Page 248 - Physical chemistry understanding our chemical world

PHASE EQUILIBRIA AND COLLIGATIVE PROPERTIES 215

We now see why the melting-point temperature decreases following contamination,
when its mole fraction deviates from unity. Conversely, the mole fraction does not
change at all if the two components within the mixed-melting-point experiment are
the same, in which T (melt) remains the same.

Justification Box 5.3

When we formulated the total differential of G (Equation (4.30)) in Chapter 4, we only
considered the case of a pure substance, saying

∂G ∂G

dG = dp + dT
∂p ∂T
We assumed then the only variables were temperature and pressure. We must now
rewrite Equation (4.30), but we add another variable, the amount of substance n i in a
mixture:

∂G i ∂G i ∂G i
dG = dp + dT + dn i (5.13)
∂p ∂T ∂n i
We append an additional subscript to this expression for dG to emphasize that we refer
to the material i within a mixture. As written, Equation (5.13) could refer to either the
host or the contaminant – so long as we define which is i.
The term ∂G i /∂n i occurs so often in second law of thermodynamics that it has its
own name: the ‘chemical potential’ µ, which is defined more formally as

∂G i
µ i = (5.14)
∂n i
p,T,n j
where the subscripts to the bracket indicate that the variables p, T , and the amounts of
all other components n j in the mixture, each remain constant. The chemical potential
is therefore seen to be the slope on a graph of Gibbs function G (as ‘y’) against the
amount of substance n i (as ‘x’); see Figure 5.19. In general, the chemical potential
varies with composition, according to Equation (5.12).
The chemical potential µ can be thought of as the constant of proportionality between
a change in the amount of a species and the resultant change in the Gibbs function of
a system.
The way we wrote ∂G in Equation (5.13) suggests the chemical potential µ is the
Gibbs function of 1 mol of species i mixed into an infinite amount of host material.
For example, if we dissolve 1 mol of sugar in a roomful of tea then the increase in
Gibbs function is µ (sugar) . An alternative way to think of the chemical potential µ is to
consider dissolving an infinitesimal amount of chemical i in 1 mol of host.