Thermal pressure coefficient

The terms p, T, and v are characteristic reducing parameters which may be obtained by fitting pressure-volume-temperature data (density, thermal expansion coefficient, and thermal pressure coefficient) for each pure component in the mixture (3,12). Values of p, v, and T are given in Tables I and II. 

Since the glass transition corresponds to a constant value of the relaxation time [15], dTjdP is just the pressure coefficient of Tg. Comparing Equations 24.10 and 24.13, we see that the scaling exponent is related to quantities—thermal pressure coefficient, thermal expansion coefficient, Tg, and its pressure coefficient—that can all be determined from PVT measurements... 

Instead of measuring the force-temperature dependence at constant volume and length, one can measure this dependence at constant pressure and length but in this case it is necessary to introduce the corresponding corrections. The corrections include such thermomechanical coefficients as iso-baric volumetric expansion coefficient, the thermal pressure coefficient or the pressure coefficient of elastic force at constant length 22,23,42). 

Of course, the vapour pressure is very temperature dependent, and reaches P° = 101.325 kPa at the normal boiling point, Tb. The isochoric thermal pressure coefficient, dp/dT)v = otp/KT, can be obtained from the two quantities on the right hand side listed in Table 3.1. Except at T it does not equal the coefficient along the saturation line, (dp/dT)a, which is the normal vapour pressure curve. The latter temperature dependence is often described by means of the Antoine equation ... 

The inverse of the thermal pressure coefficient, (dp/dT)c along the saturation line (dTh/8P)a, determined... 

Here, b is a constant and is a function of T only. For this gas, determine expressions for the isothermal compressibility k and the thermal pressure coefficient (dP/dT)v. These expressions should contain only T, P, 0, d9/dT, and constants. 

Pf (of pure components only) is obtained from the thermal pressure coefficient, y ... 

The thermal pressure coefficients of the pure polymers can be estimated from the ratio of thermal expansion coeffident, a, to isothermal compressibility, as y = oc/Pj. 

Fig. 25. Hypothetical simulated spinodal curves for the phase separation of two polymers on heating, illustrating the effect of the thermal expansion coefficient (a) and thermal pressure coefficient (y). The curves are all simulated using values of S /Sj = rj/fj = 1 X j = = 0 Vf = 100,000 cm ...

The thermal pressure coefficients of the pure polymers can be estimated from the ratio of thermal expansion coefficient, a, to isothermal compressibility, Px, as y = a/px. In the case where y is not available from the literature it can be calculated from solubility parameters which themselves are related to the cohesive energy density (C.E.D.) and hence to the strength of the internal pressure of the structural molecules u5). The binary parameter, S2/S, is obtainable from the method of group contribution given by Bondi116. It can alternatively be calculated by casting shadows of models of the molecules for various orientations, where the area for the monomer unit is estimated from the area of the projections. 

Fig. 25. Hypothetical simulated spinodal curves for the phase separation of two polymers on heating, illustrating the effect of the thermal expansion coefficient (a) and thermal pressure coefficient (y). The curves are all simulated using values of S2/St = r2/r, = 1 Xl2 = Q12 = 0 V = 100,000 cm3 xmor1. With Yj = y2 = 1 (J cm 3K-1) and oij = otj the polymers will always be miscible but if oij = 5 x 10-4 K 1 and a, = 4.7 x 10-4 phase separation occurs with a spinodal as shown (1). Increasing both y values to 1.1 moves the phase diagram to lower temperatures (2). Making y2 = 0.8 and y2 = 1,2 (i.e. keeping the sum the same but increasing the difference) moves the spinodal to one side in composition...

Incidentally, the thermal expansivity measured at constant temperature and pressure, viz. Oj and a, and the thermal pressure coefficients measured at constant temperature and volume, viz. and P, remain the same in arrested as well as in internal equilibrium, i.e. a.j. = ttp = a and p.j, = p = p. In non-equilibrium, i.e. in the states between the two extremes of arrested and internal equilibrium, one finds in general that ol and p.. p. ... 

Free Energy (kcal/mol) Internal Energy (kcal/mol) Entropy (cal-mol deg ) Pressure (atm) Thermal Pressure Coefficient (atm/deg) Heat Capacity (cal-mol deg" )... 

Thermal expansion coefficient a, isothermal compressibility c, and thermal pressure coefficient y can be elucidated from the PVT data observed by the conventional method ... 

An accurate determination of a and c in the low-pressure region is crucially important. To avoid the effect of the well-known anomaly in the immediate vicinity of the NI transition point, the values estimated in the stable region are extrapolated to the phase boundaries. Thermal pressure coefficients can thus be calculated from the relation... 

FIGURE 7.14 Variation of thermal expansion coefficient a, isothermal compressibility k, and thermal pressure coefficient y as a function of temperature, (a-c) The values for MBBE-6 were estimated for an ordinary pressure, (d-f) Since the PVT data reported for 5CB at 0.1 MPa (squares) do not cover a wide range of the nematic state, the analysis was also attempted at 80 MPa (circles). For both compounds, discrete changes oi a, k, and y are observed at the phase boundaries. The phase transition temperatures (Tcn and T n) are indicated by the dashed line in each diagram. 

Orwoll, R. A., Sullivan, V. J., and Campbell, G. C., Thermal pressure coefficients and specific volumes of cyanobiphenyls and their transition entropies at constant volume. Mol. Cryst. Liq. Cryst., 149, 121-140 (1987). 

Response of P and v to changes in T. The response of pressure to a constant-volume change in temperature defines the thermal pressure coefficient,... 

Both of these class 1 derivatives are intensive measurable state functions. The thermal pressure coefficient is the slope of an isomet on a PT diagram and is positive for both liquids and gases. But Yj,-values for liquids are much greater than those for gases representative values are given in Table 3.2. 

The isothermal compressibility (3.3.25), the thermal pressure coefficient (3.3.5), and the volume expansivity (3.3.6) satisfy a triple product rule (Appendix A) ... 

To evaluate other isometric residual properties, we will need the thermal pressure coefficient (3.3.5). Applying its definition to the equation of state (4.5.38), we obtain... 

This is another mean-value theorem on a PT diagram, the slope of the vapor-pressure curve is the mean of the values of the thermal pressure coefficient along the van der Waals loop. Combining (8.2.25) and (8.2.26) gives Clapeyron s equation... 

Table 1 Thermal pressure coefficients y, transition volumes A 7, and volume-dependent transition entropies ASy of n-alkanes and polyoxyethylene (POE) for the crystal-isotropic (Cl) phase transitions conformational entropy changes estimated by the rotational isomeric state approximation are included for comparison...

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