Gibbs molar enthalpy

The partial molar enthalpy for every component i is found from an appropriate form of the Gibbs-Helmholtz equation... 

G = Gibbs molar free energy S = molar entropy F = Helmholtz free molar energy H = molar enthalpy U = molar internal energy... 

Gibbs free energy or Gibbs molar free energy molar flow of gas phase acceleration of gravity enthalpy, molar enthalpy, weight enthalpy Henry s constant Planck s constant height horsepower radiation intensity molar flux... 

Figure A2.5.2. Schematic representation of the behaviour of several thennodynamic fiinctions as a fiinction of temperature T at constant pressure for the one-component substance shown in figure A2.5.1. (The constant-pressure path is shown as a dotted line in figure A2.5.1.) (a) The molar Gibbs free energy Ci, (b) the molar enthalpy n, and (c) the molar heat capacity at constant pressure The fimctions shown are dimensionless...

Figure A2.5.4 shows for this two-component system the same thennodynamic fimctions as in figure A2.5.2, the molar Gibbs free energy (i= + V2P2> the molar enthalpy wand the molar heat capacity C. , again all at...

The partial molar entropy of a component may be measured from the temperature dependence of the activity at constant composition the partial molar enthalpy is then determined as a difference between the partial molar Gibbs free energy and the product of temperature and partial molar entropy. As a consequence, entropy and enthalpy data derived from equilibrium measurements generally have much larger errors than do the data for the free energy. Calorimetric techniques should be used whenever possible to measure the enthalpy of solution. Such techniques are relatively easy for liquid metallic solutions, but decidedly difficult for solid solutions. The most accurate data on solid metallic solutions have been obtained by the indirect method of measuring the heats of dissolution of both the alloy and the mechanical mixture of the components into a liquid metal solvent.05... 

Thermodynamics gives limited information on each of the three coefficients which appear on the right-hand side of Eq. (1). The first term can be related to the partial molar enthalpy and the second to the partial molar volume the third term cannot be expressed in terms of any fundamental thermodynamic property, but it can be conveniently related to the excess Gibbs energy which, in turn, can be described by a solution model. For a complete description of phase behavior we must say something about each of these three coefficients for each component, in every phase. In high-pressure work, it is important to give particular attention to the second coefficient, which tells us how phase behavior is affected by pressure. 

To find a numerical value for AHi, we need to know ArH° at one temperature, while evaluation of I requires ArG° at one temperature. The usual choice is to obtain ArH° and ArG° at T = 298.15 K from standard molar enthalpies of formation and standard molar Gibbs free energies of formation. Earlier in this chapter we referred to examples of these quantities. It is now time to define AfH° and AfG° explicitly and describe methods for their measurement. 

Standard molar enthalpies of formation, AfH°m, and standard molar Gibbs free energies of formation, Af(7, are useful, since they can be used to calculate ArH° and ArG°. The relationships are... 

FIGURE 8.33 On the left of the semipermeable membrane is the pure solvent with its characteristic molar enthalpy, entropy, and Gibbs free energy. On the right is the solution. The molar Gibbs free energy of the solvent is lower in the solution (an entropy effect), and so there is a spontaneous tendency for the solvent to flow into the solution. 

In open systems consisting of several components the thermodynamic properties of each component depend on the overall composition in addition to T and p. Chemical thermodynamics in such systems relies on the partial molar properties of the components. The partial molar Gibbs energy at constantp, Tand rij (eq. 1.77) has been given a special name due to its great importance the chemical potential. The corresponding partial molar enthalpy, entropy and volume under the same conditions are defined as... 

The same type of polynomial formalism may also be applied to the partial molar enthalpy and entropy of the solute and converted into integral thermodynamic properties through use of the Gibbs-Duhem equation see Section 3.5. 

In this chapter, we shall consider the methods by which values of partial molar quantities and excess molar quantities can be obtained from experimental data. Most of the methods are applicable to any thermodynamic property J, but special emphasis will be placed on the partial molar volume and the partial molar enthalpy, which are needed to determine the pressure and temperature coefficients of the chemical potential, and on the excess molar volume and the excess molar enthalpy, which are needed to determine the pressure and temperature coefficients of the excess Gibbs function. Furthermore, the volume is tangible and easy to visualize hence, it serves well in an initial exposition of partial molar quantities and excess molar quantities. 

The molar Gibbs free energy of the phase at P and T of interest is obtained by combining equations 2.94, 2.95, and 2.96 for known values of molar enthalpy and molar entropy at 2) and P reference conditions H% p and S% respectively). 

Temperature, Heat capacity. Pressure, Dielectric constant. Density, Boiling point. Viscosity, Concentration, Refractive index. Enthalpy, Entropy, Gibbs free energy. Molar enthalpy. Chemical potential. Molality, Volume, Mass, Specific heat. No. of moles. Free energy per mole. 

