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110 Thermochemical Processes: Principles and Models Another property of gases which appears in the Reynolds and the Schmidt numbers is the viscosity, which results from momentum transfer across the volume of the gas when there is relative bulk motion between successive layers of gas, and the coefficient, , is given according to the kinetic theory by the equation D .1/3/c D 3 mcd2 D 1.81 ð 10 5 .MT/1/2 poise where is the density of the gas, and d is in angstroms .10 8 cm/. The viscosity of a gas mixture, mix, can be calculated from the equation xiim1/2 mix xim1/2 The viscosity increases approximately as T1/2, and there is, of course, no vestige of the activation energy which characterizes the transport properties of condensed phases. The thermal conductivity is obtained in terms of and c through the equation D .1/3/Cvc D Cv where Cv is the specific heat at constant volume. The heat capacity at constant volume of a polyatomic molecule is obtained from the equipartition principle, extended to include not only translational, but also rotational and vibrational contributions. The classical values of each of these components can be calculated by ascribing a contribution of R/2 for each degree of freedom. Thus the transla-tional and the rotational components are 3/2R each, for three spatial compo-nents of translational and rotational movement, and .3n 6/R for the vibra-tional contribution in a non-linear polyatomic molecule containing n atoms and .3n 5/R for a linear molecule. For a diatomic molecule, the contributions are 3/2Rtrans C Rrot C Rvib. The classical value is attained by most molecules at temperatures above 300K for the translation and rotation components, but for some molecules, those which have high heats of formation from the constituent atoms such as H2, the classical value for the vibrational component is only reached above room temperature. Consideration of the vibrational partition function for a diatomic gas leads to the relation E E0 Rxe x T 1 e x Vapour phase transport processes 111 where E0 is the zero point energy and x is equal to h/kT. By differentiation with respect to temperature the heat capacity at constant volume due to the vibrational energy is ∂E Rx2 v ∂T V 2.cosh x 1/ This function approaches the classical R value of 8.31Jmol 1 K 1, when x is equal to or less than 0.5. Above this value, the value of Cv decreases to four when x reaches 3 (Figure 3.6). 5 4 3 2 1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 T/q Figure 3.6 The heat capacity of a solid as a function of the temperature divided by the Debye temperature Treating the atomic vibration as simple harmonic motion yields the expres- sion s D 2 where k is the force constant and is the reduced mass defined by 1 1 1 mA mB in the AB molecule. The force constant is roughly inversely proportional to the internuclear distance, the product, kd2, having the value about 8 ð 10 2 Nnm 1 for the hydrogen halide molecules. 112 Thermochemical Processes: Principles and Models It follows from this discussion that all of the transport properties can be derived in principle from the simple kinetic theory of gases, and their inter-relationship through and c leads one to expect that they are all characterized by a relatively small temperature coefficient. The simple theory suggests that this should be a dependence on T1/2, but because of intermolecular forces, the experimental results usually indicate a larger temperature dependence even up to T3/2 for the case of molecular inter-diffusion. The Arrhenius equation which would involve an enthalpy of activation does not apply because no ‘acti-vated state’ is involved in the transport processes. If, however, the temperature dependence of these processes is fitted to such an expression as an algebraic approximation, then an ‘activation enthalpy’ of a few kilojoules is observed. It will thus be found that when the kinetics of a gas–solid or liquid reaction depends upon the transport properties of the gas phase, the apparent activation enthalpy will be a few kilojoules only (less than 50kJ). Some typical results for the physical properties of common gases which are of industrial importance are given in Table 3.3. The special position of hydrogen which results from the small mass and size of the H2 molecule should be noted. Equations of state for ideal and real gases The equation of state for a