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Another important application of first
passage ideas is to chemical kinetics,
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where one is often interested in how
efficiently do diffusing reactants
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actually undergo a reaction.
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This basic problem goes by the name of
reaction rate theory.
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Here is the setup of the problem.
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Consider a sphere of radius a,
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that is immersed in a fluid that contains
diffusing particles.
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The particles are some unknown,
some fixed concentration c,
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and each particle diffuses around,
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and if one of these diffusing particles
hits the sphere, it is absorbed.
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And each particle is diffusing according
to some diffusion coefficient D.
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So, the basic thing that we want to
calculate here is how quickly,
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are particles absorbed on the surface
of the sphere.
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This is embodied by something known
as the reaction rate,
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which comes by the letter k.
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So this is defined as the reaction rate,
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and it's defined as the number of
particles absorbed per unit time.
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This definition is not quite complete,
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because clearly, if we double the
concentration,
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we're going to double the number of
particles that are absorbed,
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so to get an intrinsic quantity, we should
divide by the overall concentration.
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Without doing any calculations,
this simple formula already contains
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a number of surprising results.
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Let's just do dimensional analysis,
and ask,
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"What is the dimensions
of the reaction rate?"
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So, the dimensions of particles,
there's no dimensions,
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there's a 1/time, and the concentration is
1/volume,
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and so the dimensions of the
reaction rate are
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a length scaled to the power dimension
of space,
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divided by time.
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On the other hand,
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what can the reaction rate depend on?
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And clearly, the only two parameters
that describe the system itself,
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is the radius of the sphere, and the
diffusion coefficient of the particles.
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So the reaction rate should be a function
of the diffusion coefficient,
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and the radius of the absorbing sphere.
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Now, one more thing to note is, is that
the dimensions of the diffusion coefficient
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we've discussed this earlier in this
tutorial,
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the dimension of the diffusion coefficient
is a length squared, divided by the time.
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So, armed with the information that we
know the dimensions of the
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diffusion coefficient, and the dimensions
of the radius of the sphere,
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we can infer how this reaction rate
depends on the parameters in this system.
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we know that the reaction rate
scales as 1/time.
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The only place that time appears is
in the diffusion coefficient, so clearly,
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the reaction rate should be a linear
function of the diffusion coefficient.
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If we use that fact, then we have two
powers of length in the numerator,
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whereas the reaction rate has d powers
of length in the numerator,
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so the only way we can engineer things to
have d powers of length in the numerator,
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is to have d-2 powers of a in the
numerator.
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So from these simple considerations,
we infer that the reaction rate
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should be proportional to the diffusion
coefficient times the radius of the sphere,
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to the power d-2 power.
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So, this is the basic result of reaction
rate theory,
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that the reaction rate has this dependence
on the diffusion coefficient,
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and the radius of the sphere.
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Let's just look at this in a little bit
more detail,
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because this already has some very
amazing results.
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Let's first of all, think about three
dimensions, physical space.
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In this case, the reaction rate is
proportional to the radius of the sphere.
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Now, let me put an exclamation point next
to this, to emphasize the fact that
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it's not proportional to the cross-
sectional area of the sphere,
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but to the radius of the sphere.
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So it has an anomalous dimension
dependence.
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On the other hand, if we're below two
dimensions,
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then something even weirder seems to occur
because it says that below two dimensions,
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this is a function such that the
reaction rate increases as a decreases.
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And again, let's put an exclamation point
next to that, because clearly,
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this is a nonsensical result.
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It's not possible to have an increase
in the reaction rate,
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as you decrease the size of the
absorbing sphere.
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So, the point is that below two dimensions
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There is a new dependence
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on system parameters.
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Let's now work out the reaction rate
theory quantitatively in three dimensions.
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So we have our familiar setup of an
absorbing sphere of radius a,
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that is immersed in a fluid of diffusing
particles, that are each diffusing with
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diffusion coefficient D, whenever one of
these diffusing particles
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hits the surface of the sphere, it is
absorbed.
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So, to compute the reaction rate, what
we first have to compute is the
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concentration profile of particles outside
the sphere,
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and then from that, we can then compute
the flux of particles to the surface
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of the sphere itself. So, for the first
part of the problem,
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to find the concentration profile,
we have to solve the following
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classic boundary value problem, we
have to solve the diffusion equation
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dcdt is equal to D, laplacian of c,
subject to
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appropriate initial boundary conditions.
And an appropriate initial condition is
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that exterior to the surface of the sphere,
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at initial time, the concentration, which
we can set it to be anything,
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let's set it to be equal to 1.
