Coefficient of thermal expansion aP

From the equation of state in the form V = V(P, T), we can now define two important derivative properties of the substance the coefficient of thermal expansion aP,... 

The coefficient of thermal expansion ap provides us with a measure of volume change of the system due to a change in temperature at constant pressure. 

Figure 1. Examples of water s thermodynamic anomalies. Dependence on temperature of (a) the isothermal compressibility Kt, (b) the isobaric specific heat Cp, and (c) the coefficient of thermal expansion ap. The behavior of water is indicated by the solid line that of a typical liquid by the dashed line. Data from Ref. [5]. Bottom Schematic illustration of different temperature domains, at atmospheric pressure, of H2O. Only one domain is stable the others are metastable.

The coefficient of thermal expansion ap can also be determined with the knowledge of the variation of the volume with temperature according the equation 10 ... 

For most solids, one can neglect the difference between Pp f (ap f/3 for an isotropic body) and the coefficient of thermal expansion at constant P is usually used. Therefore, we may use P and a without subscripts. Assuming that E and p are independent of temperature and ignoring the change in lateral dimensions during defonnation (i.e. we take the Poisson s ratio p = 0, because this simplification gives effects of only the second order of smallness), one can arrive at relations similar to Eqs. (17)—(21). To do this, it is necessary to replace in Eq. (16) the volume deformation e by e, the modulus K by E and a by p (see Fig. 1). For the simple deformation of a Hookean body the characteristic parameter r is also inversely dependent on strain, viz. r = 2PT/e and sinv = —2PT. It is interesting to note that... 

In Eq. (7) p is the solution density, v is the sound velocity, ap is the coefficient of thermal expansion, Cp is the specific heat, and T is the concentration dependence of the equilibrium. Neglecting activity... 

V)(dV/dT)p is the coefficient of thermal expansion for the pure solvent. The additional temperature derivative is d ne/dT)p —4.3 x 10 (Uematsu and Eranck, 1980), at the standard point indicated above, and ap 3x 10 " K (Eisenberg and Kauzmann, 1969). This entropy contribution is negative and has a magnitude of a small multiple of lcalK mol This magnitude is about a power of ten smaller than typical experimental results. Again, notice that this doesn t make a comparison that would warrant detailed discussion of a standard state for a particular experiment (Friedman and Krishnan, 1973). 

X 10" K" and ap(1000 K) = 10.4 x 10 K". As can be seen in Table 4, these are in much better agreement with the experimental data than are the fluctuation formula results. Particularly striking is the fact that the ratios of both the calculated and the experimental values at the two temperatures are now nearly the same, 1.4 (calc.) vs 1.36 (exp.). (With the fluctuation formula, this ratio is 2.2.) It should also be mentioned that when the coefficient of thermal expansion is obtained from its definition, which involves the volume and not its fluctuation, the convergence is much more rapid than with the fluctuation method. Figure 12 shows that the final volume is reached within a few picoseconds. 

In EQs 1, 2, and 3 k is a constant, denotes wall modulus and its coefficient of thermal expansion. The pressure drop Ap across the preconverter depends on flow rate Q, hydraulic diameter Djj, and open frontal area OFA, as follows ... 

The derivative dv/dP)j can be derived in algebraic form with either a v-explicit EOS or (as its reciprocal) from aP- explicit EOS. But dv/dT)p cannot be easily derived algebraically with a P-explicit EOS. All the commonly used EOSs (see section 2.11 and Appendix F) are P-explicit, and cannot be solved to give simple algebraic expressions for dv/dT)p. For liquids and solids these two derivatives are equal to the coefficient of thermal expansion and the isothermal compressibility, (see Appendix D). Various numerical techniques... 

Kg, gas film coefficient A, surface area of water body 7), diffusion coefficient of compound in air W, wind velocity at 2 m above the mean water surface v, kinematic viscosity of air a, thermal diffusion coefficient of air g, acceleration of gravity thermal expansion coefficient of moist air AP, temperature difference between water surface and 2 m height APv virtual temperature difference between water surface and 2 m height. 

As in the case of and cv, ap does not differ significantly from av at low temperatures. For anisotropic solids, there are two or three (depending on the symmetry of the crystal) principal linear coefficients. For isotropic solids, the volumetric thermal expansion is j8 = 3a. 