gas consisting of non-interacting point particles has the form PV D RT for one mole of gas The assumptions involved in this equation clearly do not accurately describe real gases in which the atoms or molecules interact with one another, and occupy a finite volume in space the size of which is determined by the complexity and mass of the particles. The first successful attempt to improve on the ideal equation was that of van der Waals .P C a/V2/.V b/ D RT The correcting term in the pressure reflects the diminution in the impact velocity of atoms at the containing walls of the gas due to the attraction of the internal mass of gas, and the volume term reflects the finite volume of the molecules. Data for these two constants are shown in Table 3.4. The interaction forces which account for the value of a in this equation arise from the size, the molecular vibration frequencies and dipole moments of the molecules. The factor b is only related to the molecular volumes. The molar volume of a gas at one atmosphere pressure is 22.414lmol 1 at 273K, and this volume increases according to Gay–Lussac’s law with increasing Vapour phase transport processes 113 Table 3.3 Thermophysical properties of common gases Temp Viscosity (K) (micropoise) Sp. heat at constant pressure .Jg 1 K 1/ Thermal conductivity .Wcm 1 K 1 ð 105/ H2 300 84 14.48 166.53 1100 210 14.72 447.69 Ar 300 209 0.519 15.90 1100 550 0.519 43.93 N2 300 170 1.046 5.65 1100 415 1.138 64.85 O2 300 189 0.941 24.10 1100 500 1.067 75.32 CO 300 166 1.046 22.97 1100 450 1.142 67.36 CO2 300 139 0.994 14.35 1100 436 1.025 71.96 SO2 300 116 0.678 8.53 800 310 0.786 33.93 H2OŁ 400 125 1.924 23.89 700 241 2.025 55.23 Note the smaller range of temperature for SO2 and H2O. This was due to lack of high temperature viscosity data. Table 3.4 van der Waals constants for some common gases H2 a D 0.244l2 atmosmol 2 O2 1.36 H2O 5.46 CO 1.49 CO2 3.59 HCl 3.8 SO2 6.7 b D 26.6 ð 10 3 lmol 1 31.8 30.5 39.9 42.7 41.0 56.0 temperature. At a temperature T(K) VT D V273T/273 Clearly the effects of the van der Waals corrections will diminish significantly at 1000K, and the ideal gas approximation will become more acceptable. The 114 Thermochemical Processes: Principles and Models effects will also diminish considerably as the pressure is decreased below one atmosphere. Expressing the deviation from the ideal gas laws by the parameter z, so that PV D zRT the value of z for SO2 at 1000K and one atmos pressure, at which temperature the molar volume is 82.10l, is less than 1.001, compared with 1.013 at 298K. The effects of the constants in the van der Waals equation become more marked as the pressure is increased above atmospheric. Early measurements by Regnault showed that the PV product for CO2, for example, is considerably less than that predicted by Boyle’s law P1V1 D P2V2 the value of this product being only one quarter, approximately, of the pre-dicted value at 100 atmos using one atmosphere data, i.e. the molar volume is 22.414 litres at room temperature. The parameter z can be obtained from Regnault’s results and these show a value of z of 1.064 for hydrogen, 0.9846 for nitrogen, and 0.2695 for carbon dioxide at room temperature and 100 atmospheres pressure. These values are related to the corrections introduced by van der Waals. Molecular interactions and the properties of real gases The classical kinetic theory of gases treats a system of non-interacting parti-cles, but in real gases there is a short-range interaction which has an effect on the physical properties of gases. The most simple description of this interac-tion uses the Lennard–Jones potential which postulates a central force between molecules, giving an energy of interaction as a function of the inter-nuclear distance, r, ( 12 6) E.r/ D 4ε r r where dc is the collision diameter, and ε is the maximum interaction energy. The collision diameter is at the value of ε.r/ equal to zero, and the maximum interaction of the molecules is where ε.r/ is a minimum. The interaction of molecules is thus a balance between a rapidly-varying repulsive interaction at small internuclear distances, and a more slowly varying attractive interaction as a function of r (Figure 3.7). Chapman and Enskog (see Chapman and Cowling, 1951) made a semi-empirical study of the physical properties of gases using the Lennard–Jones ... - tailieumienphi.vn
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