Metrics

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 .