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PROFESSOR: So,
as an aside for more

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advanced students,
let's try to fill

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in some mathematical
details to provide a theory

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to support or interpret
the Lin-Marr hypothesis

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of disinfection
kinetics having to do

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with the concentration
of solutes during drying

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and their effect on
deactivating viruses.

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So, to put it in
mathematical terms,

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if we have a certain number
of viruses Nv in a droplet,

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then we'll postulate
that d Nv dt is

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minus lambda v0,
the deactivation

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rate per solute
virion collision,

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times the volume fraction
of disinfecting solutes

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we'll call phi d, which
is time dependent,

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having to do with the size
of the droplet, times Nv.

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The volume fraction
of disinfecting

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solutes we'll write as alpha
D, a constant, times phi

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s, which is the total volume
fraction of solutes present.

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And that might be, for example,
the fraction of solutes

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that are sodium chloride
or some other salt that

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might be causing the damage
to the virus, as opposed

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to the mucins or other
macromolecules that

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may be present.

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Then we can-- as the droplet is
shrinking with a radius R of t,

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then it's simply
the volume of phi s

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that is getting re-scaled
relative to the initial value,

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phi 0, as R0, the
initial radius,

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divided by R of t cubed.

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So that's just simply the
changing of the volume.

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Now let's recall
some of our results

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from the past earlier
part of this chapter

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having to do with Wells'
theory of evaporation.

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So, if we consider
diffusion-limited droplets,

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we've shown that the radius
of the droplet versus time

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relative to the
initial radius R0

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is square root of 1
minus t over tau e, where

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tau e is the evaporation
time, R0 squared divided

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by d bar, a constant with
units of diffusivity, times 1

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minus RH, the relative humidity.

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Now that predicts
pure liquid droplets

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that shrink all the way to
nothing and evaporate away,

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but, when there's
solute present, there's

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a cutoff, which we've also
discussed that gives you

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an equilibrium stable
size of the drop,

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R equilibrium, relative to
R0, which is given by phi s0,

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the solid volume fraction--
or solute volume fraction

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initially divided by 1 minus
RH raised to the 1/3 power.

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By writing that as square root
of 1 minus tau over tau e,

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we can also define the time tau
when you reach the equilibrium

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size by a diffusion-limited
evaporation process.

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So that's sort of the time to
form a stable droplet nucleus.

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Now let's start combining
all these equations,

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and we can write what is the
volume fraction of disinfecting

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solutes, phi d of t.

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Well, from this equation
here, it'll be alpha d times

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phi s of t, which
is phi s0, times

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this ratio, R0 over R cubed.

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So, using this expression for
diffusion-limited kinetics,

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this would give me a 1 minus
t over tau e to the 3/2.

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And, if we look at
the ultimate limit

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here that they'll get from when
it's a solute, when tau goes--

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or when t goes to tau,
the evaporation time,

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so when you've reached
the droplet nucleus stage,

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we're left with just
alpha d times 1 minus RH.

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So that tells us sort of the
fraction of solutes which

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are present as a function
of relative humidity,

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but, also, as a function of
time, as drying is going on.

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So now let's go back to
this dynamical equation.

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And let's go ahead and solve it.

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So this is a first-order,
separable-order differential

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equation.

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So what we can do is write this
as minus d Nv over lambda v0.

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Nv is equal to phi d of t dt.

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So we've put all
the N's on one side

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and the t's on the other side.

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And so we can actually then
integrate this equation.

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And so the integral of dN over
N is the natural log of N.

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So we can write this as
minus 1 over lambda v0

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natural log of Nv
over Nv0, which

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is the initial value of Nv.

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And, in time, we're integrating
from the initial time

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0 up to the droplet nucleus
time tau of phi d of t dt.

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So, substituting our
expression right here,

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we then see that we have
alpha d phi s0 times

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the integral from 0 to tau
dt over 1 minus t over tau e

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to the 3/2.

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And we can do that integral
and get alpha d phi s0.

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And then let's see.

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To get the integration
variable, we

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need to have a tau e here and
write that as dt over tau e.

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And, doing the integral,
we would get 2 times 1

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over square root of 1 minus
tau over tau e minus 1,

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evaluating at the two
limits of integration,

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taking into account the
integral of the-- antiderivative

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of integrand there is 1
over 1 minus t over tau e

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to the 1/2 power times 2.

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So, putting all
this together then,

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we can write the viability.

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So we can write the log
of Nv over Nv0 as minus--

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we have all this stuff here--

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2 alpha d phi s0 lambda
v0, putting the lambda

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v0 back on the other
side with the minus sign.

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And then we have
times two factors.

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So, first, there's
the factor, which

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we know has units of time,
which is R0 squared over d bar.

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So that's, essentially, kind
of a water vapor diffusion time

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that comes into the
evaporation time, tau e.

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So that sets the timescale here.

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But then what we're
really interested in

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is the relative humidity effect.

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So that would be--

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let's see here.

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So we have this factor,
and then we also have the--

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let's see.

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The 1 minus RH is
coming in where?

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Sorry, so, 1 over square
root of tau, this one

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is from right here.

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That's R over R of tau.

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And R of tau is, by
definition, R equilibrium.

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So it's this factor here.

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So we get 1 minus RH over
phi s0 to the 1/3 minus 1.

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And then we also
have this factor

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of 1 minus RH that comes,
yes, from the tau e

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because the tau e has this
sort of basic timescale,

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but there's also a factor of
1 minus RH that I've included.

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So the point of
all this theory was

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to try to understand
what is the dependence

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on relative humidity, which is
what I've shown here in white.

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And, if you plot this
function, then what you find

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is a function of
relative humidity.

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Then, if you do here
log of Nv over Nv0--

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so this is our relative
viability of the virus,

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and the 0 here corresponds
to Nv0, the initial--

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then, this white function,
what this looks like

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is something, which
decays like this.

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It kind of reaches
a minimum around 80

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or in this range from
sort of 60 to 80,

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depending on what the values
of this parameter phi s0

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is in fact.

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And then it goes back up again.

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So, basically, we get a
shape for the dependence

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of the relative humanity
that nicely matches

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the experimental data and is
consistent with the hypothesis

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of disinfection kinetics that
was postulated by Lin and Marr.