Isobaric coefficient of thermal expansion

Here Q(t) denotes the heat input per unit volume accumulated up to time t, Cp is the specific heat per unit mass at constant pressure, Cv the specific heat per unit mass at constant volume, c is the sound velocity, oCp the coefficient of isobaric thermal expansion, and pg the equilibrium density. (4) The heat input Q(t) is the laser energy released by the absorbing molecule per unit volume. If the excitation is in the visible spectral range, the evolution of Q(t) follows the rhythm of the different chemically driven relaxation processes through which energy is... 

The coefficient of isobaric thermal expansion is defined as the fractional change in volume of a liquid as temperature changes under constant pressure. 

When a value of thermal expansion is reported, it must include the pressure and temperature range for which it is valid. Thermal expansion as defined here must not be used interchangeably with the coefficient of isobaric thermal expansion defined above. 

EXAMPLE 8-6 A sample of reservoir oil was placed in a laboratory cell at 5000psig and 76°F. The volume was 54.74 cc. Temperature was increased to 220°F and pressure was held constant by increasing cell volume to 59.55 cc. Calculate the coefficient of isobaric thermal expansion and calculate the thermal expansion. 

First, calculate the coefficient of isobaric thermal expansion. 

The density of the pseudoliquid is adjusted to reservoir pressure using the coefficient of isothermal compressibility and is adjusted to reservoir temperature using the coefficient of isobaric thermal expansion. 

Since the lattice parameters depend significantly on the temperature (Table 2), it is possible to estimate the coefficient of isobaric thermal expansion roughly to about 2.8x10 K ... 

Some of polymer properties change at the glass transition in discontinuous way (the coefficient of isobaric thermal expansion a, coefficient of isothermal compressibility kj, specific heat, etc), whereas the other ones change continuously (volume V, enthalpy H, and entropy S). As it is schematically shown in Figure 3, the onset of solidlike rigidity in an amorphous polymer at Tg is accompanied by sharp reductions in heat capacity Cp, thermal expansion coefficient up, and compressibility coefficient /cj (22). 

Irvine. 1988. Experimental measurements of isobaric thermal expansion coefficients of Non-Newtonian fluids. Heat Transfer, 1, 2, 155-163. 

We may first assume that isothermal compressibility fiy and isobaric thermal expansion coefficient a are independent, respectively, of T and P. Equations 1.91 and 1.99, integrated on T and P, respectively, give... 

We should remember that certain partial derivatives are quantities that can be measured conveniently the isobaric thermal expansivity (also known as the coefficient of thermal expansion) ... 

Fig. 1. Isobaric relationship between volume and temperature in the liquid, glassy, and crystalline states. Fm is the melting temperature, and Fga and Fgb are the glass transition temperatures corresponding to slow (a) and fast (b) cooling rates. The lower diagram shows the behavior of the thermal expansion coefficient corresponding to curve b. (From Debenedetti, 1996.)...

From the MF analysis, when = 0 the model coincides with the one proposed in which gives rise to the SF scenario (Fig. 3a). When > 0 the model displays a phase diagram with a LLCP (Fig. 3b) [13]. For 0, keeping J and the other parameters constant, we find that Tc 0, and the power-law behavior of Kt and the isobaric thermal expansion coefficient Op is preserved. Further, we find for the entropy S that, for any value of J , (dS/dT)p ... 

The temperature dependence of yields an estimate of the isobaric thermal expansion coefficient parallel to the director axis... 

The temperature dependence of the order parameter profiles yields the isobaric thermal expansion coefficient an, according to Eq. (7). The data from the mixed phase collected in the last column of Table 1 are in broad agreement with thermal expansion eoefficients obtained from X-ray diffraction experiments in pure phospholipid bilayers [22,23] indicating that in the L,i-phase the phospholipid acyl chain motion is not constrained by an increasing proportion of the surfactant. 

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... 

The isobaric thermal expansion coefficients, P = 1 / V(dV/dT)p, for organic crystals are of the order of 2 (a more detailed discussion is in Section 11.4.1), of course much smaller than the corresponding values for liquids. Sample results of a detailed calculation of the state equation of an organic molecule by MD, carried out in conjunction with a temperature-dependent experimental determination of the crystal densities by X-ray diffraction [20], are shown in Fig. 9.6 for succinic anhydride. 

Nuclear motion drags along the electronic cloud, so that as temperature rises, molecular envelopes oscillate more and more. If the intermolecular potential were perfectly harmonic, the overall volume effect would be nil, because the compressions and expansions would average out but the potential is much steeper on the compression side (Fig. 4.4), so expansion is hindered less than contraction and molecules effectively occupy more and more space as mobility increases. So thermal expansion is very strictly dependent on the shape of the potential curve, that is on the strength and anisotropy of the intermolecular potential, in a typical structure-property relationship. The simple equation that defines the isobaric thermal expansion coefficient a is... 

Figure 4.9 shows the temperature dependence of the isobaric thermal expansion coefficient, Up, below, at, and above the critical pressure. The dashed line is the ideal gas result. Below the critical pressure a jump occurs when the gas-liquid saturation line (cf. Fig. 4.4 right panel) is crossed. At the critical point we observe a divergence. Above the critical point a maximum marks the smooth continuation of the gas-liquid saturation line, which sometimes is called Widom line. 

Fig. 4.9 Temperature dependence of the isobaric thermal expansion coefficient for different pres-sures...

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 ). ... 

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