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And we also have the boundary condition
that if a particle hits the sphere,
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at any positive time, it is absorbed,
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which corresponds to the absorbing
boundary condition, of concentration 0.
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So, this is the problem that we have
to solve.
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This is a classic boundary value problem,
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the solution can be written in terms of
vessel functions,
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and with a little bit of cleverness,
one can actually transform this to a
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one-dimensional problem, where the
solution can be written in terms of
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elementary functions, namely the error
function, but I'm going to try to avoid
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all of that, by taking advantage of the
fact that in three dimensions,
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a diffusing particle, or a random walker,
is transient.
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That is, the probability that a particle starting at some distance away from the sphere,
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actually hits the sphere, is less than 1.
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What this means is that the depletion of
particles near the surface of the sphere
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is not very severe. Moreover, it also means
that particles from infinity can come in
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and replenish the particles that are
depleted because of the absorption.
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And because of this feature, I'm going to
hypothesize,
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and in fact it's a true statement, that in
the long time limit,
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this concentration profile actually goes
to a steady state.
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So, instead of solving the time dependent
problem, I'm going to solve the time-
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independent, or the steady-state version
of the same problem,
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that is, I'm going to solve not the diffusion
equation, but the laplace equation,
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D, laplacian of c, is equal to 0.
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Namely, I'm going to set the time
derivative to 0.
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We have to solve the same problem
with the corresponding boundary conditions
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as with the time-dependent problem.
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And that is that, at the surface of the
sphere, the steady state concentration
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should be equal to 0, and moreover, if
we go to infinity,
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the concentration should be 1. So, this
is the problem that we're going to solve
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instead of the full time-dependent problem
and I will argue, and in fact it is true that,
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this time-independent problem corresponds
to the infinite time limit
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of the time-dependent problem. The nice
feature of solving this time-independent
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problem is that it's just the good old
laplace equation at the surface of the
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spherical conductor, and so we know the
solution from our freshman physics.
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And for this set of boundary conditions,
the solution is very simple.
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c of r, is equal to 1 minus a/r.
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Now, in addition to the fact that the
solution is simple,
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it also has a very nice interpretation in
terms of random walks.
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earlier in the discussion of first passage
ideas, we learnt that the exit probability
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or the hitting probability satisfies the
laplace equation with an appropriate
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boundary condition. If one looks here
at what we're trying to solve,
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it looks very similar to the hitting
probability problem,
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namely the probability that the random
walk will eventually hit the sphere,
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except the boundary conditions is a
little bit backwards,
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because we have a boundary condition where
if you're at the surface of the sphere,
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the concentration is 0.
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So, in fact, what we're solving is not
the hitting probability,
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we're solving the classic escape
probability.
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Namely, what is the probability that a
random walk could have start at radial
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position r, escape to infinity without
ever hitting the surface of the sphere?
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And these boundary conditions just mirror
the escape probability.
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If I start at the surface of the sphere,
by definition, I cannot escape.
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And if I start infinitely far away, then
I have escaped, by definition.
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So, this is nothing more than the
eventual escape probability,
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and notice it has this nice feature that
if you start at distance a above the
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surface of the sphere, or namely,
r is 2a,
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the escape probability is 1/2,
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and the eventual hitting probability
is also equal to 1/2.
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And finally, let's now come to the problem
which is to calculate the reaction rate.
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So the reaction rate is the flux of
to the surface of the sphere,
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so we have to calculate the diffusive
flux, D gradient of c,
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we want to compute its dot product with
the surface area of the sphere,
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and integrate over the surface of the
sphere,
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and we have to have with a minus sign,
because we have the inward surface model,
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so this is a very simple integral to do,
because it is spherically symmetric,
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so we have D coming out in front,
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the concentration field has an a out
in front,
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when I take the gradient of 1/r,
I get 1/(r^2),
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so we're integrating over the surface
of the sphere of radius a, of 1/(r^2),
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which is 1/(a^2), times dS, and this is
nothing more than
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4, the surface area of the sphere,
of radius a, which is 4 pi a^2,
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and so what we get here is 4 pi D a.
And so, as advertised,
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the reaction rate is proportional
to the diffusion coefficient
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of the diffusing particles, and it's
linearly proportional to the length of
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the radius of the sphere.
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Notice again that it's not proportional to
the cross-sectional area of the sphere.
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If one solves the full time-dependent
problem,
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it turns out that there is a correction,
due to the time dependence,
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it goes away in the long-time limit,
and so the correct answer here,
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including the full time dependence,
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is 1 + a/sqrt(pi Dt).
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So, this is a fundamental result in
chemical reaction kinetics,
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the reaction rate of the sphere of
radius a.