Partial molal free energy

Here G, S, T, and p are the Gibbs free energy, the entropy, the temperature, and the (total) pressure, respectively. The partial molal free energy of species number i is fit, and nt is the number of moles of species number i in the system. If it is assumed.that... 

It is a characteristic of two component systems forming one phase at a certain temperature that at this temperature the curve relating the partial molal free energy to the relative concentration of the components becomes horizontal. In other words at the critical point... 

In these equations is the partial molal free energy (chemical potential) and Vj the partial molal volume. The Mj are the molecular weights, c is the concentration in moles per liter, p is the mass density, and z, is the mole fraction of species i. The D are the multicomponent diffusion coefficients, and the are the multicomponent thermal diffusion coefficients. The first contribution to the mass flux—that due to the concentration gradients—is seen to depend in a complicated way on the chemical potentials of all the components present. It is shown in the next section how this expression reduces to the usual expressions for the mass flux in two-component systems. The pressure diffusion contribution to the mass flux is quite small and has thus far been studied only slightly it is considered in Sec. IV,A,6. The forced diffusion term is important in ionic systems (C3, Chapter 18 K4) if gravity is the only external force, then this term vanishes identically. The thermal diffusion term is impor-... 

The relation between the partial molal free energy and the fugacity for the mixed gas system has been obtained. 

The partial molal free energies of transfer are related to the fugacities in pure liquid, f° (vapor pressure corrected for deviation from die perfect gas law ), and in solution, by the equations,... 

Partial molal free energy of compound j in phase l (chemical potential)... 

The constant KV) which is called the ionic product of water, may be computed from equations (V-44) and (V-46), if partial molal free energies (potentials) of formation of all the reaction components in the corresponding standard states are known. For such standard states we select both the state of a hypothetical ideal solution with molal concentration of hydrogen and hydroxyl ions equalling unity and the state of hypothetical, absolutely undissociatcd pure water. Since in actual diluted solutions the activity of undissociated water hardly differs from the activity in its standard state, aji2 in the equation (V-51) may be considered as equalling unity so that then Km = K . The following expression is valid for a temperature of 25° C ... 

The partial molal free energy of a spray of finely dispersed droplets is not so simply related to escaping tendency as it is in systems usually treated by thermodynamic methods. Even the definition of the partial molal free energy is not entirely devoid of difficulty. To discuss equilibrium we wish to think of the transfer of a small amount of adsorbent from each droplet in the spray to another phase— e.g., the gas phase (16). If a little material leaves each droplet, there must be a concomitant decrease in surface area, A, and an increase in specific area, quantity related to escaping tendency of material is not the derivative of free energy with respect to n, the number of moles in the spray at either constant area or constant specific area. The condition is that the number of droplets, v, be constant. (This means that area divided by mass to the 2/3 power is nearly invariant.)... 

This quantity is, of course, a partial molal free energy. The fact that the extensive free energy depends upon v means that the free energy of each drop is not proportional to the amount of material in the drop. Hence, when the amount of material per drop is small, the extensive free energy, G, is not equal to nG. 

Another definition of partial molal free energy avoids this particular difficulty. Consider the addition of an increment of substance to the spray of finely... 

A change in partial molal free energy of the adsorbent calculated from adsorption isotherms is an average for the exterior and interior adsorbent because we have taken the total number of moles of adsorbent as rq. This average, multiplied by the total number of moles of adsorbent, is an extensive free energy function. Its temperature variation yields a heat of adsorption which is also extensive. This heat divided by the number of moles of adsorbent is the average heat of adsorption per mole. Fortunately, it is this quantity which is most directly determined in a calorimeter. 

Adsorption isotherms for the system BaS04-H20 at three temperatures have been obtained. Thermodynamic study of these data reveals that part of the free energy decrease in the adsorption process involves changes in the partial molal free energy of the adsorbent. From the three isotherms differential and integral heats of adsorption were derived and compared with new calorimetric determinations of the same thermodynamic functions. In both kinds of measurements exactly the same system and exactly the same materials were used. 

The partial molal entropies are calculated from the partial molal free energies and enthalpies. The change in the partial molal entropy of water increases monotonically to a value near zero with increasing amount adsorbed. The partial molal entropy of barium sulfate decreases with increasing amount adsorbed. 

The sum of the partial molal free energies multiplied by their appropriate mole fractions gives the total molal free energy ... 

Similarly, although there is a maximum difference in cadmium partial molal free energy of a little less than 2 kcal. per mole between the extrapolated upper structure, a, of Figure 1 and the most stable observed microphases, the difference in total molal free energy is less than 10 cal. per mole. (The calculation is based on the path 133 to 209 with assumed two-phase equilibrium pressures of cadmium. This path closes on microphase H.) The difference in these two values emphasizes that the vapor pressure measurements have reflected the concentration and bonding of cadmium species which are minor fractions of the total cadmium present and which did not much alter the average bonding of the system. 

One should note that Mi m is really a partial molal free energy, in the notation of Section 1.22. Equation (1.23.1) may be rewritten as... 

The quantity AE can be related to the dimensionless partial molal free energies y of each species / in each phase j [reaction (1)] ... 

Partial Molar Free Energy Chemical Potential.—The partial molal free energy is an important thermodynamic property in connection with the study of electrolytes it can be represented either as G, where G is employed for the Gibbs, or Lewis, free energy, or by the symbol /i, when it is referred o as the chemical potential thus the appropriate form... 

Concentration Cells with a Single Electrolyte Amalgam Concentration Cells.—In the concentration cells already described the e.m.p. is a result of the difference of activity or chemical potential, i.e., partial molal free energy, of the electrolyte in the two solutions it is possible, however, to obtain concentration cells with only one solution, but the activities of the element with respect to which the ions in the solution are reversible are different in the two electrodes. A simple method of realizing such a cell is to employ two amalgams of a base metal at different concentrations as electrodes and a solution of a salt of the metal as electrolyte thus... 

Markin et al. (38) have determined the e.m.f. of high temperature galvanic cells involving the plutonium oxide-oxygen system. The plots of partial molal free energy of oxygen vs. temperature show a profound change in the composition interval 1.691 and 1.812. In many respects the behavior of PuO is quite similar to CeO. ... 

The chemical potential is the partial molal free energy, and by modifying Equation (2-3), we obtain... 

Some authors use the term partial molal free energy for the quantity defined in this way. This term was invented by G. N. Lewis loc. cit). 

In terms of the partial molal free energy G t of the zth component we have... 

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