Heat capacity Cp and thermal expansion coefficients

TABLE 11.2 Measured Thermodynamic Properties (in SI Units) of Some Common Fluids at 20° C, 1 atm Molar Heat Capacity CP, Isothermal Compressibility jS7, Coefficient of Thermal Expansion otp, and Molar Volume V, with Monatomic Ideal Gas Values (cf. Sidebar 11.3) Shown for Comparison... 

Figure 5 (a, b) shows typical heat capacities and thermal expansion coefficient curves for some epoxy-aromatic amine networks. Table 1 gives some numerical values for networks with different component ratios P Cp and P values for the glassy state do not practically depend on the chemical composition of networks and are very... 

The glass transition is characterized in part by an observed second-order transition distinguished by a discontinuity of the Gibbs free energy with respect to the aforementioned state variables, but by continuity of entropy, volume, and enthalpy. Hence, heat capacity, Cp, as well as the thermal expansion coefficient, a, as defined below, both exhibit a discontinuity at the glass transition temperature (McKenna, 1989). 

The coefficient of thermal expansion a of a condensed substance is related to the molar heat capacities cp at constant pressure. The above equation (dv/dT)p = (ds/dp)T for one mole can be differentiated with respect to T and combined with (ds/dT)p = cp/T to obtain the following equation ... 

Fig. 5a and b. Heat capacities Cp (a) and cubic thermal expansion coefficient P (b) for DGER-mPhDA network of different P.T = 5-15 K./min... 

The result for the thermal expansion coefficient, a, which is equal to (dV/dT)/V, is shown in Fig. 13.36 for the cooling and heating process. In the cooling process a decreases gradually from tq to ag. Hysteresis in the volume causes in the subsequent heating process an anomalous effect in the thermal expansion coefficient, depicted by undershoot and overshoot, as also shown in Fig. 13.36. A similar effect occurs in enthalpy H and accordingly in cp, the specific heat capacity, equal to dH/dT. This effect is frequently observed in DSC (Differential Scanning Calorimetry) experiments. 

The ratio dpjdT) is thus determined essentially by the ratio of the configurational heat capacity to the configurational thermal expansion coefficient. This equation may be applied in particular to the case where there is a discontinuity in the heat capacity at the Curie point 1 = 0, = 0 (c/. fig. 19.7). If this discontinuity is written Cp-Cp, and the corresponding discontinuity in the coefficient of expansion is ol - a then at the Curie point... 

To calculate the thermodynamic functions for pure metals, one needs the thermal heat capacity C at ambient pressure. Above the Debye temperature, Cp consists of three parts a) the Dulong-Petit value of 3/cB, b) an additional linear increase proportional to temperature, which can also be seen in the thermal expansion coefficient, and c) an additional amount close to the melting temperature Tm, which results from the formation of defects (mainly vacancies). The last part can be approximated for small concentration as... 

Further properties include the isothermal bulk modulus (Kt), the thermal expansion coefficient (/ ) and the constant pressure heat capacity (Cp). The isothermal bulk modulus is calculated first (or from corrected elastic constants as discussed in Section 6.2) and is usually defined as the Reuss bulk modulus... 

Properties of water p = density, Cp = heat capacity, a = thermal expansion coefficient, A = thermal conductivity, t] = viscosity coefficient 402 Density />/kg m of water at different temperatures and pressures 404 Specific heat capacity Cp/kJ kg of water at different temperatures and pressures 405... 

AH, heat of vaporization Cp heat capacity x, isothermal compressibility a, coefficient of thermal expansion p, dipole moment D, diffusion constant and s, dielectric constant. 

Partial derivatives (dfcan be transformed to (df/Byi)yjjiyi with the help of functional determinants (Jacobi transformation) if the functions jc, = x, (yy ) are known. For practical use all partial derivatives of energy functions U, H, A, and G and of entropy S are reduced to functions of the tabulated material properties a (thermal expansivity coefficient), p (isothermal compresibility coefficient), and Cp (heat capacity at constant pressure) ... 

The material properties here are heat capacity Cp, thermal expansion coefficient a, and isothermal compressibility p. 

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