Coefficients of thermal expansion and compressibility

The dependence of the volume of a solid or liquid on temperature at constant pressure can be expressed by the equation 

Fg is a function of pressure. Experimentally, it is found that the relation between volume and pressure is given by 

The coefficients a and k are usually given more general definitions than are implied by Eqs. (5.1) and (5.3). The general definitions are 

According to Eq. (5.4), a is the relative increase (dV/V) in volume per unit increase in temperature at constant pressure. Similarly, k is the relative decrease in volume ( — dV/V) per unit increase in pressure at constant temperature. 

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In this connection, it is very interesting that the volume and intrachain changes obtained by various experimental methods 24,29,85) [Eq. (101)] agree well with Eq. (56) following from the Tobolsky-Shen semiempirical equation of state or the related phenomenological Eq. (76). The values of y determined from the data are rather small (0.1-0.3). As has been mentioned above, according to the semiempirical approach by Tobolsky and Shen one can formally suggest that the front-factor in Eq. (28) is pressure dependent. If it is really so, then the parameter y for rubbers can be considered as an experimental coefficient similar to the coefficient of thermal expansion and compressibility 29). 

Generally, cation-anion bonding is soft for large cations and anions. In other words, the coefficients of thermal expansion and compressibility are both larger for the A X bond than for the B-X bond. Thus, not only temperature but also pressure should be considered to affect the bond length matching to be estimated in terms of tolerance factor. 

The equations of state discussed so far, the ideal gas law, the van der Waals equation, and others, were relations between p, V, and T obtained from empirical data on the behavior of gases or from speculation about the effects of molecular size and attractive forces on the behavior of the gas. The equation of state for a liquid or solid was simply expressed in terms of the experimentally determined coefficients of thermal expansion and compressibility. These relations applied to systems at equilibrium, but there is a more general condition of equilibrium. The second law of thermodynamics requires the relation, Eq. (10.19),... 

Densities, Coefficients of Thermal Expansion, and Compressibilities of Amorphous Polymers... 

By an assortment of thermodynamic manipulations, the quantities dn/dp and [N (d G/dp )o] can be eliminated from Eq. (10.48) and replaced by the measurable quantities a, /3, and dn/dT the coefficients of thermal expansion, isothermal compressibility, and the temperature coefficient of refractive index, respectively. With these substitutions, Eq. (10.48) becomes... 

The transition between crystalline and amorphous polymers is characterized by the so-called glass transition temperature, Tg. This important quantity is defined as the temperature above which the polymer chains have acquired sufficient thermal energy for rotational or torsional oscillations to occur about the majority of bonds in the chain. Below 7"g, the polymer chain has a more or less fixed conformation. On heating through the temperature Tg, there is an abrupt change of the coefficient of thermal expansion (or), compressibility, specific heat, diffusion coefficient, solubility of gases, refractive index, and many other properties including the chemical reactivity. 

The quantity (dV/dT)P is the coefficient of thermal expansion and [dV/dP)T is the coefficient of compressibility of the liquid. For many liquids, the internal pressure is in the range 2000 to 8000 atm. Because the internal pressure is so much greater than the external pressure,... 

An extension of the procedure for calculating the deton velocities to include those expls which.yield solid carbon as a reaction product has been accomplished by the same investigators (See Ref 32) on the assumption that the volumes of solid and gas are additive, that the gas obeys eq 23 and that the solid has zero coefficients of thermal expansion and basic compression. The composition of the reaction products was assumed to be that of chemical equilibrium at the temp and pressure immediately behind the deton wave, and a numerical procedure, involving successive approximations, was developed for the determination of the composition from a consideration of the simultaneous equilibria involved. This method of calculation was briefly discussed in Ref 39, pp 86-7... 

From the interatomic distances the conclusion is to be drawn that the bonds in the hexagonal layers of atoms in these metals are stronger than those between layers. This conclusion is substantiated by the properties of the crystals, which show basal cleavage and have larger values of the compressibility, coefficient of thermal expansion, and electrical resistance in the direction perpendicular to the basal plane than in this plane. Moreover, measurements of the intensities of... 

The transition at 19° C involves an expansion of 0.0058 cm3/g (Clark and Muus). Sincethe transition temperatureincreaseswith pressure by about 0.013° C per atmosphere (Beecroft and Swenson), the latent heat is about 3.2 cal/g. These values are for the crystal and would be reduced in proportion to the crystalline content. The transition at 30° C is only about one-tenth as large. The over-all increase in entropy at these transitions is about 0.0108 cal deg-1g-1. The portion due to the increase in volume is (a// ) A V, where a is the volumetric coefficient of thermal expansion and / is the compressibility. Since the compressibility of the crystal is not known, this quantity is somewhat uncertain. Using the average of the values of a (Quinn, Roberts, and Work) and p (Weir, 1951) for the whole polymer above and below the transitions, it appears that (a/P)A V is about 0.0041 cal deg 1g 1. The entropy of the transition corrected to constant volume is, therefore, about 0.0067 cal deg g-1. 

