The Gibbs energy of a mixture

The fact that the pi are intensive properties implies that they can depend only on other intensive properties such as temperature, pressure, and intensive composition variables such as the mole ratios, or the mole fractions. Since the pi depend on the mole numbers only through intensive composition variables, an important relation is easily derived. 

The Pi were taken out of the integrals because, as we have shown above, each Pi must have the same value everywhere throughout a system at equilibrium. Now we allow our initial small boundary to shrink to the limit of enclosing zero volume then rif = 0, and G = 0. This reduces Eq. (11.8) to 

The addition rule in Eq. (11.9) is a very important property of chemical potentials. Knowing the chemical potential and the number of moles of each constituent of a mixture, we can compute, using Eq. (11.9), the total Gibbs energy, G, of the mixture at the specified temperature and pressure. If the system contains only one substance, then Eq. (11.9) reduces to G = np, or 

the p of a pure substance is simply the molar Gibbs energy for this reason the symbol p was introduced for molar Gibbs energy in Section 10.8. In mixtures, pj is the partial molar Gibbs energy of the substance i. 

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Although generally used for pure fluids, Equation (4.307) is sometimes also used to find the fugacity of a mixture, (() , =fjp- The fugacity of a mixture is just an altered representation of the Gibbs energy of a mixture, RT nf = g- g. ... 

Analyzing the Gibbs Energy of a Mixture to Detennine Whether It Is an Ideal Mixture... 

The Gibbs energy of a mixture of two enantiomorphs (optical isomers of the same substance) is given by... 

On further cooling, the tangent breaks away from the liquid curve and becomes a straight line below the liquid curve, giving the Gibbs energy of a mixture of diopside and anorthite crystals just as in the section at T. The difference is that now it is completely below the liquid curve, and therefore a mixture of crystals is the stable configuration of the system. 

Discuss the relationships among chemical potential, activity and the Gibbs energy of a mixture. 

This equation relates the Gibbs energy of a mixture of substances to the amounts and chemical potentials of those substances. As argued above for GnvA/ the chemical potential p.A relates the change in G to a change in the amount of substance A. 

Our interest in chemical potentials (and activities) arises from the fact that the Gibbs energy of a mixture can be expressed in terms of the amounts of the substances and their chemical potentials, as shown by equations (13.31) and (13.32). These two equations, and the second law of thermodynamics, can be used to develop the criterion for predicting the direction of spontaneous chemical change. 

Analogous to the defining equation for the residual Gibbs energy of a mixture, G G — G , is tne definition of a partial molar residual Gibbs energy ... 

The first term, G°, corresponds to the Gibbs energy of a mechanical mixture of the constituents of the phase,... 

Figure 16.5. Supersaturation behavior, (a) Schematic plot of the Gibbs energy of a solid solute and solvent mixture at a fixed temperature. The true equilibrium compositions are given by points b and e, the limits of metastability by the inflection points c and d. For a salt-water system, point d virtually coincides with the 100% salt point e, with water contents of the order of 10-6 mol fraction with common salts, (b) Effects of supersaturation and temperature on the linear growth rate of sucrose crystals [data of Smythe (1967) analyzed by Ohara and Reid, 1973],...

At one atm pressure, the Gibbs energy of a two-phase mechanical mixture containing riA moles of component A and Ub moles of B, in which there is no solubUity between... 

The condition that the Gibbs energy of a system at a given temperature and pressure be a minimum at equilibrium is applied to determine the equilibrium composition of the reacting mixture (2). 

Figure 4.4 shows the Gibbs energy of a binary mixture at a constant T and p with the datum state of either component selected to be the pure fluid at the given T and p. The equilibrium states are on the curve. The nonequilibrium states are above the curve and seek to descend to the equilibrium curve. For instance, to form a mixture of composition m from the pure fluids, the initial state of the two fluids upon being put together before molecular mixing takes place is m with G =... 

These definitions are also valid for mixtures, provided the values of U, H, and 5-used are those for the mixture. That is. the Helmholtz and Gibbs energies of a mixture bear the same relation to the mixture internal energy, enthalpy, and entropy as the pure component Gibbs energies do to the pure component internal energy, enthalpy, and entropy. 

Since the Gibbs energy of a multicomponent mixture is a function of temperature, pressure, and each species mole number, the total differential of the Gibbs energy function can be written as... 

Because of the instability a decrease in the Gibbs energy of a binary liquid mixture can occur because of the formation of phase splitting. Using the excess Gibbs energy for the binary liquid mixture,... 

Now we replace each G(NaCl), G(HOH), G(NaOH), G(HCl) with the expression for the Gibbs energy of a substance in a mixture ... 

Hie first tenn of the right-hand side corresponds to the Gibbs energy of a mechanical mixture of the components, the second me corresponds to die entropy of mixing for an ideal solution and the third term, the so called excess term, represents all the de iations fiom ideality. 

When the excess molar volume and entropy are set equal to zero, the model describes what is called a regular solution. The excess molar Gibbs energy of a mixture is = + pVm m- Using the expression of Eq. 11.1.31 with the further assumptions that and 5 are zero, this model predicts the excess molar Gibbs energy is given by... 

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