Over
Thanksgiving break, I was lucky enough to enjoy pancakes at one of my favorite
diners. One of the main reasons that this is my favorite diner is because of
the diverse selection of syrups they offer their patrons. As a risky move, I
decided to add some chocolate chips to the small container of syrup before
drowning my pancakes in it. While I was pouring the syrup, I noticed how slowly
the syrup and chocolate chips in it moved out of the container and onto the
pancakes. Since the syrup was a moving fluid (and thus more complicated to
analyze), I decided to analyze the diffusion of the chocolate chip in the
static maple syrup (when it’s just sitting in the container).

D=K

_{B}T/(6πnr)
K

_{B}=1.38 x 10^{-23}
T (room
temperature) = 293 K

n
(coefficient of viscosity)= 2-

**3**Pa *s
d=.5
cm, r=.25 cm=.0025 m

D=K

_{B}T/(6πnr)
D= (1.38 x 10

^{-23})( 293)/( 6π*3*.0025)= 2.86*10^{-20}m^{2}/s
Assume
movement of chocolate chip in 3-D. Sitting in syrup for 12 seconds.

<r

^{2}>=6*D*t= 6*2.86*10^{-20}*12= 2.06 * 10^{-18}m^{2}
r=1.43*
10

^{-9}m= 1.43 * 10^{-11 }cm
Versus
if I choose really, really viscous chocolate syrup:

D=K

_{B}T/(6πnr)
K

_{B}=1.38 x 10^{-23}
T (room
temperature) = 293 K

n
(coefficient of viscosity)= 10-

**25**Pa *s
d=.5
cm, r=.25 cm=.0025 m

D=K

_{B}T/(6πnr)
D= (1.38 x 10

^{-23})( 293)/( 6π*25*.0025)= 3.43 * 10^{-21 }m^{2}/s
Assume
movement of chocolate chip in 3-D. Poured syrup for 12 seconds.

<r

^{2}>=6*D*t= 6*3.43 * 10^{-21}*12= 2.47* 10^{-19}m^{2}
r=4.97*10

^{-10}m= 4.97 * 10^{-12 }cm
How much more
does the chocolate chip in syrup move due to diffusion than the chocolate chip
in chocolate syrup?

r

_{maple}/r_{chocolate= }1.43 * 10^{-9 }cm/4.97 * 10^{-12 }cm= 287~300
Due
to diffusion, the chocolate chip diffuses nearly 300 times more in the maple
syrup than it does in chocolate chip! Both substances are clearly very viscous
in comparison to something like water, but since the chocolate syrup is even
more viscous than maple syrup, the chocolate chip does not diffuse as much in
the syrup. If you assume that the chocolate chip was JUST moving by diffusion,
how long would it take the chocolate syrup to diffuse the same distance?

6*3.43
* 10

^{-21}*t= 2.06 * 10^{-18 }
t=100.1
s

If
you assume that the chocolate chip is moving just by diffusion, the chocolate
syrup would have to remain static for about 100s. This is a lot longer than how
long the maple syrup would have to remain static for it to diffuse the same distance
(12 s).

Ultimately,
the distance the chocolate chip moves in either maple syrup or chocolate syrup
is extremely small, and would probably be undetectable. In part, this is
because the chocolate chip is so small and thus the diffusion coefficient is
really small. If you wanted the chocolate chips to diffuse a greater distance
you could: (1) choose a less viscous solvent like water (2) choose mini
chocolate chips or (3) increase the amount of time that the chocolate chip is
sitting in the static fluid.

Sources:

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