TABLE 11.2 Measured Thermodynamic Properties (in SI Units) of Some Common Fluids at 20°C, 1 atm Molar Heat Capacity CP, Isothermal Compressibility pr, Coefficient of Thermal Expansion />, and Molar Volume V, with Monatomic Ideal Gas Values... 

This isothermal bulk modulus (Kj) measured by static compression differs slightly from the aforementioned adiabatic bulk modulus (X5) defining seismic velocities in that the former (Kj) describes resistance to compression at constant temperature, such as is the case in a laboratory device in which a sample is slowly compressed in contact with a large thermal reservoir such as the atmosphere. The latter (X5), alternatively describes resistance to compression under adiabatic conditions, such as those pertaining when passage of a seismic wave causes compression (and relaxation) on a time-scale that is short compared to that of thermal conduction. Thus, the adiabatic bulk modulus generally exceeds the isothermal value (usually by a few percent), because it is more difihcult to compress a material whose temperature rises upon compression than one which is allowed to conduct away any such excess heat, as described by a simple multiplicative factor Kg = Kp(l + Tay), where a is the volumetric coefficient of thermal expansion and y is the thermodynamic Griineisen parameter. 

A study of table 3.04 shows, as of course follows immediately from equation (3.02), that lattice energy increases as interionic distance decreases Other properties also show a systematic dependence on this distance. Thus, as the distance between the ions increases and the lattice energy is reduced the melting point and hardness of the crystals fall progressively, while conversely the coefficients of thermal expansion and of compressibility increase. 

A rubber-like solid is unique in that its physical properties resemble those of solids, liquids, and gases in various respects. It is solidlike in that it maintains dimensional stability, and its elastic response at small strains (<5%) is essentially Hookean. It behaves like a liquid because its coefficient of thermal expansion and isothermal compressibility are of the same order of magnitude as those of liquids. The implication of this is that the intermolecular forces in an elastomer are similar to those in liquids. It resembles gases in the sense that the stress in a deformed elastomer increases with increasing temperature, much as the pressure in a compressed gas increases with increasing temperature. This gas-like behavior was, in fact, what first provided the hint that rubbery stresses are entropic in origin. 

Two other important quantities are the isobaric expansivity ( coefficient of thermal expansion ) and the isothermal compressibility ic defined as... 

The subscript is added to show which factor has been supposed constant during the differentiation. Note the change of dvfdT to dw/dT at constant pressure. The first of equations (4) states that the change in the volume of a gas when heated is equal to the ratio of the increase of pressure with temperature at constant volume, and the change in the elasticity of the gas the second tells us that the ratio of the coefficients of thermal expansion and of compressibility is equal to the change in the pressure of the gas per unit rise of temperature at constant volume. 

In order to have the first term of Eq. (3.28) negative, the pressure should decrease with increasing volume. The second term in brackets does not play any role in the statement of the second law. Because of the negative sign and the square, it is always negative. This means that the statement of the second law in this form explicitly allows any sign of the coefficient of tension and related coefficients such as the ratio of thermal expansion and compressibility coeflBcient. Recall the relation... 

PubI 2005 covers mortars and concrete applied by hand, recasting, spraying, overlaying and patching. Requirements covered include thermal compatibility, elastic modulus, skid resistance, coefficient of thermal expansion and capillary absorption. Materials are divided into structural and non-structural based on compressive strength adhesion and elastic modulus... 

The effect of temperature and pressure on volume is quantified by two coefficients, the volumetric coefficient of thermal expansion, and the coefficient of isothermal compression. The volumetric coefficient of thermal expansion is defined as... 

Maxwell relations. Table 1.1 shows the coefficients of thermal expansion and isothermal compressibility of gas state for some compounds at 298.15 K at 1 atm. 

Table 1.1 Coefficients of Thermal Expansion and Isothermal Compressibility of Some Gas Compounds at 7=298.15 K and P= 1 atm...

This procedure however leads to the omstant volume capacity. The constant pressure heat capacity may be computed from the standard relation Cp—Cv=Va T/Pc where (x is the coefficient of thermal expansion and P is the compressibility. Since the elastic and com iance constants can be found as above, the compressibility may found from the relation... 

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