Conditions for Critical Effects in the Mass Action Kinetics Equations
for Water Radiolysis
Richard S. Wittman,* Edgar C. Buck, Edward J. Mausolf, Bruce K. McNamara, Frances N. Smith,
and Chuck Z. Soderquist
Energy and Environment Division, Pacific Northwest National Laboratory, Richland, Washington 99352, United States
ABSTRACT: We report on a subtle global feature of the mass action kinetics equations for
water radiolysis that results in predictions of a critical behavior in H
2
O
2
and associated
radical concentrations. While radiolysis kinetics have been studied extensively in the past, it
is only in recent years that high-speed computing has allowed the rapid exploration of the
solution over widely varying dose and compositional conditions. We explore the radiolytic
production of H
2
O
2
under various externally fixed conditions of molecular H
2
and O
2
that
have been regarded as problematic in the literaturespecifically, “jumps” in predicted
concentrations, and inconsistencies between predictions and experiments have been
reported for α radiolysis. We computationally map-out a critical concentration behavior for
α radiolysis kinetics using a comprehensive set of reactions. We then show that all features of
interest are accurately reproduced with 15 reactions. An analytical solution for steady-state concentrations of the 15 reactions
reveals regions in [H
2
] and [O
2
] where the H
2
O
2
concentration is not uniqueboth stable and unstable concentrations exist.
The boundary of this region can be characterized analytically as a function of G-values and rate constants independent of dose
rate. Physically, the boundary can be understood as separating a region where a steady-state H
2
O
2
concentration exists from one
where it does not exist without a direct decomposition reaction. We show that this behavior is consistent with reported α
radiolysis data and that no such behavior should occur for γ radiolysis. We suggest experiments that could verify or discredit a
critical concentration behavior for α radiolysis and could place more restrictive ranges on G-values from derived relationships
between them.
■
INTRODUCTION
It is notable that the basis of the chemical kinetics equations, the
Law of Mass Action, dates back to the 1860s
1,2
before the
atomic theory of interacting molecules was established. As in
thermodynamics, experiment and theory led to general principles
that have mostly survived modern developments in physics and
chemistry. This survivability is not surprising because the laws of
mass action and thermodynamics are primarily the consequence
of the statistical nature of many interacting degrees of freedom
in our case, interacting molecular species.
In radiolysis, many of the species are very short-lived, rapidly
de-exciting or interacting with surrounding species. On the time
scale of interest for radiolysis kinetics reactions, the formation of
short-lived radical species and their reactions in water have been
extensively studied.
3-8
As in the case of many other authors,
3-9
we take these reactions and generation rates as a starting point,
but we emphasize that we are not competing with those works or
proposing a more sophisticated radiolysis model.
Our goal here is to highlight a subtle numerical behavior of the
radiolysis kinetics equations in general that has not previously been
recognized and then give the analytical basis for such behavior.
Whether such behavior actually occurs in nature is an open
question to be answered experimentally.
As in previous work, we notice a particular dependence of
H
2
O
2
production on the concentrations of O
2
and H
2
O
2
.
5-7
The
conditions considered are particularly applicable to the H
2
O
2
-
driven corrosion of spent nuclear fuel in an environment
depleted of O
2
and in the presence of H
2
overpressure generated
from structural iron reacting with water.
10
While many complex
chemical and physical processes can be imagined to operate at
the exposed UO
2
surface,
9-13
we focus on radiolysis of water
from a uniform α dose.
Unlike previous work, we identify a reduced reaction set
exhibiting the same global features analytically and then show
how [H
2
O
2
] behaves like an order parameter that characterizes
distinct phases of the system. A critical point in H
2
O
2
concentration is predicted that implies a ([H
2
], [O
2
]) region
with a discontinuous boundary that may be the source of
disagreement between model predictions and data.
7
This work
attempts to explain how a ([H
2
], [O
2
]) region with a boundary,
discontinuous in [H
2
O
2
], can emerge in radiolysis kinetics.
The next section describes the approach taken to solve the
radiolysis kinetics equations numerically and outlines the
reasoning for working with a reduced reaction set analytically
at steady-state. It is then shown explicitly how a critical jump in
H
2
O
2
concentration arises from stability conditions on model
parameters. Finally, we summarize the conditions for global
critical jumps in radiolysis kinetics and offer suggestions on how
experiments could confirm or possibly correct model parameters.
Received: October 1, 2014
Revised: November 24, 2014
Article
pubs.acs.org/JPCA
© XXXX American Chemical Society A dx.doi.org/10.1021/jp509856g | J. Phys. Chem. A XXXX, XXX, XXX-XXX