Thermodynamic formulas

These equations may be used directly to predict the effect of pressure on the chemical reactions preceding or following the electron transfer step and, by use of standard thermodynamic formulae, they may be modified to allow a consideration of the electron transfer step itself. For example, the electrode reaction... 

All the solubilities so far mentioned have been presented in weight or concentration units. These may be converted via standard thermodynamic formulas into Gibbs free energies of solution. One group of authors, however, has preferred to present their data in this form, for YC13, LaClj, and six other lanthanide trichlorides (184). 

The enthalpy and entropy of formation at the melting point are calculated from Eq. (127) using standard thermodynamic formulas. 

TABLE 13.3 T-dependent Equilibrium Constant (KT), Gibbs Free Energy of Reaction (AGT), and Overall Entropic Shift (AAG = AG12oo — AG90o) for the Water Gas Shift Reaction (cf. Tables 13.1, 13.2, and Text), as Determined from Theoretically ( B3LYP ) or Empirically ( Hill ) Evaluated Statistical Thermodynamic Formulas Versus Experiment ( Exp. )... 

The chemical potential pB of species B in binary solution is found by the standard thermodynamic formula... 

Bridgman, P. W. (1936). Condensed Collection of Thermodynamic Formulas. Cambridge, MA Harvard University Press. 

Because for systems, unlike particles, there are no requirements for them to be indistinguishable, we use the thermodynamic formulas analogous to those for distinguishable particles. For systems containing N particles and having volume... 

When the number of spins is very large (i.e. in a magnetic material) the statistical thermodynamical formulae involving the partition function Z are the following ... 

In any case, even with much more complicated systems, the work done1 will have an analogous form for Eq. (1.1) is simply a force (P) times a displacement (rfF), and we know that work can always be put in such a form. If there is occasion to set up the thermodynamic formulas for a more general type of force than a pressure, we simply set up dW in a form corresponding to Eq. (1.1), and proceed by analogy with the derivations which we shall give here. 

The Elementary Partial Derivatives.—We can set up a number of familiar partial derivatives and thermodynamic formulas, from the information which we already have. We have five variables, of which any two are independent, the rest dependent. We can then set up the partial derivative of any dependent variable with respect to any independent variable, keeping the other independent variable constant. A notation is necessary showing in each case what are the two independent variables. This is a need not ordinarily appreciated in mathematical treatments of partial differentiation, for there the independent variables are usually determined in advance and described in words, so that there is no ambiguity about them. Thus, a notation, peculiar to thermodynamics, has been adopted. In any partial derivative, it is obvious that the quantity being differentiated is one of the dependent variables, and the quantity with respect to which it is differentiated is one of the independent variables. It is only necessary to specify the other independent variable, the one which is held constant in the differentiation, and the convention is to indicate this by a subscript. Thus (dS/dT)P, which is ordinarily read as the partial of S with respect to T at constant P, is the derivative of S in which pressure and temperature are independent variables. This derivative would mean an entirely different thing from the derivative of S with respect to T at constant V, for instance. 

Methods of Deriving Thermodynamic Formulas.—We have now introduced all the thermodynamic variables that we shall meet P, W T, S, i/, H, A, G. The number of partial derivatives which can be formed... 

This is the same value for the entropy change obtained earlier from the classical thermodynamic formula for ideal gases. 

With a knowledge of the vibrational frequencies for the normal modes of N2 and CO2 and the appropriate statistical thermodynamic formulae (see Exp. 37), one can calculate quite accurate C values for N2 and CO2. 

The thermodynamic functions for the first and second ionization processes were evaluated from Equations 6, 7, and 8 (for pKi) and 12, 13, and 14 (for pK2) in 10, 30, and 50 mass % THF-H20, respectively, using the customary thermodynamic formulae. The values of the standard Gibbs energy (AG°), enthalpy (AH°), entropy (AS°), and heat capacity... 

This will be recognized as the classical result with the additional normalizing factor in the denominator. It is valid only when fcT h /SmP, where the volume of the gas V = Since the presence of constants in Q does not affect the relative values of thermodynamic quantities, the quantized molecules will have the same thermodynamic formula for the... 

Ionic solutions were at first treated in the same way as nonelectrolytic solutions, though the latter do not contain (interacting) charged species. The starting point was the classical thermodynamic formula for the chemical potential q,- of a nonelectrolyte solute... 

In this way he showed that the variation of the transition temperature with pressure is in close agreement with the thermodynamical formula (deduced in Chapter VII. p. 221) for the dependence of the melting point and the transition point on... 

The corresponding excess entropy, enthalpy, volume, energy, heat capacity and compressibility are readily obtained by use of the general thermodynamic formulae. 

The thermodynamic formulae for thermochromatography will be discussed in more detail later in this chapter. The one essential for the purpose is [4]... 

Temperature values to be used for the currently valid thermodynamic formulas (EOS-80) of seawater (Fofonoff and Millard, 1983 Mamayev et al., 1991) need to be expressed in the International Practical Temperature Scale of 1968 (IPTS-68). Practically measured temperatures are assumed to be in IPTS-68 before January 1, 1990, and in the International Temperature Scale of 1990 (ITS-90) afterward (Mamayev et al., 1991). Temperatures required for the thermodynamic potential functions for water, ice, and seawater need to be expressed in ITS-90, see Section 20.2.1. 

In order to calculate AGciay or AGsoln (Eq. [8] and [9]) with the free energy perturbation technique, it is necessary to compute and sum the relative free energy differences between adjacent mutation steps. A procedure that can be used for this purpose is readily derived from standard statistical thermodynamic formulae. 

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