Environmental Engineering Reference
In-Depth Information
voltammogram is the derivative of U with respect to time:
dU
dt ¼ dU
du
dQ
dQ
dt
(3 : 8)
du
Here, dU/dt is the linear sweep rate of the experiment and dQ/dt is the measured cur-
rent. With a transfer of one electron per adsorbed H, the relation between the charge
transferred per area Q, and the coverage u is Q ¼ Q tot u, where Q tot is e times the
density of Pt atoms in the surface layer. Introducing +K for the sweep rate and i(t)
for the current, we obtain
1
dU = du ¼ +KQ tot
du
dU
i(t) ¼ +KQ tot
(3 : 9)
Equations (3.7) and (3.9) imply that we need only the hydrogen coverage dependence
of the reaction free energy of Reaction (3.4) as input in the theoretical CV calculation.
This reaction energy should be calculated for H adsorbed at the platinum electrode
under realistic conditions for electrochemical measurements, i.e., in the presence of
both the electrolyte and the electric field. To assess the importance of these effects,
we could first look at the effect of water and electric field on the reaction energy.
As was mentioned previously, and as can be seen from Fig. 3.1a, it turns out that
the presence of both water and electric field has a very minor effect on the reaction
energy of Reaction (3.4). Hence, we can simply do the calculations in a normal surface
science setup, i.e., with vacuum above the metal surface. In Fig. 3.1a and b, the reac-
tion energy for Reaction (3.4) is shown for Pt(111) and Pt(100) versus the number of
nearest neighbors and versus coverage, respectively.
Before inserting the coverage dependence shown in Fig. 3.1 into the formulas
above, we must ensure that we have taken all temperature effects into account.
Whereas the reaction free energy, as defined in Equation (3.5), already contains an
entropy term, when considering the macroscopic coverage u, we will also have to
include differential configurational entropy. In principle, this can be done in two
ways. We can use the differential configurational entropy of noninteracting particles,
DS conf ¼ k B ln 1 u
u
(3 : 10)
in which case we obtain the mean field, or Frumkin, isotherm. Alternatively, we
can model the system by use of a lattice gas model and solve for the free energy as
a function of coverage using, for instance, Metropolis Monte Carlo. In the latter
case, the configurational entropy of interacting particles is taken into account to the
limits of the accuracy of the lattice gas model. We will not go into all details
here (see [Karlberg et al., 2007a]), but rather go directly to the resulting CVs shown
in Fig. 3.2. The agreement with experiments, such as the CVs reported by
[Markovic et al., 1997], is good, a result that is in line with a similar investigation
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