Friday, December 2, 2016

Chocolate Chips in Syrup: A Case of (Very Little) Diffusion

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=KBT/(6πnr)

KB=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=KBT/(6πnr)
D= (1.38 x 10-23)( 293)/( 6π*3*.0025)= 2.86*10-20 m2/s

Assume movement of chocolate chip in 3-D. Sitting in syrup for 12 seconds.
<r2>=6*D*t= 6*2.86*10-20 *12= 2.06 * 10-18 m2
r=1.43* 10-9 m= 1.43 * 10-11 cm

Versus if I choose really, really viscous chocolate syrup:
D=KBT/(6πnr)

KB=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=KBT/(6πnr)
D= (1.38 x 10-23)( 293)/( 6π*25*.0025)= 3.43 * 10-21 m2/s

Assume movement of chocolate chip in 3-D. Poured syrup for 12 seconds.
<r2>=6*D*t= 6*3.43 * 10-21*12= 2.47* 10-19 m2
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?

rmaple/rchocolate= 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.

 

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