Besides equilibriumconstants, additional thermodynamic data were included, if available, although little emphasis was put on their completeness. The data for primary master species comprise the standard molar thermodynamic properties of formation from the elements (AfG standard molar Gibbs energy of formation AfH°m standard molar enthalpy of formation ApSm- standard molar entropy of formation), the standard molar entropy (5m), the standard molar isobaric heat capacity (Cp.m), the coefficients Afa, Afb, and Afc for the temperature-dependent molar isobaric heat capacity equation... 

The conventional thermodynamic standard state values of the Gibbs energy of formation and standard enthalpy of formation of elements in their standard states are A(G — 0 and ArH = 0. Conventional values of the standard molar Gibbs energy of formation and standard molar enthalpy of formation of the hydrated proton are ArC (H +, aq) = 0 and Ar// (H +, aq) = 0. In addition, the standard molar entropy of the hydrated proton is taken as zero 5 (H+, aq) = 0. This convention produces negative standard entropies for some ions. 

Table 2.2 Standard molar Gibbs energies of formation and standard molar enthalpies of formation for some anions at 25 "C (in kJ mol - )...

In equation 33, the superscript I refers to the use of method I, a T) is the activity of component i in the stoichiometric liquid (si) at the temperature of interest, AHj is the molar enthalpy of fusion of the compound ij, and ACp[ij] is the difference between the molar heat capacities of the stoichiometric liquid and the compound ij. This representation requires values of the Gibbs energy of mixing and heat capacity for the stoichiometric liquid mixture as a function of temperature in a range for which the mixture is not stable and thus generally not observable. When equation 33 is combined with equations 23 and 24 in the limit of the AC binary system, it is termed the fusion equation for the liquidus (107-111). 

Note that gw0 is the molar Gibbs energy of formation of the standard hydrate at the reference conditions (To and Pq), ht is the molar enthalpy, and vt is the molar... 

In equation (11.31), AHm and A Vm are the molar enthalpy and molar volume changes associated with the change in phase, and dp/dT gives the slope of the equilibrium lines. Point b is the triple point for CO2. It is an invariant point, since, according to the Gibbs phase rule... 

Figure 11. (a) The calculated partial molar entropy of oxygen (sQJ and (b) the calculated partial molar enthalpy of oxygen (fto2) as a function of 8 for La02Sr08Fe0 55Tio4503 s. Symbols are calculated by the Gibbs-Helmholtz equation. Lines correspond to the partial molar quantities calculated by statistical thermodynamics. 

Equations (2.28) and (2.29) indicate that the molar number of the products caused by the oxidation of hydrogen H2 and carbon monoxide CO are smaller than the total molar number of the reactants. Concerning the oxidation of CH4, the molar number of the products and reactants are equal. Thus, there is theoretically no change in the entropy for the last case. This is the reason for the low dependency of the Gibbs free enthalpy of the methane oxidation from the temperature. 

Thermodynamic considerations are applied to understand the processes of energy conversion in SOFCs. The reversible work of a fuel cell, represented by the Nemst voltage, can be calculated by the Gibbs free enthalpy of the reaction. The consideration of the electrical effects shows that the molar flow of the spent fuel is proportional to the electric current and that the reversible work is proportional to the reversible voltage. A coupling between the thermodynamic data and the electrical data is only possible by using the quantities power or heat flow and not by using work and heat. 

For open systems the change in the mole number that corresponds to the addition or removal of material from the system would be done at constant temperature and pressure. The molar enthalpy, the molar entropy, and the chemical potential would then be constant and the change in the enthalpy, entropy, and Gibbs energy of the system would be the product of the molar quantities and the change in the number of moles. Again, such changes cannot be calculated, because the absolute values are not known. However, the concept of the operation is used in later chapters. 

The corresponding molar enthalpy is obtained by use of the Gibbs-Helmholtz equation H = — T2[ d(G/T)/dT P. Applying the Gibbs-Helmholtz equation to equations 8.6-2 and 8.6-5 yields... 

Gibbs functions for a real salt solution and the corresponding ideal salt solution containing m2 moles of salt in a kilogram of solvent. GE can be calculated for many aqueous salt solutions from published values of 0 and y . In the same way, the corresponding excess enthalpy HE can be defined and this equals the apparent partial molar enthalpy. Thus the properties of salt solutions can be examined in plots of GE, HE, and T SE against m2, where SE is the... 

Here, M, may represent the partial molar internal energy Uit the partial molar enthalpy H(, the partial molar entropy Sif the partial molar Gibbs energy Gt, etc. Comparison of Eq. (10.1) with Eq. (11.2) written for the Gibbs energy shows that the chemical potential and the partial molar Gibbs energy are identical, that... 

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