Help, my bottle is freezing! (Part 1)

Long winter runs become extra miserable if you're a sports-drink-nutrition kind of person and if your bottles freeze while you are out frolicking in the freeze.

Facing my (first ever) 20 miler in mid-February, and also looking at temperatures in the teens (Farenheit), I dug through my ancient chemistry memory to figure out if I was going to have a Skratch-pop before the end of the run.

The principle that I wanted to use was "freezing point depression." I know that adding salt to roads helps lower the freezing point so the roads are less icy, and I know I'm drinking an electrolyte mix, so how low does the freezing point of Skratch get?

TL;DR: Even a double-concentrated Skratch mix will give you a Skratch-sicle, sorry!

1. Assumptions:
• Pure water, acting like pure water
• Wild guesses on the concentration of the ingredients that come from the Skratch mix 
• No accounting for jostling that will actually come from running
• No accounting for body heat keeping the bottle warm (hopefully most of my heat stays inside my jacket)
2. Background:
•  According to my sister, Dr. Chemist, and wikipedia, the formula I was trying to remember is 
  • ΔT_F = K_F · b · i
• DeltaT_F is the change in the solvent's freezing point temperature (in Celsius) 
  
• K_f is called the cryoscopic constant, and the internet believes that you should mostly look this up in books if you are trying to figure this out for things other than water, or like, ethanol (spoiler alert for part 2 of this post)
• b is the place where some chemistry is going to really kick in because I have to calculate the "molality" of the stuff. Flashback to Mole Day celebrations, which are way less delicious than Pi Day celebrations
  
• i is the "Van 't Hoff factor" and has to do with how many ions the dissolved stuff splits into.
  
3. K_F - For water, this value is 1.86. A bigger K_F means that the same mixture (assuming same dissolution) would have a bigger temperature shift, so too bad Benzene is toxic (K_F of 5.12)
4. b - Molality is moles of "stuff" per kilogram solvent. For Skratch, I have decided there are three main kinds of stuff, aka solute.



Reference source for wild guesses for solutes

• To calculate molality, I had to first find the number of moles of each of the compounds I'm interested in.
• The molar mass of a compound comes from adding up the atomic weight of the elements, or from looking it up on the internet, and it's units are grams/mole.
• My ingredients are
• Sugar - 20 grams
• Sodium Citrate -  0.36 grams
• Potassium Citrate - 0.04 grams
• From the nutrition information from Skratch, and from wikipedia, I was able to calculate the number of moles of each as shown in the table below

Skratch Content	Grams	Molar mass (moles/gram)	Moles (grams / molar mass)
Sugar	20	342.3	0.05
Sodium Citrate	0.36	258.0	0.0013
Potassium Citrate	0.04	306.4	0.00013

• Next, I have to figure out how many grams of water I am dissolving these in. I'll be making the Skratch with 2 scoops in a 20 oz bottle, or 1 scoop per 10 oz. 10 oz of water weighs about 300 g. I've updated the table with a new column for this data. The tiny values are making me not feel too optimistic about this calculation

Skratch Content	g	m	Moles (grams / molar mass)	Molality, (mol/kg), b
Sugar	20	342.3	0.05	0.19
Sodium Citrate	0.36	258.0	0.0013	0.0046
Potassium Citrate	0.04	306.4	0.00013	0.0004

5. i is the number of ions the things split into. For this, I drew on the expertise of my sister again
• Sugar - 1 - no splitting
• Sodium Citrate - 3 - Three sodium
• Potassium Citrate - 3 - Three potassium
6. Calculation

Skratch Content	Molality, (mol/kg), b	i	ΔTF = KF · b · i
Sugar	0.19	1	0.19
Sodium Citrate	0.0046	3	0.013
Potassium Citrate	0.0004	3	0.001

• ΔT_F = 0.19 + 0.01 + 0.001 = 0.39 degrees Celsius change
• Sadly this is only 31.3 degrees Farenheit, so given Saturday's weather was in the teens, I would have had a Skratch-sicle

Next up, will a Skratch-cocktail solve my problem? Will I still be able to run 20 miles without getting tipsy?