Page 22 - Battery Reference Book

Electromotive force ln

among them or arranged in an interpenetrating lattice. will thus be proportional to their concentration (is.
An ion in the surface layer of the metal is held in its to the number near the metal) and to the rate at
position by the cohesive forces of the metal, and before which these acquire sufficient kinetic energy. The
it can escape from the surface it must perform work rate of deposition can thus be expressed as Q2 =
in overcoming these forces. Owing to their thermal kl‘c exp(-WZlkT).
agitation the surface ions are vibrating about their Q1 and Qz are not necessarily equal. If they are
equilibrium positions, and occasionally an ion will unequal, a deposition or solution of ions will take place
receive sufficient energy to enable it to overcome the and an electrical potential difference between the metal
cohesive forces entirely and escape from the metal. and the solution will be set up, as in Nenast’s theory.
On the other hand, the ions in the solution are held to The quantities of work done by an ion in passing from
the adjacent water molecules by the forces of hydration Q to S or R to S are now increased by the work
and, in order that an ion may escape from its hydration done on account of the electrical forces. If VI is the
sheath and become deposited on the metal, it must have electrical potential difference between Q and S, and
sufficient energy to overcome the forces of hydration. VI’ that between S and R, so that the total electrical
Figure 1.4 is a diagrammatic representation of the potential difference between Q and R is V = V’ + V”,
potential energy of an ion at various distances from the total work done by an ion in passing from Q to
the surface of the metal. (This is not the electrical S is W1 - neV’ and the total work done by an ion in
potential, but the potential energy of an ion due to the passing from R to S is Wz + neV”, where n is the
forces mentioned above.) The equilibrium position of valency of the ion and e the unit electronic charge. V’
an ion in the surface layer of the metal is represented is the work done by unit charge in passing from S to
by the position of minimum energy, Q. As the ion is Q and V” that done by unit charge in passing from R
displaced tovvards the solution it does work against to S. The rates of solution and deposition are thus:
the cohesive forces of the metal and its potential
energy rises while it loses kinetic energy. When it 81 = k’exp [-(Wl - nV’)/kT]
reaches the point S it comes within the range of the 82 = k’lcexp [-(Wz + nV”)/kT]
attractive forces of the solution. Thus all ions having
sufficient kinetic energy to reach the point S will For equilibrium these must be equal, i.e.
escape into the solution. If W1 is the work done in
reaching the point S, it is easily seen that only ions k’exp [-(Wl - nV’)ikT] = kl’cexp [-(W2 + nY”)/kT]
with kinetic energy W1 can escape. The rate at which
ions acquire this quantity of energy in the course of or
thermal agitation is given by classical kinetic theory
as Q1 = k‘ exp(-W1lkT), and this represents the rate
of solution of metal ions at an uncharged surface.
In the same way. R represents the equilibrium If No is the number of molecules in the
position of a hydrated ion. Before it can escape from gram-molecule, we may write:
the hydration sheath the ion must have sufficient
kinetic energy to reach the point S, at which it comes No(W1 - Wz) = AE
into the region of the attractive forces of the metal. Noe = F
If Wz the difference between the potential energy
is
of an ion at FL and at S, it follows that only those ions Nok = R
that have kinetic energy greater than Wz can escape and we then have
from their hydration sheaths. The rate of deposition v = - - lnc + - (G)
AE
RT
RT
+
tr nF nF nF In
The final term contains some statistical constants
which are not precisely evaluated, but it is evident
that, apart from this, V depends mainly on AE, the
difference of energy of the ions in the solution and in
the metal.
Comparing this with the Nernst expression we see
that the solution pres P is
LQ * (1.14)

Distance from surface One of the difficulties of Nernst’s theory was that
Figure 1.4 Potential energy of an ion at various distances from the values of P required to account for the observed
the surface of a metal potential differences varied from enormously great to