The Gibbs energy

The new function A is called the Helmholtz energy.2 Since E, T, and S are functions of the state of the system, A is also a function of the state of the system, and its differential is exact. The change in the value of the Helmholtz energy in going from some initial state to some final state is independent of the path. However, the determination of this change must be obtained by the integration of Equation (4.3) along any reversible path between the two states. 

Any physical interpretation of the Helmholtz energy must be based on interpreting Equation (4.3). Thus, for an isothermal change of state, the equation becomes 

The algebraic sum of the terms P dV and dW T is the total work done reversibly by the system for a differential change of state. Hence, dA is the negative of the maximum work that the system can do on the surroundings for an isothermal differential change of state. 

Although the integration of this equation is limited to reversible paths, the change of enthalpy in going from one state to another is still independent of the path. 

the entropy appears as an independent variable. It is therefore convenient to define another function so that the temperature appears as an independent variable. This is done by subtracting Equation (4.2) from Equation (4.6)  

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The enthalpy of fomiation is obtained from enthalpies of combustion, usually made at 298.15 K while the standard entropy at 298.15 K is derived by integration of the heat capacity as a function of temperature from T = 0 K to 298.15 K according to equation (B 1.27.16). The Gibbs-FIehiiholtz relation gives the variation of the Gibbs energy with temperature... 

Table 2.10 shows the effect of substituents on the endo-exo ratio. Under homogeneous conditions there is hardly any substituent effect on the selectivity. Consequently the substituents must have equal effects on the Gibbs energies of the endo and the exo activated complex. 

Equations 54 and 58 through 60 are equivalent forms of the fundamental property relation apphcable to changes between equihbtium states in any homogeneous fluid system, either open or closed. Equation 58 shows that ff is a function of 5" and P. Similarly, Pi is a function of T and C, and G is a function of T and P The choice of which equation to use in a particular apphcation is dictated by convenience. Elowever, the Gibbs energy, G, is of particular importance because of its unique functional relation to T, P, and the the variables of primary interest in chemical technology. Thus, by equation 60,... 

For the Gibbs energy of an ideal gas mixture, — T the parallel relation for partial properties is equation 149 ... 

The chemical potential, plays a vital role in both phase and chemical reaction equiUbria. However, the chemical potential exhibits certain unfortunate characteristics which discourage its use in the solution of practical problems. The Gibbs energy, and hence is defined in relation to the internal energy and entropy, both primitive quantities for which absolute values are unknown. Moreover, p approaches negative infinity when either P or x approaches 2ero. While these characteristics do not preclude the use of chemical potentials, the appHcation of equiUbrium criteria is faciUtated by the introduction of a new quantity to take the place of p but which does not exhibit its less desirable characteristics. 

Each reactant and product appears in the Nemst equation raised to its stoichiometric power. Thermodynamic data for cell potentials have been compiled and graphed (3) as a function of pH. Such graphs are known as Pourbaix diagrams, and are valuable for the study of corrosion, electro deposition, and other phenomena in aqueous solutions.Erom the above thermodynamic analysis, the cell potential can be related to the Gibbs energy change... 

For convenience, the three other fundamental property relations, Eos. (4-16), (4-80), and (4-82), expressing the Gibbs energy and refated properties as functions of T, P, and the are collected nere ... 

C (diamond) — C (graphite) has the Gibbs energy change, at one atmosphere... 

These terms are obtained from die equation above by differentiation with respect to r, and setting the resultant equal to zero. This is equivalent to taking the point on the graph of the Gibbs energy of nucleus formation versus the size of the nucleus where the tangent has zero slope. 

Once a number of nuclei ai e formed on the surface of the substi ate, the next stage of dre film formation process involves the U ansport of nuclei or their constituent atoms across the surface in order to cover the ai ea available to form the complete film. It is clear from the relationship between the Gibbs energy... 

This equation is derived by considering die transfer of material from a flat surface to a droplet. For the U ansfer of a small mass Sm from the flat surface of vapour pressure p° to the droplet of vapour pressure p, the Gibbs energy of transfer is... 

In the case of hydrogen, for example, at a teiuperamre of 2500 K, the equilibrium constant for dissociation has the value, calculated from the tlrermo-dynamic relation between the Gibbs energy of formation and the equilibrium constant of 6.356 x 10 " and hence at a total pressure of 10 atmos, the degree of dissociation is 0.126 at 2500 K, which drops to 8.32 x 10 at 2000 K. 

The Gibbs energy of formation of tire more important molcules are,... 

A simple example of the analysis of multicomponent systems will suffice for the present consideration, such as the calculation of the components in a gaseous mixture of oxygen, hydrogen and sulphur. As a first step, the Gibbs energy of formation of each potential compound, e.g. S2, H2S, SO, SO2, H2O etc. can be used to calculate the equilibrium constant for the formation of each compound from the atomic species of the elements. The total number of atoms of each element will therefore be distributed in the equilibrium mixture in proportion to these constants. Thus for hydrogen with a starting number of atoms and the final number of each species... 

There is a significant difference between rhodium and the odier metals in that rhodium forms a relatively stable oxide, RI12O3. The Gibbs energy of formation of this oxide is given by the equation... 

These Huctuations lead to two more components to the Gibbs energy changes in the process. The hrst arises from the fact that the Huctuations represent a local gradient in composition, and hence there is a change in the number of... 

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