The Chemical Potential

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The thermodynamic terms heat and work can be viewed as the product of a capacity factor (or quantity of something) and a difference in a potential. Table 7.1 lists several examples of this breakdown of heat/work expressions for mechanical, electrical, thermal, and chemical processes.

Although rate processes are not within the purview of thermodynamics, they involve the same potentials as those responsible for producing heat or work. The basic rate laws are of the form: flux = coefficient x potential gradient

Table 7.1 Work/heat as a capacity times a potential difference

Process

Capacity

Potential

Work/heat

Lower a weight

mass

gravitational

mgAh

pumping a fluida

volume

pressure

VAp

electrical

charge

electrostatic

qAO

thermal

energy

temperature

nCpAT

chemicalb

moles of i

chemical

niA^i

a See Eq (4.13a); adiabatic process b Isothermal, isobaric process a See Eq (4.13a); adiabatic process b Isothermal, isobaric process

Table 7.2 shows the four common rate laws of this type.

Table 7.2 Flux as a kinetic coefficient times a potential gradient

Rate Process

Rate coefficient

potential

flux

name of law

heat conduction

thermal conductivity

temperature

-kVT

Fourier's

momentum transfer(fluid)

viscosity

pressure

-^Vp

Newton's

Electricity flow

electrical conductivity

electrostatic

-kVO

Ohm's

Diffusion

diffusion coefficient

chemical

-(Dci/kT) V^i

Fick's

Chemical reactions and interphase mass transfer are also driven by imbalances of the chemical potentials of species in the system. The chemical potential is as important a thermodynamic driving force as are temperature and pressure. This potential drives individual chemical species from one phase to another, from one molecular form to another, or from regions of high concentration to regions of low concentration.

The chemical potential is directly related to the Gibbs free energy of a system. For a one-component system, the chemical potential is identical to the molar Gibbs free energy of the pure substance. In solutions or mixtures, the chemical potential is simply another name for the partial molar Gibbs free energy. The discussion in Sect. 7.3, in which enthalpy was used to illustrate partial molar and excess properties, applies to the Gibbs free energy; one need only replace h everywhere by g.

The reason that the partial molar Gibbs free energy (g) is accorded the special name "chemical potential" is not only to shorten a cumbersome five-word designation. More important is the role of the chemical potential in phase equilibria and chemical equilibria when the restraints are constant temperature and pressure. Instead of the symbol g, the chemical potential is designated by The connection between the Gibbs free energy of a system at fixed T and p and the equilibrium state is shown in Fig. 1.18. In the remainder of the present chapter, the relation between the Gibbs free energy of a multicomponent system and the chemical potentials of its constituents is developed.

The chemical potential is embedded in the equation for the differential of the Gibbs free energy of a solution at fixed T and p analogous to Eq (7.13) for the enthalpy:

The partial derivatives that serve as coefficients of dn are the partial molar Gibbs free energies, or the chemical potentials, of each component of the solution:

For a one-component system, G = nigi, where gi is the molar free energy. Consequently, for the pure substance, Eq (7.25a) reduces to:

Following the lines of the treatment using h and hi in Sect. 7.3, the following fundamental relations between g and * are obtained:

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