Page 73 - Battery Reference Book

1/58 Introduction to battery technology
the right-hand side of Equation 1.115 must also be standard free energy change AFo must depend upon
constant. The gas constant R has, of course, a definite the particular choice of standard states.
value and so it follows that, if the temperature T is Attention may be drawn to the fact that although
constant, AFo depends on the particular standard states that are
chosen, the value of AF is as it should be, independent
(u6 x a; x . . .)
= constant = K (1.116) of their nature. This may be readily seen by writing
(UZ x a; x . . .) Equation 1.120 in the simple form
This is the exact expression for the equilibrium con- AF=RTln- Qa (1.121)
stant K, as given in Equation 1.113; it has been derived K
from thermodynamic considerations alone, without the from which it is evident that AF is determined by
assumption of the law of mass action. It may be sim- the ratio of Qa to K. Consequently, as long as both
plified for systems which do not depart appreciably of these quantities are expressed in terms of the same
from ideal behaviour, i.e. for reactions in solution standard states, that is, in terms of the same units, the
mole fractions. In dilute solutions, concentrations may result will be independent of the particular standard
be employed. In general, however, with concentrated state employed.
solutions such as cell electrolytes, it is best to use For reactions in dilute solution, the standard state
activities. is chosen as the (hypothetical) ideal solution of unit
concentration, i.e. 1 mol (or 1 gram-ion) per litre, or of
1.23.1 The reaction isotherm unit molality, i.e. 1 mol/1000 g of solvent. Under these
conditions the standard free energy is given by
If the symbol K for the equilibrium constant, as given
by Equation 1.116, is substituted in Equation 1.115, AF: = -RTlnK, (1.122)
the result is
and the reaction isotherm becomes
AFo = -RTInK (1.117)
Qc
AF = -RTlnKc + RTInQ, = RTln - (1.123)
This is a very important' equation, as it relates the Kc
standard free energy change of a reaction to the where Qc is the arbitrary reaction quotient with the
experimentally determinable equilibrium constant. If states of the reactants and products expressed in terms
this value for AFo is now inserted in Equation 1.113, of concentrations in their ideal solutions. If the solu-
it follows that the free energy change for the reaction tions are sufficiently dilute, the actual concentrations
with reactants and products in any arbitrary state is may be employed in place of the ideal values.
given by
ui x a$ x . . . 1.23.2 Criteria of spontaneous reaction
AF = -RTInK+RTIn
ai x af; x . . . The essential importance of the reaction isotherm lies
in the fact that it provides a means of determining
where the a values refer to the activities in these whether a particular reaction is possible or not, under a
arbitrary states. The arbitrary reaction quotient in terms given set of conditions. For a thermodynamically irre-
of activities may be represented by the symbol Qa, versible process taking place at constant temperature
that is and pressure, AF must be negative; that is, the free
energy of the system diminishes. If a particular phys-
u;xu;x ...
Qa = (1.119) ical or chemical change is to be theoretically possible
...
u~xa~x it must be able to occur spontaneously; spontaneous
processes are, however, irreversible in the thermody-
so that Equation 1.118 becomes namic sense and hence it follows that a reaction can
AF = -RTlnK+RTlnQ, (1.120) take place spontaneously only if it is accompanied by
a decrease of free energy, i.e. AF must be negative, at
which is a form of what is known as the reaction constant temperature and pressure. This result applies
isotherm. It is evident that, if the arbitrary states hap- to any process, physical or chemical; it is immaterial
pen to correspond to those for the system at equilib- whether the latter is reversible, in the chemical sense,
rium, Qa will become identical with the equilibrium or if it goes to virtual completion.
constant K, since the expressions for both these quanti- If the value of AF under a given set of conditions
ties are of exactly the same form (see Equations 1.1 16 is positive, the reaction cannot possibly occur spon-
and 1.119). By Equation 1.120 the value of AF would taneously under those conditions, although it may be
then be zero, as indeed it should be for an equilibrium able to do so if the conditions are altered. By writing
system. the reaction isotherm in the general form
The standard state of unit activity may be defined Q
in any convenient manner and so it is obvious that the AF=RTln-
K