The approach to the critical point, from above or below, is accompanied by spectacular changes in optical, thermal, and mechanical properties. These include critical opalescence (a bright milky shimmering flash, as incident light refracts through intense density fluctuations) and infinite values of heat capacity, thermal expansion coefficient aP, isothermal compressibility /3r, and other properties. Truly, such a confused state of matter finds itself at a critical juncture as it transforms spontaneously from a uniform and isotropic form to a symmetry-broken (nonuniform and anisotropically separated) pair of distinct phases as (Tc, Pc) is approached from above. Similarly, as (Tc, Pc) is approached from below along the L + G coexistence line, the densities and other phase properties are forced to become identical, erasing what appears to be a fundamental physical distinction between liquid and gas at all lower temperatures and pressures. 

For most particulate composites the mismatch between the particles and the matrix is more important than the anisotropy of either component (though alumina/aluminium titanate composites provide a notable exception and are described below). The main features of the stresses can therefore be understood in terms of a simple elastic model assuming thermoelastic isotropy and consisting of a spherical particle in a concentric spherical shell of matrix with dimensions chosen to give the appropriate volume fractions. The particles are predicted to be under a uniform hydrostatic stress, ap after cooling. If the particles have a larger thermal expansion coefficient than the matrix, this stress is tensile, and vice versa. For small particle volume fractions the stress... 

Fig. 6a and b. Heat capacity ACP(TS) and cubic thermal expansion coefficient jumps AP(Tg) for some epoxy-amine systems.. Experimental results O DGER-mPhDA networks of different P, completely cured and annealed (T = 0.5°/min) DGEBA-mPhDA networks of different P, completely cured and annealed DGEBA-mPhDA unreacted mixture (P = 1) DGER-mPhDA unreacted mixture (P = 1) solid lines — the best fit... 

The characteristic velocity is determined by the ratio of the characteristic tangential (Marangoni) stress, 0(PAT/L), which drives this motion to the viscous forces ()(p,uc/d) that derive from this motion. The definition (6 212) also allows us to return to the condition for neglect of buoyancy forces compared with Marangoni forces as a potential source of fluid motion in the thin cavity. To do this, we introduce the thermal expansion coefficient, which we denote as a, so that the characteristic density difference Ap = O(paAT). Then the condition (Apge2t2/puc) 1 can be expressed in the form... 

Bond strengths have been measured for the APS chromium oxides, and a study of the effects of various spray parameters (powder composition, powder feed gas, powder feed rate, spray velocity and gun/nozzle selection) has been completed (see previous reports). Property measurements (microhardness, tensile strength and elastic modulus at room temperature and 320 C, thermal expansion coefficient from 25 - 540microstructural characterization of the UTRC coatings have been completed. 

Since LDL and HDL are two different liquids, the behavior of their thermodynamic response functions are quite different. The response functions of a system quantify how a given property, such as pressure, changes under the perturbation of a second property, such as T, under specific conditions, for example, constant volume and mole numbers. The basic response functions of a single component system are the isobaric specific heat, Cp T, P), isobaric thermal expansion coefficient, ap T, P), and isothermal compressibility, Kp T, P), all other response... 

Figure 6. Thermodynamic and structural quantities for the YK fluid with a = 3.3. Left column thermal expansion coefficient ap (units of k /e), isothermal compressibility Kp (units of o /e) and constant-pressure specific heat Cp (units of b) as a function of T along the isobar P = 2.5. For conventional liquids, oip, Kp, and Cp monotonically increase with T and ap > 0. Right column translational order parameter —sz (units of ks), bond-order parameter ge [89). and self-diffusion coefficient D ((units of cr (e/m / )), where m is the particle mass) as a function of P along the isotherm T = 0.06. For conventional liquids, —sz and ge increase with P while D decreases monotonically. Data are from Ref. [88].

The cubic thermal expansion coefficient, determined from the specific volume vs. temperature plot, both before and after the densification process, (run II), is ap = 2.0 10 " (7), typical of that for glassy polymers. 

Close to the gas-liquid critical point one can show that the quantities Sp, Sv, and St, which are small deviations from the critical point in terms of the variables p, V, and t, are simply related to each other as well as to thermodynamic functions like the isobaric thermal expansion coefficient, ap, and the isothermal compressibility, Kt (as well as all others ). ... 

Example Thermal Contraction. According to experience most materials expand when their temperature increases, i.e. their thermal expansion coefficient, ap, cf.(2.5), is positive. However, take a rubber band fixed af one end and sfrefched to some extend by a weight attached to its other end. Upon heating of the rubber band, using a blow torch or a hair dryer capable of producing sufficient heat, a significant contraction is observed. Why does this happen Once again Eq. (5.16) helps to find the answer. First however we must tie the entropy to the thermal contraction just described. 

---