Entropy as function of T and

Enthalpy and Entropy as Functions of T and P At constant composition the molar thermodynamic properties are functions of temperature and pressure (Postulate 5). Thus... 

Thermodynamics. (21) This is to determine the vdume and entropy as functions of T and P by using equation 8. It involves the eos and the temperature dependent conOgimitional heat capacity Cn at atmospheric pressure. The polymer of choice was Bisphend-A-Polycaibonate. The eos and positron data are available. (22,23) The total heat capacity Cp of the glass is linear in T. (24) The assumption then is a corresponding expression for Cp (config.). 

Equations (4-34) and (4-35) are general expressions for the enthalpy and entropy of homogeneous fluids at constant composition as functions of T and P. The coefficients of dT and dP are expressed in terms of measurable quantities. 

The most useful property relations for the enthalpy and entropy of a homogeneous phase result when these properties are expressed as functions of T and P. What we need to know is how H and S vary with temperature and pressure. This information is contained in the derivatives (BH/BT)P, (dS/dT)P, (BH/BP)t, and (dS/dP)T. 

Piezoelectric solids are characterized by constitutive relations among the stress t, strain rj, entropy s, electric field E, and electric displacement D. When uncoupled solutions are sought, it is convenient to express t and D as functions of t], E, and s. The formulation of nonlinear piezoelectric constitutive relations has been considered by numerous authors (see the list cited in [77G06]), but there is no generally accepted form or notation. With some modification in notation, we adopt the definitions of thermodynamic potentials developed by Thurston [74T01]. This leads to the following constitutive relations ... 

The enthalpy is useful in considering isentropic and isobaric processes, but often it becomes necessary to rather deal with isothermal and isobaric processes. In such case one needs a thermodynamic function of T and P alone, defining the Gibbs potential G = U(T, P, Nj) as the Legendre transform of U that replaces entropy by temperature and volume by pressure. This transform is equivalent to a partial Legendre transform of the enthalpy,... 

In lambda transitions, no discontinuity in enthalpy or entropy as a function of T and/or P at the transition zone is observed. However, heat capacity, thermal expansion, and compressibility show typical perturbations in the lambda zone, and T (or P) dependencies before and after transition are very different. 

So far we have considered only the stability of a phase with respect to perturbations at constant U and F. A similar calculation may be made keeping T and V constant. In this case, instead of calculating the perturbation of the entropy we consider the Helmholtz free energy, F, and write the molar free energy / as a function of T and v. The molar volumes are perturbed by b v and h"v as in (15.21) but since the perturbation is isothermal 8 T — b"T = 0. We then obtain a formula... 

Figure 32. Entropy as a function of temperature for different modifications of water equilibrium (5), degassed structured (S), degassed with disordered structure (S). The heat capacities of water Cp and Cv as functions of T are plotted as well. (From Ref. 359.)...

In general, volume V and entropy S of the supercooled liquid can be treated as functions of P and T. We may therefore write. 

Since Tf is also a function of T and P, as both the entropy terms are, one can calculate the heat capacity of a glass in the region of glass transition as... 

Statistical mechanics provides relations between macroscopic thermodynamic quantities and microscopic molecular properties. Thermodynamics, on the other hand, provides only relationships between various thermodynamic quantities. The multitude of these relationships arise from the freedom we have in choosing the independent variables to describe a thermodynamic system. For instance, we can choose the variables T, V, N to describe the system. Hence, all the other variables such as energy, entropy, pressure, etc., are viewed as functions of these independent variables or we could choose T, P, N to describe the system and view all other variables, such as energy, entropy, volume, etc., as functions of T, P, N. 

While such problems are not insurmountable, not everyone can nor does determine the phase ratios. An alternative approach involves determining the separation factor, a, where a = k n/k s Here the subscripted R and S refer to the Kahn-Prelog-Ingold stereochemical descriptors for enantiomers. Equation (6) can now be converted to a linear equation in terms of a as a function of T and differential enthalpies and entropies of binding, AAH and AAS, as ... 

Chemical potential was introduced as a partial property of the Gibbs free energy to solve the phase equilibrium problem. Similar partial properties may be considered for other extensive properties, as volume, enthalpy, entropy, etc. It would be useful to generalise the approach. Let s consider again that M represents the mean molar value of a property. For the whole system we have nM = f P,P,n,n2,nj,... The derivation of the nM) as function of T, P, and composition gives ... 

The stated goal for this chapter is to obtain equations for the enthalpy and entropy in terms of pressure and temperature.5 We approach this as a calculus problem if enthalpy is taken to be a function of T and P, the general form of its differential must be,... 

The importance of eqs. f and f 5.40 ) is that they relate residual enthalpy and entropy to the equation of state. In the above form, these equations are useful if the equation of state can be expressed as function of pressure and temperature. Cubic equations express pressure in terms of volume and temperature to calculate residual properties from such equations, eqs. and must be converted so that the independent variables are V and T. The mathematical manipulations are shown on next page. 

Here, the subscripts E and M denote ethane and methane, respectively. In (4.7.5), we view each quantity as a function of T and P hence, the corresponding entropy change for this particular process is... 

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