In the previous section we noted that convexity applies to functions,
such as and
, defined for all vectors, such as
and
, rather
than restricting these vectors to unit vector, such as
to give
,
for which
and for the molar free
energy,
, for which
. The two expressions
for the unit vectors
are fundamentally different; area and the number of moles are examples
of different metrics.
Metrics that measure the distance of a point from the origin are
simple examples of a convex functions.
The most familiar metric for vectors (including the area vector
) is
the Euclidean
metric, also known as
,
Convexity for this metric is just the triangle inequality; the length of any
side of a
triangle is not more than the sum of the lengths of the other two
sides.
has this metric.
Other metrics are appropriate in other physical situations.
Consider the driving distance between two intersections in
a city, the appropriate distance is the metric:
sometimes called the Manhattan metric, since
such
distances apply to travelers who can
only travel on a rectangular grid, like the streets of Manhattan.
A vector,
in
with
components which are the
amounts (here taken to be number of moles) of each of the constituents
is an indication of the size of a chemical system. The Manhattan
"length" of this vector is the total number of moles. This length is
the factor
used to convert
to
.
A simple way of extending the concept of metrics is to give different
weights to the components in the sums that define a metric. A
weighted metric for area can account for some anisotropy, but we
know of no useful application. The weighted Manhattan metric, occurs
quite naturally if mass and weight percent, rather than number of
moles
and mole percent, become the variables. The mass of a system
is
, where
is the molecular
weight of
the
species.
One limit of the weighted metrics, that gives zero weight to all but one of
the components, is used for both surfaces and chemical systems.
This limit provides a link between the mathematics of surfaces and chemical
systems, and
permits other convexification methods to be used.
For chemical systems, this weighting is used for molal concentrations,
the number of moles of solutes for a fixed amount of solvent (usually
one mole or one Kg). [45] Molal concentrations are defined as (or
with
, e.g. for aqueous
solutions). The size of the system (length of the vector) is then
defined as the amount of solvent alone, regardless of the amounts of
the other components. Note that
The unit length
is one mole or 1 kg. The
molal free energy is
Molal concentrations are on a special planar cut of the space of all
and thus the
convexification applied to
gives the same common tangents
and equilibria that would be obtained from
. Molal free energies
also provide the same spinodal stability limits from the same
curvature criteria.
For vicinal surfaces, area is often defined as the area
projected along some symmetry axis, giving no weight to other
components of the area vector.[52][51]
If we define the components of a
new orientation vector as without regard to the
sign or size of this ratio, we have extended
the concepts of vicinal surfaces to all orientations, and the analogy with
molal concentrations is kept. Note that
lives in the space
of
on a planar cut perpendicular to one of the axes in the same way
as
molal concentrations do in the space of
. We will denote the
surface free
energy per unit projected area projected along the
direction as
. Convexity applies to
.
Using an metric
to describe the size of a
chemical system makes little physical sense, but, as we shall see, it opens
up
some useful surface techniques for chemical thermodynamics. By defining a
Euclimolar metric
, we can define a
Euclimolar free energy
, which is
evaluated
on the unit sphere
.