et al.,"" Daubert and Danner,JANAF thermochemical tables,TRC thermodynamic tables,and Stull et al. If C" values are not in the preceding sources, there are several estimation techniques that require only the molecular structure. The methods of Thinh et al. and Benson et al. " are the most accurate but are also somewhat complicated to use. The equation of Harrison and Seaton " for C" between 300 and 1500 K is almost as accurate and easy to use ... 

Example Q Using Eq. 2-48 to estimate the ideal gas heat capacity of acetone (C HgO) at 600 K ... 

Cp may be assumed to be the ideal gas heat capacity, Cp. Average errors can be expected to be less than 5 percent. 

The most satisfactory calciilational procedure for thermodynamic properties of gases and vapors requires PVT data and ideal gas heat capacities. The primary equations are based on the concept of the ideal gas state and the definitions of residual enthalpy anci residual entropy ... 

The correction of the ideal gas heat capacity to account for real conditions of temperature and pressure was discussed in Chapter 3, Section 3.7. 

Table 8.4. Group contributions to ideal gas heat capacities, kJ/kmol°C (Rihani and Doraiswamy, 1965)...

For the compounds listed below, estimate the constants in the equation for ideal gas heat capacity, equation 3.19, using the method given in Section 8.9.2. 

One problem remains. The reference enthalpy must be defined at temperature T and pressure P0. The reference state for enthalpy can be taken as an ideal gas. At zero pressure, fluids are in their ideal gaseous state and the enthalpy is independent of pressure. The ideal gas enthalpy can be calculated from ideal gas heat capacity data3 ... 

Table 10-3 Ideal Gas Heat Capacity Coefficients for Common Fnel Cell Gases...

We define the standard state of a real gas so that Eq. (51) is general (i.e., so that it also applies to ideal gases). For ideal gases, the standard state is at 1.0 bar pressure. For real gases, we also use a 1.0-bar ideal gas as the standard state. We find the standard state by the two-step process shown in Fig. 6. First we extrapolate the real gas to very low pressure, where / —> P and the gas becomes ideal (Step I). We then convert the ideal gas to 1.0 bar (step II). The convenience of an ideal gas standard state is that it allows temperature conversions to be made with ideal gas heat capacities (which are pressure independent). Conversion to the real gas state is then made at the temperature of interest. 

Mujtaba (1989) used CMH model to simulate the operations considered by Domenech and Enjalbert (1974). Since the overall stage efficiency in the experimental column was 75%, the number of theoretical plates used by Mujtaba was 3. The column was initialised at its total reflux steady state values. Soave-Redlich-Kwong (SRK) model was used for the VLE property calculations. Vapour phase enthalpies were calculated using ideal gas heat capacity values and the liquid phase enthalpies were calculated by subtracting heat of vaporisation from the... 

The minimum information covers chemical formula, molecular weight, normal boiling point, freezing point, liquid density, water solubility and critical properties. Additional properties are enthalpies of phase transitions, heat capacity of ideal gas, heat capacity of liquid, viscosity and thermal conductivity of liquid. Computer simulation can estimate missing values. The use of graphs and tables of properties offers a wider view and is strongly recommended. 

As shown in Chap. 6, ideal-gas heat capacities, rather than the actual heat capacities of gases, are used in the evaluation of thermodynamic properties such as internal energy and enthalpy. The reason is that thermodynamic-property evaluation is conveniently accomplished in two steps first, calculation of ideal-gas values from ideal-gas heat capacities second, calculation from PVT data of the differences between real-gas and ideal-gas values. A real gas becomes ideal in the limit as P - 0 if it were to remain ideal when compressed to a finite pressure, its state would remain that of an ideal-gas. Gases in these hypothetical ideal-gas states have properties that reflect their individuality just as do real gases. Ideal-gas heat capacities (designated by Cf and Cy) are therefore different for different gases although functions of temperature, they are independent of pressure. 

Figure 4.1 Ideal-gas heat capacities of argon, nitrogen, water, and carbon dioxide as functions of temperature.

The effects of temperature on C or Cy are determined by experiment, most often from spectroscopic data and knowledge of molecular structure by the methods of statistical mechanics. Where experimental data are not available, methods of estimation are employed, as described by Reid, Prausnitz, and Sherwood.t Ideal-gas heat capacities increase smoothly with increasing temperature toward an upper limit, which is reached when all translational, rotational, and vibrational modes of molecular motion are fully excited. 

Although ideal-gas heat capacities are exactly correct for real gases only at zero pressure, real gases rarely depart significantly from ideality up to several bars, and therefore C p and C y are usually good approximations for the heat capacities of real gases at low pressures. 

Since the equations of thermodynamics which derive from the first and second laws do not permit calculation of absolute values for enthalpy and entropy, and since all we need in practice are relative values, the reference-state conditions T0 and P0 are selected for convenience, and values are assigned to H 0a and S 9 arbitrarily. The only data needed for application of Eqs. (6.45) and (6.46) are ideal-gas heat capacities and PVT data. Once V, H, and S are known at given conditions of T and P, the other thermodynamic properties follow from defining equations. 

The ideal-gas heat capacity of isobutane vapor in the temperature range of interest is given by... 

The generalized correlations for HR and SR, together with ideal-gas heat capacities, allow calculation of enthalpy and entropy values of gases at any temperature and pressure by Eqs. (6.45) and (6.46). For a change from state 1 ... 

Estimate the ideal-gas heat capacity C° of 2-methyl-1,3-butadiene and n-methyl-2-pyrrolidone at 527°C (800 K, or 980°F) using the group-contribution method of Rihani and Doraiswamy. The Rihani-Doraiswamy method is based on the equation... 

TABLE 1.3 Group Contributions to Ideal-Gas Heat Capacity... 

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