Heat-Capacity Ratios for Gases

The ratio CJCy of the heat capacity of a gas at constant pressure to that at constant volume will be determined by either the method of adiabatic expansion or the sound velocity method. Several gases will be studied, and the results will be interpreted in terms of the contribution made to the specific heat by various molecular degrees of freedom. 

In considering the theoretical calculation of the heat capacities of gases, we shall be concerned only with perfect gases. Since Cp = Cy+ R for an ideal gas (where Cp and are the molar quantities C ln and CJn), our discussion can be restricted to C . 

The number of degrees of freedom for a molecule is the number of independent coordinates needed to specify its position and configuration. Hence a molecule of Adatoms has 3A/ degrees of freedom. These could be taken as the three Cartesian coordinates of the N individual atoms, but it is more convenient to classify them as follows. 

Translational degrees of freedom Three independent coordinates are needed to specify the position of the center of mass of the molecule. 

Vibrational degrees of freedom One must also specify the displacements of the atoms from their equilibrium positions (vibrations). The number of vibrational degrees of freedom is 2 N — 5 for linear molecules and 3AA— 6 for nonlinear molecules. These values are determined by the fact that the total number of degrees of freedom must be 3A. For each vibrational degree of freedom, there is a normal mode of vibration of the molecule, with characteristic symmetry properties and a characteristic harmonic frequency. The vibrational normal modes for CO2 and H2O are illustrated schematically in Fig. 1. 

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Table 13.8 Heat capacity ratio for various gases.

When actual data are not available, a useful approximate rule for ordinary temperatures and pressures, is to take y as 1.67 for monatomic gases, 1.40 for diatomic gases, 1.30 for simple polyatomic gases, such as water, carbon dioxide, ammonia and methane. It may be noted that the heat capacity ratio for hydrogen gas increases at low temperatures toward the vfdue for a monatomic gas. This matter will be explained in Chapter VI. 

Van t Hoff and Kekule were not alone in recognizing the inherently dynamic character of molecules—relatively few of their contemporaries were "naive ball-and-stick guy[s]." The problem was how to use that awareness to solve chemical problems. On the one hand, chemists were not able to derive very much of practical value from treating molecules as dynamic objects for example, the kinetic theory of gases wasn t even capable of predicting correct heat capacity ratios for polyatomic molecules (Sackur, 1917, pp. 154-166). On the other hand, a large body of previously confusing experimental data readily made sense if one treated molecules as more or less rigid objects. 

An interesting feature of Equation 4-49 is that for ideal gases the choked pressure is a function only of the heat capacity ratio y. Thus ... 

Values for the heat capacity ratio y for a variety of gases are provided in Table 4-3. 

CA 62, 12965(1965). Following is its abstract A criterion for unidimensional instability of detonation of gases is derived, taking into account the thermal effects of the reaction. Instability occurs when [(y -1)/ y](E/RT)[1 /(1 + cg/c) qU>l, where y is the heat capacity ratio, E is the energy of activation, Cg and c are the velocities of sound in the burned and unburned gases, resp, q is the ratio of thermal effect of combustion to the internal energy of the unburned gas, and M is the ratio of the burning rate to the velocity of sound in the unburned gas... 

The adiabatic expansion method is not the best method of determining the heat capacity ratio. Much better methods are based on measurements of the velocity of sound in gases. One such method, described in Part B of this experiment, consists of measuring the wavelength of sound of an accurately known frequency by measuring the distance between nodes in a sonic resonance set up in a Kundt s tube. Methods also exist for determining the heat capacities directly, although the measurements are not easy. 

Heat capacity ratio, k = Cp/c, = 1.04 for gases with molar mass > 100. The value of k increases to 1.67 as the molar mass decreases. For air =1.4 and for such gases as ethylene, carbon dioxide, steam, sulfur dioxide, methane, ammonia = 1.2-1.3. Temperature rise between feed 1 and exit 2 ... 

The ratio of heat capacities y for monoatomic gases is given by... 

For most gases, the heat capacity ratio k is about 1.4, and the ratio of the power needed for adiabatic and isothermal compression is given by ... 

Equations (2.15) through (2.17) also require a value of k, the heat capacity ratio. Table 2.6 provides selected values. For monotonic ideal gases, li = 1.67,for diatomic gases, k = 1.4 and for triatomic gases, k = 1.32. API (1996) recommends a value of 1.4 for screening purposes. 

TABLE 2.6. Heat Capacity Ratios k for Seiected Gases ... 

For gases with specific heat ratios of approximately 1.4, the critical pressure ratio is approximately 0.5. For hydrocarbon service, this means that if the back-pressure on the relief valve is greater than 50% of the set pressure, then the capacity of the valve will be reduced. In other words, if the pressure in the relief piping at the valve outlet is greater than half (he set pressure, then a larger relief valve will be required to handle the same amount of fluid. 

The heat capacity of a subshince is defined as the quantity of heat required to raise tlie temperature of tliat substance by 1° the specific heat capacity is the heat capacity on a unit mass basis. The term specific heat is frequently used in place of specific heat capacity. This is not strictly correct because traditionally, specific heal luis been defined as tlie ratio of the heat capacity of a substance to the heat capacity of water. However, since the specific heat of water is approxinuitely 1 cal/g-°C or 1 Btiiyib-°F, the term specific heal luis come to imply heat capacity per unit mass. For gases, tlie addition of heat to cause tlie 1° tempcniture rise m iy be accomplished either at constant pressure or at constant volume. Since the mnounts of heat necessary are different for tlie two cases, subscripts are used to identify which heat capacity is being used - Cp for constant pressure or Cv for constant volume. Tliis distinction does not have to be made for liquids and solids since tliere is little difference between tlie two. Values of heat capacity arc available in the literature. ... 

Table 13.8 gives the ratio of heat capacities CP/CV for a number of common gases. 

Cp - Cy equals [P + (dU/dV)T](dV/dT)p. The dUldV term is often referred to as the internal pressure and is large for liquids and solids (See Internal Pressure). Since ideal gases do not have internal pressure, Cp - Cy = nP for ideal gases. The ratio of the heat capacities, Cp/Cy, is commonly symbolized by y. 

Effect of Addition of Inert Diluents. The addition of inert gases to an explosive mixture will have two major effects. It will increase the heat capacity of the mixture, and depending upon the nature of the added gas, it will change the mixture thermal conductivity. Equation 26 shows that an increase in the heat capacity of the mixture will tend to increase the induction period. The addition of a high thermal conductivity gas such as helium will increase the limiting pressure. Rearranging Equation 18 shows that for a given vessel diameter, reactant concentration, and furnace temperature, the ratio... 

A similar argument is used to deal with s this is based on the empirical observation that in an adiabatic process involving a noble gas at low pressures, the product PV7 is virtually constant. Here 7 is a fixed quantity (which will later turn out to be the ratio of molar heat capacities at constant pressure and volume) whose exact significance is irrelevant at this stage near room temperature and for monatomic gases 7 has a value close to 5/3. We therefore use the product PV7 as a measure of the empirical entropy through the simple relation... 

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