This is a difficult challenge, so I am
going to up the ante considerably. The BMA math/science department
is offering a $100 cash prize for a mathematically and scientifically
correct answer that is reasonable given real-world uncertainties.
As I wait for Eastern Cups to happen,
I'm looking up at Nolan Kasper's slalom course. They placed
salt-bags every seven gates. Each bag is 50 pounds of Sodium
Chloride. Make assumptions about some unknown information and
calculate the temperature of the snow that is treated in this manner.
The depth, detail, quality and
presentation of your solution matters much more than the actual
answer you come up with.
This contest is open to students,
alums, parents, faculty, everybody. It's an open challenge.
Deadline for submission of solutions is
5pm, April 15th.
Submissions must be emailed to
diverson@burkemtnacademy.org
The problem doesn't give us much data to work on so let's make some assumptions: Bags of salt are 7 gates apart. Assume 10 meters between gates (a bit long but we need a number) 7x10m is 70 linear meters. Then assume the salted path is 2m wide 140 square m to salt. Finally assume penetration 2 inches, 5 cm or 0.05 m gets us 7 cubic meters of snow to treat.
ReplyDeleteThe change in freezing point depends on solute/Kg solvent so we need the density of the snow. Google gives a packed snow density of 481 Kg/m3, so that gets us about 3.4x103 Kg of snow to treat.
Now the salt; 50 lb bag is 23 Kg or 2,3x104 g. Molar mass of NaCl is 58 so a bag of salt is about 4x102 moles.
Freezing depression is a function of molality: moles of salt/Kg of solvent (water:ice) 4x102 moles/3.4x103 Kg water = 0.12 molal
Now for the final change in FP -1.86 C x 0.12 m x 2 (ions per "molecule" of NaCl) = -0.45 degrees C
Most people will understand this better in F. That's about a decrease of 0.8 F so the new temp of the snow is 31 degrees F.
That may not seem like much but here is the key. Because melting the upper layer draws energy out of the underlying snow, this is sufficient to harden and freeze the surface for racing. To refreeze the snow melted by the ice, 80 calories of heat are drawn out for every gram was melted. This can lower the temperature of the underlying snow by almost 20 degrees for a hard and raceable surface