Entropy as a function of temperature and pressure

Figure 1.4 The entropy of ideal He gas as a function of pressure and temperature is restricted to the surface shown in the figure. Thus, specifying p and T fixes 5m.

In Fig. XI-6 we show the entropy of water in its three phases, as a function of pressure and temperature, computed as we have described above. We are struck by the resemblance of this figure to that giving the volume, Fig. XI-4 the entropy, like the volume, increases with increase of temperature or decrease of pressure. Lines of constant pres-... 

Among the most fundamental knowledge about the thermodynamic properties of hydrogen is the equation of state (EoS), namely, the volume as a function of pressure and temperature V p,T). Once the EoS is known, all the thermodynamic quantities can be calculated. The Gibbs energy G(p,T) and entropy S p,T) can be obtained by integration... 

Thus, in conclusion, we have the following procedure for calculation of the entropy S T,p) for solid, liquid and gaseous substances as a function of pressure and temperature... 

The entropy is most easily determined as a function of volume and temperature from the equation (dS/dT)v = Cv/T. At the absolute zero of temperature, the entropy of a solid is zero independent of its volume or pressure. The reason goes back to our fundamental definition of entropy... 

This thermodynamic diagram provides descriptions of fluid properties as a function of pressure and enthalpy. One can follow lines of constant temperature, density, or entropy. The (mostly) horizontal curves represent constant density, and the (mostly) vertical curves are lines of constant entropy. 

Figure 10. LCT configurational entropy ScT as a function of the reduced temperature 5T = (T — To)/To for low and high molar mass F-F and F-S polymer fluids at constant pressure of P = 1 atm (0.101325 MPa). The product ScT is normalized by the thermal energy k To at the ideal glass transition temperature To- (Used with permission from J. Dudowicz, K. F. Freed, and J. F. Douglas, Journal of Physical Chemistry B 109, 21350 (2005). Copyright 2005 American Chemical Society.)...

Figure 22. The configurational entropy Sc per lattice site as calculated from the LCT for a constant pressure, high molar mass (M = 40001) F-S polymer melt as a function of the reduced temperature ST = (T — To)/Tq, defined relative to the ideal glass transition temperature To at which Sc extrapolates to zero. The specific entropy is normalized by its maximum value i = Sc T = Ta), as in Fig. 6. Solid and dashed curves refer to pressures of F = 1 atm (0.101325 MPa) and P = 240 atm (24.3 MPa), respectively. The characteristic temperatures of glass formation, the ideal glass transition temperature To, the glass transition temperature Tg, the crossover temperature Tj, and the Arrhenius temperature Ta are indicated in the figure. The inset presents the LCT estimates for the size z = 1/of the CRR in the same system as a function of the reduced temperature 5Ta = T — TaI/Ta. Solid and dashed curves in the inset correspond to pressures of P = 1 atm (0.101325 MPa) and F = 240 atm (24.3 MPa), respectively. (Used with permission from J. Dudowicz, K. F. Freed, and J. F. Douglas, Journal of Physical Chemistry B 109, 21350 (2005). Copyright 2005, American Chemical Society.)...

Finally, reference is made to a series of articles by Griskey et al. (1966,1967) mentioning values for enthalpy and entropy as a function of temperature and pressure for a number of commercial plastics. 

The rates of substitution of the latter ligand by TU, DMTU and TMTU were followed as a function of nucleophile concentration, temperature and pressure by s.f. spectrophotometry. The reaction was first order in both platinum complex and nucleophile concentrations. From the form of the rate law and the negative entropies and volumes of activation it was concluded that the mechanism is an associ-atively activated substitution. It prevailed that substitution in the terpy parent ligand did not affect significantly the kinetic parameters. The reaction was slower when a carbon a-donor was in the cis position than when an N a-donor occupies this position, indicating a different situation from the effect of a Pt-C bond in the trans position. 

This simply shows that there is a physical relationship between different quantities that one can measure in a gas system, so that gas pressure can be expressed as a function of gas volume, temperature and number of moles, n. In general, some relationships come from the specific properties of a material and some follow from physical laws that are independent of the material (such as the laws of thermodynamics). There are two different kinds of thermodynamic variables intensive variables (those that do not depend on the size and amount of the system, like temperature, pressure, density, electrostatic potential, electric field, magnetic field and molar properties) and extensive variables (those that scale linearly with the size and amount of the system, like mass, volume, number of molecules, internal energy, enthalpy and entropy). Extensive variables are additive whereas intensive variables are not. 

A related thermodynamic method is to fit the temperature and density dependence of simulation data to equations of state for the solid and liquid.[158] The explicit forms of the pressure P(T, p) and internal energy U T, p) equations for each phase are used to calculate the entropy and free energy. The condition AG = 0 determines the solid-liquid equilibrium temperature as a function of pressure. 

Expanding the entropy as a function of temperature and pressure results in... 

These equations require the enthalpy and entropy of the system and its various streams. The problem, then is how to calculate the properties of mixture. The properties of pure substance are functions of pressure and temperature. For mixtures, in addition to pressure and temperature we must consider composition. The goal in this chapter is to formulate equations for the properties of mixtures as a function of pressure, temperature and composition. Essentially, the task will be to incorporate composition as a new independent variable. We will express composition in terms of moles of each component, therefore, when we refer to composition as an independent variable we will understand not a single variable but a set of variables. First we will develop formal mathematical expressions for properties, their differentials and their partial derivatives in terms of the independent variables, temperature, pressure and moles of component i. Next we will show how we can use equations of state to calculate the volume, enthalpy and entropy of mixtures. The mathematical formulation is based on the material developed in Chapter r. A review of that chapter is recommended, as we will make frequent references to results obtained there. 

Phase transitions are classified according to the partial derivatives of the Gibbs energy. Ordinary phase transitions such as vaporizations, freezings, and so on, are called first-order phase transitions, which means that at least one of the first derivatives dG /dT) p or (dGta/dP)T is discontinuous at the phase transition. In most first-order transitions, both of these derivatives are discontinuous. From Chapter 4 we know that (9Gm/9r)/> is equal to -5m and that (9Gm/9F)r is equal to Fm- Figure 5.7 shows schematically the molar volume as a function of pressure as it would appear for a solid-liquid or a solid-solid transition. Figure 5.8 shows schematically the molar entropy as a function of temperature as it would appear for a liquid-vapor transition. 

FIGURE 3.5 Coverage of species A plotted as a function of pressure at three different values of adsorption enthalpy, A//. The temperature is 300 K and the standard adsorption entropy is... 

The use of entropy in solving problems requires a mathematical relationship for entropy as a function of temperature and pressure. We return to the first law,... 

A more recent compilation includes tables giving temperature and PV as a function of entropies from 0.573 to 0.973 (2ero entropy at 0°C, 101 kPa (1 atm) and pressures from 5 to 140 MPa (50—1400 atm) (15). Joule-Thorns on coefficients, heat capacity differences (C —C ), and isochoric heat capacities (C) are given for temperatures from 373 to 1273 K at pressures from 5 to 140 MPa. 

Values for the free energy and enthalpy of formation, entropy, and ideal gas heat capacity of carbon monoxide as a function of temperature are listed in Table 2 (1). Thermodynamic properties have been reported from 70—300 K at pressures from 0.1—30 MPa (1—300 atm) (8,9) and from 0.1—120 MPa (1—1200 atm) (10). 

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