Thermochemical Processes Episode 11

Physical properties and applications of liquid metals 293 given by Furth’s equation p.j/dj D Cj2 exp [E.j//kT]dj where C is a constant, and the average distance being Z 1 ,Z 1 j D 0 j4 exp kT dj 0 j2 exp kT dj Swalin then uses a Maclaurin expansion of the Arrhenius term to obtain the energy of formation of a cavity in the liquid which permits a small jump, to obtain the equation 2 3ZN0kT 16Hv˛ and hence using Einstein’s equation for the diffusive movement of a particle moving randomly in three dimensions, as in Brownian motion, 2 D D 6 where D h Z with obtained from a transition state theory equation for the movement of a particle over an energy barrier with Z neighbours, results in 3Z2N0k2T2 96Hvh˛ The Morse function which is given above was obtained from a study of bonding in gaseous systems, and this part of Swalin’s derivation should prob-ably be replaced with a Lennard–Jones potential as a better approximation. The general idea of a variable diffusion step in liquids which is more nearly akin to diffusion in gases than the earlier treatment, which was based on the notion of vacant sites as in solids, remains as a valuable suggestion. Further support for this approach is provided by modern computer studies of molecular dynamics, which show that much smaller translations than the average inter-nuclear distance play an important role in liquid state atom move-ment. These observations have confirmed Swalin’s approach to liquid state diffusion as being very similar to the calculation of the Brownian motion of suspended particles in a liquid. The classical analysis for this phenomenon was based on the assumption that the resistance to movement of suspended particles in a liquid could be calculated by using the viscosity as the frictional force in the Stokes equation F D 3dU 294 Thermochemical Processes: Principles and Models where d is the particle diameter, and U is the (constant) velocity of the particle through the liquid of viscosity . This, when combined with a diffusion compo-nent obtained from a random walk description yields the Stokes–Einstein equation for Brownian movement. This calculation was then extended to the movement of atoms in liquids, by substituting the diameter of atoms for the diameter of the particles. kT 3d The viscosity therefore replaces the restraint on diffusion arising from the interaction of atoms expressed by the Morse potential in Swalin’s treatment. The introduction of molecular dynamical considerations suggests that the the use of the atomic diameter in the Stokes–Einstein equation should be replaced by an expression more accurately reflecting the packing fraction of atoms in liquids, i.e. the volume available to an atom in a close-packed arrangement compared to that which is occupied in a liquid. An average value of this function for liquid metals is about 0.47, corresponding to a ratio of the distance of closest approach of the atoms in a liquid metal to the atomic radius of about 1.55. Each atom must be considered as moving in a ‘cage’ of nearest neighbours which is larger than that afforded by close packing, as in a solid. Thermophysical properties of liquid metals Viscosities of liquid metals The viscosities of liquid metals vary by a factor of about 10 between the ‘empty’ metals, and the ‘full’ metals, and typical values are 0.54 ð 10 2 poise for liquid potassium, and 4.1 ð 10 2 poise for liquid copper, at their respective melting points. Empty metals are those in which the ionic radius is small compared to the metallic radius, and full metals are those in which the ionic radius is approximately the same as the metallic radius. The process was described by Andrade as an activated process following an Arrhenius expression D 0 expQvis/RT poise where Qvis has a value of about 5–25kJ, and Eyring et al. have suggested that the viscosity is determined by the flow of the ion cores and if the energy for the evaporation of metals Evap is compared with that of viscosity, .Evap/Qvis/ ð .rion/rmetal/3 D 3 to 4 Physical properties and applications of liquid metals 295 A further empirical expression, due to Andrade, for the viscosity of liquid metals at their melting points, which agrees well with experimental data is D 5.1 ð 10 4.MTM/0.5V poise where M is the molecular weight in grams, TM is the melting point, and V is the molar volume in cm3. A further point to note is that the viscosities of liquid metals are similar to that of water at room temperature, about 10 2 poise, and so useful models of the behaviour of high-temperature processes involving liquid metals can be made easily visible at room temperature using water to substitute for metals, and a suitable substitute for other phases, usually liquid salts or metallurgical slags, which can have up to 10 poise viscosity. In connection with the earlier consideration of diffusion in liquids using the Stokes–Einstein equation, it can be concluded that the temperature dependence of the diffusion coefficient on the temperature should be T.exp. Qvis/RT// according to this equation, if the activation energy for viscous flow is included. Surface energies of liquid metals A number of experimental studies have supplied numerical values for these, using either the classical maximum bubble pressure method, in which the maximum pressure required to form a bubble which just detaches from a cylinder of radius r, immersed in the liquid to a depth x, is given by pmax D xmaxg C 2 where is the surface energy, and max is the maximum radius of the bubble just before detachment, or the Rayleigh equation for the oscillation frequency ω in shape of a freely suspended levitated drop of mass m in an electromagnetic field, which is related to the surface energy by D 3mω2 The resulting data for liquid metals indicate a systematic relationship with the bonding energy of the element, which is reflected in the heat of vaporization Hvap. Skapski suggested an empirical equation Hvap V2/3 where K is a universal constant, and Vm is the molar volume, which provides a fair correlation among the data for elements. Because of the relatively high diffusion coefficients in liquids, and the probability of the rapid convection current distribution of solute elements to their equilibrium sites, the surface energies of liquid metals are found to be very sensitive to the presence 296 Thermochemical Processes: Principles and Models of surface active elements, and to be substantially reduced by non-metallic elements such as sulphur and oxygen, in the surrounding atmosphere. Great care must therefore be taken in the control of the composition of the gaseous environment to assure accurate data for liquid metals. Table 10.2 shows some representative results for elements which should be compared with the data for the corresponding solids (Table 10.2). Table 10.2 Surface energies of liquid elements Element Sodium Antimony Bismuth Lead Indium Magnesium Germanium Gallium Silicon Zinc Aluminium Silver Copper Uranium Iron Cobalt Surface energy (mJm 2) 197 371 382 457 556 577 607 711 775 789 871 925 1330 1552 1862 1881 liquid vap (kJmol 1) 104 244 198 190 237 138 325 267 400 123 320 274 326 522 401 408 It will be observed that the surface energy is also approximately proportional to the melting point. Surface energies of liquid iron containing oxygen or sulphur in solution yield surface energies approximately one-half of that of the pure metal at a concentration of only 0.15 atom per cent, thus demonstrating the large change in the surface energy of a metal when a small amount of some non-metallic impurities is adsorbed to the surface of the metal. Thermal conductivity and heat capacity The conduction of heat by liquid metals is directly related to the electronic structure. Heat is carried through a metal by energetic electrons having Physical properties and applications of liquid metals 297 translational energies above the energy distribution of the metal ion cores. The conductivity can therefore be calculated using the Lorenz modification of the Wiedemann–Franz ratio T D a constant where the constant for liquid metals is about 2.5WohmK 2. For liquid silver near the melting point, this value is 2.4, and the corresponding value for the solid metal is approximately the same. The thermal conductivity would therefore be about 3.8Wcm 1 K 1. 90 = Sodium = Zinc 80 = Tin 70 = Lead 60 50 40 30 20 10 300 400 500 600 700 800 900 1000 Temperature (K) 35 30 25 2300 500 700 900 = Sodium = Zinc = Tin = Lead Temperature (K) Figure 10.1 Thermal conductivities and heat capacities of the low-melting elements Na, Zn, Sn and Pb ... - tailieumienphi.vn