Page 35 - Battery Reference Book

1/20 Introduction to battery technology

mole fraction for an ideal solution in the free energy Equation (1.5 1) becomes
equation. (1.53)
Although the definition of activity as represented a, = CO
by Equation 1.49 has been derived with particular that is, the activity of the solute is equal to its molar
reference to the solvent, an exactly similar result is concentration. The standard state of unit activity may
applicable to the solute. If F' is the free energy of 1 mol thus be defined as a hypothetical solution of unit molar
of solute in one solution, and F" is the value in another concentration possessing the properties of a very dilute
solution, the increase of free energy accompanying the solution. The word 'hypothetical' is employed in this
transfer of 1 mol of solute from the first solution to the definition because a real solution at a concentration of
second is then given by Equation 1.49, where a' and 1 mol (or gram-ion) per litre will generally not behave
a" are, by definition, the activities of the solute in the ideally in the sense of having the properties of a very
two solutions. dilute solution.
Equation 1.49 does not define the actual or abso- Another standard state for solutes that is employed
lute activity, but rather the ratio of the activities of especially in the study of galvanic cells is that based
the particular substances in two solutions. To express on the relationships
activities numerically, it is convenient to choose for
each constituent of the solution a reference state or a=ym or y=- a (1.54)
standard state, in which the activity is arbitrarily taken m
as unity. The activity of a component, solvent or solute where m is the molality of the solute, i.e. moles (or
in any solution is thus really the ratio of its value in gram-ions) per 1000 g solvent, and y is the appropriate
the given solution to that in the chosen standard state. activity coefficient. Once again it is postulated that y
The actual standard state chosen for each component is approaches unity as the solution becomes more and
the most convenient for the purpose, and varies from more dilute, so that at or near infinite dilution it is
one to the other, as will be seen shortly. If the solution possible to write
indicated by the single prime is taken as representing
the standard state, a' will be unity, and Equation 1.49 a0 = mo (1.55)
may be written in the general form the activity being now equal to the molality. The
standard state of unit activity is consequently defined
F-FO=RT~~~ (1.50)
as a hypothetical solution of unit molality possessing
the double primes being omitted, and a superscript zero the properties of a very dilute solution. The difference
used, in accordance with the widely accepted conven- between the actual value of the activity coefficient y
tion, to identify the standard state of unit activity. This and unity is a measure of the departure of the actual
equation defines the activity or, more correctly, the solution from an ideal solution, regarded as one having
activity relative to the chosen standard state, of either the same properties as at high dilution.
solvent or solute in a given solution. In view of Equations 1.53 and 1.55 it is evident that
The deviation of a solution from ideal behaviour in the defined ideal solutions the activity is equal to
can be represented by means of the quantity called the molarity or to the molality, respectively. It fol-
the activity coefficient, which may be expressed in lows, therefore, that the activity may be thought of as
terms of various standard states. In this discussion an idealized molarity (or molality), which may be sub-
the solute and solvent may be considered separately; stituted for the actual molarity (or molality) to allow
the treatment of the activity coefficient of the solute for departure from ideal dilute solution behaviour. The
in dilute solution will be given first. If the molar activity coefficient is then the ratio of the ideal molar-
concentration, or molarity of the solute, is c moles ity (or molality) to the actual molarity (or molality). At
(or gram-ions) per litre, it is possible to express the infinite dilution both f and y must, by definition, be
activity a by the relationship equal to unity, but at appreciable concentrations the
activity coefficients differ from unity and from one
a=fc or f=f (1.51) another. However, it is possible to derive an equation
C relating f and y, and this shows that the difference
where f is the activity coefficient of the solute. Insert- between them is quite small in dilute solutions.
ing this into Equation 1.50 gives the expression When treating the solvent, the standard state of unit
activity almost invariably chosen is that of the pure
F - F: = RTln fc (1.52) liquid; the mole fraction of the solvent is then also
unity. The activity coefficient f of the solvent in any
applicable to ideal and non-ideal solutions. An ideal solution is then defined by
(dilute) solution is defined as one for which f is unity,
but for a non-ideal solution it differs from unity. Since a=f,x or fx= f (1.56)
solutions tend to a limiting behaviour as they become X
more dilute, it is postulated that at the same time f where x is the mole fraction of the solvent. In the pure
approaches unity, so that, at or near infinite dilution, liquid state of the solvent, a and x are both equal to