Page 246 - Physical chemistry understanding our chemical world

PHASE EQUILIBRIA AND COLLIGATIVE PROPERTIES 213

Earlier, on p. 181, we looked at the phase changes of a single-component system
(our examples included the melting of an ice cube) in terms of changes in the molar
Gibbs function G m . In a similar manner, we now look at changes in the Gibbs
function for each component within the mixture; and because several components
participate, we need to consider more variables, to describe both the host and the
contaminant.
We are now in a position to understand why the melting point
For a pure substance,
of a mixture is lower than that of the pure host. Previously, when
the chemical potential
we considered the melting of a simple single-component system,
µ is merely another
we framed our thinking in terms of the molar Gibbs function G m . name for the molar
In a similar way, we now look at the molar Gibbs function of each Gibbs function.
component i within a mixture. Component i could be a contami-
nant. But because i is only one part of a system, we call the value of G m for material
i the partial molar Gibbs function. The partial molar Gibbs function is also called
the chemical potential, and is symbolized with the Greek letter mu, µ.
We define the ‘mole fraction’ x i as the number of moles of component i expressed
as a proportion of the total number of moles present:

number of moles of component i
x i = (5.11)
total number of moles

The value of µ i – the molar Gibbs function of the contaminant – Themolefraction x of
decreases as x i decreases. In fact, the chemical potential µ i of the host DEcreases as
the contaminant is a function of its mole fraction within the host, the amount of contam-
according to Equation (5.11): inant INcreases. The
sum of all the mole
fractions must always
O
µ i = µ + RT ln x i (5.12)
i equal one; and the
mole fraction of a pure
where x i is the mole fraction of the species i, and µ O is its material is also one.
i
standard chemical potential. Equation (5.12) should remind us of
Equation (4.49), which relates G and G . Strictly, Equation (5.12)
O
Notice that the mole fraction x has a maximum value of unity. relates to an ideal
The value of x decreases as the proportion of contaminant in- mixture at constant p
creases. Since the logarithm of a number less than one is always and T.
negative, we see how the RT ln x i term on the right-hand side of
O
Equation (5.12) is zero for a pure material (implying µ i = µ ). At all other times,
i
x i < 1, causing the term RT ln x i to be negative. In other words, the value of µ
will always decrease from a maximum value of µ O as the amount of contaminant
i
increases.
Figure 5.18 depicts graphically the relationship in Equa- Remember:in thistype
tion (5.12), and shows the partial molar Gibbs function of the host of graph, the lines
material as a function of temperature. We first consider the heavy for solid and liquid
bold lines, which relate to a pure host material, i.e. before con- intersect at the melting
tamination. The figure clearly shows two bold lines, one each for temperature.
the material when solid and another at higher temperatures for the