Heat capacity of ideal gas

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. 

Figure 3.13 Specific isobaric heat capacities of ideal gases as functions of temperature. (Data from [17].)...

Table 19.6. Theoretical molar heat capacities of ideal gases...

TABLE A.2.1 Heat Capacity of Ideal gases Organic Compounds... 

Hea.t Ca.pa.cities. The heat capacities of real gases are functions of temperature and pressure, and this functionaHty must be known to calculate other thermodynamic properties such as internal energy and enthalpy. The heat capacity in the ideal-gas state is different for each gas. Constant pressure heat capacities, (U, for the ideal-gas state are independent of pressure and depend only on temperature. An accurate temperature correlation is often an empirical equation of the form ... 

From this equation, the temperature dependence of is known, and vice versa (21). The ideal-gas state at a pressure of 101.3 kPa (1 atm) is often regarded as a standard state, for which the heat capacities are denoted by CP and Real gases rarely depart significantly from ideaHty at near-ambient pressures (3) therefore, and usually represent good estimates of the heat capacities of real gases at low to moderate, eg, up to several hundred kPa, pressures. Otherwise thermodynamic excess functions are used to correct for deviations from ideal behavior when such situations occur (3). 

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. 

The heat capacities of real gases are a function of the pressure and thus may differ from the ideal gas case shown in the plot. However, the author s experience is that using the ideal gas k is sufficient for most engineering applications. 

The observed heat capacities of several gases are shown in Table 9.1. Notice that the monatomic gases have values of Cv equal to jR (12.47 J K-1 mol-1). Note also that as the molecules become more complex (more atoms), Cv increases. This result is expected because the presence of more atoms means that more nontranslational motions are available to absorb energy. Finally, notice that in all cases Cp — Cv = R, as expected for gases that closely obey the ideal gas law. 

A high temperature and pressure is rapidly attained in a shock heated gas, governed ideally by the pressure ratio across the shock front and the ratio of heat capacities of the gases [80-82]. The discontinuity means that a reactant gas is raised virtually discontinuously to the shocked gas temperature, Tq. The time interval available before a rapid cooling occurs, 10-1000 jLis, is kinetically significant at high temperatures. Heat losses are negligible over this reaction time interval. These are ideal circumstances in which quantitative kinetic measurements may be made. 

Monatomic ideal gases have a temperature-invariant heat capacity real gases do not. Most attempts to express the heat capacity of real gases use a power series, in either of the two following forms ... 

Cv can thus be calculated, at least for an ideal gas, and the whole construction can be put to a stringent scientific test by comparing it with measured heat capacities. No need to say, it turns out that statistical thermodynamic calculations provide extremely accurate evaluations of the heat capacities of diluted gases. In fact, these calculations are so reliable that for small molecules the CvS of gases found in thermodynamic repertories are usually calculated from experimental vibration frequencies, rather than measured. On the other hand, heat capacities for condensed phases cannot be calculated, but are much more easily measured than for gases. In this case, calculation and experiment match and complement each other perfectly. 

Appendix The Molar Heat Capacities of Gases in the Ideal Gas (Zero Pressure) State... 

In the above expression, the first term represents the accumulation and convective transport of enthalpy, where is the heat capacity of phase k. The second term is energy due to reversible work. For condensed phases this term is negligible, and an order-of-magnitude analysis for ideal gases with the expected pressure drop in a fuel cell demonstrates that this term is negligible compared to the others therefore, it is ignored in all of the models. 

Tdependence of the covolume upon temp was adopted. By trial, the value 0.3 was adopted for parameter /S. A computational procedure was devised in which "ideal values of the deton vel and temp were calcd on the assumption that the product gases obeyed the ideal gas law. The correction factors resulting from the introduction of eq 23 for the real gases were then tabulated as functions of the heat capacity of the product gases, considered ideal, and the argument... 

The expressions in (3.72) and (3.73) are valid only for monatomic ideal gases such as He or Ar, and must be replaced by somewhat different expressions for diatomic or polyatomic molecules (Sidebar 3.8). However, the classical expressions for polyatomic heat capacity exhibit serious errors (except at high temperatures) due to the important effects of quantum mechanics. (The failure of classical mechanics to describe the heat capacities of polyatomic species motivated Einstein s pioneering application of Planck s quantum theory to molecular vibrational phenomena.) For present purposes, we may envision taking more accurate heat capacity data from experiment [e.g., in equations such as (3.84a)] if polyatomic species are to be considered. The term perfect gas is sometimes employed to distinguish the monatomic case [for which (3.72), (3.73) are satisfactory] from more general polyatomic ideal gases with Cv> nR. 

The results of early experiments showed that the temperature did not change on the expansion of the gas, and consequently the value of the Joule coefficient was zero. The heat capacity of the gas is finite and nonzero. Therefore, it was concluded that (dE/dV)Tn was zero. Later and more-precise experiments have shown that the Joule coefficient is not zero for real gases, and therefore (dE/dV)Ttheoretical concepts of the ideal gas. 

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. 

Combustion products from a burner enter a gas turbine at 7.5 bar and 900°C and discharge at 1.2 bar. The turbine operates adiabatically with an efficiency of 80 percent. Assuming the combustion products to be an ideal-gas mixture with a heat capacity of 30 J mol-1 °C, what is the work output of the turbine per mole of gas, and what is the temperature of the gases discharging from the turbine ... 

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