Page 34 - Battery Reference Book

Activities of electrolyte solutions 1/19
of moles of solvent and solute, respectively, the mole of the same solvent and soiute at a different concen-
fraction xz of the solute is tration, whose vapour pressure is p". The external
pressure, e.g. 1 atm., and the temperature, T, are the
n2
x2 = ____ same for both vessels. One mole of solvent is then
ni +nz vaporized isothermally and reversibly from the first
and hence, by Raoult's law, solution at constant pressure p'; the quantity of solu-
tion is supposed to be so large that the removal of 1 mol
of solvent does not appreciably affect the concentration
(1.42)
or vapour pressure. The vaporization has been carried
out reversibly. and so every stage represents a state of
This law, namely that equilibrium. Furthermore, the temperature and pressure
Relative lowering of vapour pressure have remained constant, and hence there is no change
Mole fraction of solute =I of free energy.
The mole of vapour at pressure p' is now removed
is obeyed, at least approximately, for many and compressed or expanded at constant temperature
solute-solvent systems. There are, however, theoreti- until its pressure is changed to p", the vapour pressure
cal reasons for believing that Raoult's law could only of the second solution. If the pressures are sufficiently
be expected to hold for solutions having a heat of dilu- low for the vapour to be treated as an ideal gas without
tion of zero, and for which there is no volume change incurring serious error, as is generally the case, the
upon mixing the components in the liquid state. Such increase of free energy is given by
solutions, which should obey Raoult's law exactly at F = RT In ($1
all concentrations and all temperatures, are called ideal (1.46)
solutions. Actually very few solutions behave ide-
ally and some deviation from Raoult's law is always Finally, the mole of vapour at the constant pressure
to be anticipated; however, for dilute solutions these p" is condensed isothermally and reversibly into the
deviations are small and can usually be ignored. second solution. The change of free energy for this
An alternative form of Raoult's law is obtained by
subtracting unity from both sides of Equation 1.42; the stage, like that for the first stage, is again zero; the
total free energy change for the transfer of 1 mol of
result is
solvent from the first solution to the second is thus
P given by Equation 1.46.
- = 1 -x2 (1.43)
PO Let F' represent the actual free energy of 1 mol of
solvent in the one solution and F" the value in the
The sum of the mole fractions of solvent and solute other solution. Since the latter solution gains lmol
must always equal unity; hence, if x1 is the mole while the former loses 1 mol, the free energy increase
fraction of the solvent, and .x2 is that of the solute, F is equal to F" - F'; it is thus possible to write, from
as given above, it follows that
Equation 1.46,
xi +x2 = 1 (1.44)
F" - F' = FTln - (1.47)
Hence Equation 1.43 can be reduced to
P =xlP 0 (1.45) If both solutions behave ideally, so that Raoult's
law is applicable, the vapour pressure is proportional
Therefore, the vapour pressure of the solvent in a to the mole fraction of the solvent in the particular
solution is directly proportional to the mole fraction solution (Equation 1.45); hence, for ideal solutions,
of the solvent, if Raoult's law is obeyed. It will be Equation 1.47 becomes
observed that the proportionality constant is po, the
vapour pressure of the solvent. (1.48)
As, in fact, most cell electrolyte solutions are rel-
atively concentrated, they are non-ideal solutions and For non-ideal solutions this result is not applicable,
Raoult's law is not obeyed. To overcome this problem but the activity of the solvent, represented by a, is
the activity concept is invoked to overcome departure defined in such a way that the free energy of transfer
from ideal behaviour. It applies to solutions of elec- of lmol of solvent from one solution to the other is
trolytes, e.g. s,dts and bases, arid is equally applicable given exactly by:
to non-electrolytes and gases. The following is a sim-
ple method of developing the concept of activity when F" - F' = RT In (z)
dealing with non-ideal solutions. ( 1.49)
Consider a system of two large vessels, one con-
taining a solution in equilibrium with its vapour at the This means, in a sense, that the activity is the
pressure p', arid the other containing another solution, property for a real solution that takes the place of the