The Chemical Potential

For processes involving diffusion mass transport we can, however, define a
thermodynamic intensive driving force responsible specifically for the total energy change
accompanying the diffusion mass transfer of molecules from one region to another. This
driving force can be defined simply as a partial derivative representing the variation of the
total internal energy of a region with respect to an increment in the number of moles of one
particular species in the region when no other extensive properties are altered. This partial
derivative is an important intensive property called the chemical potential. By reason of its
definition the chemical potential is an intensive property because whenever it is multiplied
by the extensive property change in moles of a particular molecular species within a system
the result is identically the internal energy change of the system resulting only from this
change in moles, and not from the change in any other extensive property.
In elementary physics the energy per unit mass, per mole, or per particle involved in
moving the mass, mole, or particle from one region to another is generally defined as a
potential. Table I lists several types of potentials (driving forces) which are important in
thermodynamic applications. A potential therefore can always be regarded as a driving
force for a mass change. The chemical potential is a driving force of this type. Physically the
driving force represented by the chemical potential results from the same molecular actions
which give rise to a partial vapor pressure in a liquid or a partial pressure in a gas. Each of
these has the ability to expel molecules of a given type out of a multi-component phase.
The energy change within a system accompanying a change in the number of moles
of a given component of the system by molecular processes can now be defined as a type of
work which results from a difference in a chemical potential driving force between the
system and its surroundings. As is the case of other types of work, in order to evaluate
quantitatively the work of a chemical potential driving force it is first necessary to define a
system. In accordance with the principles discussed in section 14,this work is then defined
as the product of a chemical potential of a component outside the system on its external
boundary and a change in the number of moles of this component inside the system. When
the number of moles of a molecular species increases in a system, work must be done on the
system to overcome the molecular forces tending to expel molecules of this species.
Consequently, in accordance with the sign convention, the work relative to the system
receiving this increase in moles within it must be a negative number.
Although we have discussed the chemical potential as a thermodynamic driving force
for the diffusion mass transport, its utility is not confined to this particular process alone.
Because of its definition the chemical potential is a driving force for changes in moles of a
molecular species in a system not only by means of diffusion mass transfer but by any other
molecular process as well. The most important example is the role of the chemical potential
within a system as the driving force for changes in moles brought about in the system by
chemical reactions.

Energy Transport by Mass Transfer

For any region with a boundary which is penetrated by mass, a thermodynamic
analysis always requires a distinction between mass carried across the boundary by bulk
stream flow and mass carried across by diffusion processes resulting from molecular action.
In the case of bulk stream flow with no diffusion mass transport, energy is carried
into the region in two distinct ways. Part of the energy added to a region receiving mass by
stream flow is the work of a pressure which displaces a quantity of flowing fluid into the
region. The remaining part of the energy added is the internal energy content of this
quantity of fluid which enters. In a bulk stream flow process these two parts of the total
energy transport can be separated and evaluated. This is done most conveniently by
defining the system in this case as a fixed mass enclosed by moveable boundaries which are
not penetrated by mass at all. In this manner a small contiguous quantity of fluid in an
entering conduit becomes a homogeneous sub-region within the system and its energy thus
becomes a part of the total internal energy of the entire system. The boundary of this subregion
is acted upon by an external pressure which performs work on the entire system in
moving the boundary of the sub-region. When the system is defined in this way, no energy
is carried into the system in the form of the internal energy of mass crossing its boundaries.
In a region receiving mass transported by a diffusion process, part of the energy content of all molecules outside the region is used to propel some of them into the region. In
contrast to the situation in a purely bulk stream flow process, there is no way in this case to
define a system which excludes the internal energy of transported molecules from the energy
crossing the system boundary. There is no way to define a system in which the propelling
forces which induce the mass transport are a driving force for all of the energy which crosses
the system boundary in the transport process. The diffusion processes these propelling
forces result from the behavior of individual molecules and are not scalar thermodynamic
properties at all so that we cannot define an intensive thermodynamic driving force property
to represent them.

Work

Now that we have used the term "work" it is necessary to emphasize that work, like
heat, must also be regarded only as a type of energy in transition across a well defined, zero
thickness, boundary of a system. Consequently work, like heat, is never a property or any
quantity contained within a system. Whereas heat is energy driven across this boundary by
a difference in temperature, work is energy driven across by differences in other driving
forces on either side of it. Various kinds of work are identified by the kind of driving force
involved and the characteristic extensive property change which accompanied it.
Work is measured quantitatively in much the same manner as heat. Any driving
force other than temperature, located outside the system on its external boundary, is
multiplied by a transported extensive property change within the system which was
transferred across the system boundary in response to this force. The result is the numerical
value of the work associated with this system and driving force. It is important to
emphasize that the extensive property change within the system which is used in this
computation must be a transported quantity whose transfer across the system boundary
depends on a particular driving force with different values inside and outside the system.
This transported extensive property change within the system always occurs with the same
magnitude but with opposite sign in the surroundings.
Neither work nor heat results from any part of a change in an extensive property of a
system which has not been transported in this manner without alteration in magnitude
across the system boundary. A non-transported extensive property change within a system
when multiplied by an appropriate driving force property located within the system measures a form of internal energy change in the system but not work or heat.
Conventionally the quantity of work calculated by this procedure is given a positive
sign when work is done by the system on the surroundings and energy crosses the boundary
in a direction from the system to the surroundings. An energy transport in the opposite
direction, when work is done by the surroundings on the system, is given a negative sign.
It is awkward that the sign given to energy transferred as work is opposite to that given to
energy transferred as heat in the same direction, but tradition has established the convention
and it is important that it be followed consistently. Like heat, both the absolute value and the
sign of what is called work depend entirely on how the system is specified.
Several thermodynamic driving forces and their characteristic displacements are
listed in Table I. Any of these properties, other than temperature and entropy, can measure
various types of work when the driving force is located on the outer side of the system
boundary and the displacement is a transported quantity whose change is located within the
system. The product, when given the proper sign, is a type of work transfer for this system.