Wednesday, February 19, 2014

No Feels, Only Reals: Baking Soda, pH, and Dilution Edition


Hello everyone.  So an article was posted recently that caused a bit of a kerfuffle on r/NoPoo: it showed that when raw baking soda is added to water, the pH remained more or less the same, regardless of the amount of water added.  Only when she took some of the baking-soda water solution (i.e. AFTER the solution is in equilibrium) was dilution possible.

I thought it might be instructive to look at why that is.  Not entertaining, particularly, but instructive.  There's a metric butt-ton of equations in here, which is why I changed from just posting on reddit to the blog.


–Let’s do some conversions and define some terms–

1 tablespoon = 0.0148 L
1 cup = 0.237 L
2 cups = 0.473 L (Not dodgy math: the values of 1 cup and 2 cups were calculated independently and rounded to three significant figures)

Baking soda  = NaHCO3 (Take heed! Baking soda is a chemical.  It's not really even a natural one most of the time.)


–There are a few factors that complicate this reaction–

First of all, pH is the measure of acidity or basicity in aqueous solutions.  That is to say, before we can talk about what the pH of NaHCO3 “is”, we first have to make a solution with water.

Furthermore, NaHCO3 is a weak base, which means it does not dissociate completely in water (Rats! This means loads more work for us).

What happens when you mix NaHCO3 with water?

The Na+ ion is a “spectator ion”, leaving us with HCO3. This gets even trickier because HCO3 is amphiprotic, meaning it can act like an acid OR a base, depending on the reaction.  HCO3 can react to form its conjugate acid H2CO3 (carbonic acid) and OH- (what makes a solution basic), and vice-versa, although the system will eventually reach equilibrium.  We need to know the acid dissociation constant (ka) and base dissociation constant (kb).

The ka of a weak acid can be used to find the kb of its conjugate base, using ka*kb=1x10-14.  The ka of carbonic acid is 4.45x10-7, so the kb of HCO3 is 2.25x10-8.


–Let’s get started!–

We have to do a few preliminary calculations.

First, we need to know what the molarity of our baking soda solution.  Let’s assume that we are making the standard 1 tablespoon of baking soda to 2 cups of water no-poo mix, since this is the most frequently-given ratio, the one that people are advised to try first and then adjust based on results.

We need to know what 1 tablespoon of baking soda is in grams, which is a bit of a pain, because one is a measure of volume and the other of mass.  There’s no clear conversion.   If you try and find out by googling, you’ll find kind of a range of answers.  So, I figured I’d just measure a tablespoon for myself and see what I got.  It came to 12.6 grams, which is in range, so this is the figure I’ll be using for the calculation.

The molar mass of NaHCO3 is 84.007 grams so to find how many moles in a tablespoon:



Now, we need the molarity of the solution so:



We will get one mole of HCO3 from NaHCO3, so we can say that the pre-equilibrium concentration of HCO3 will be the same as above.  The OH- concentration is so small that it can, helpfully, be regarded as 0M (technically speaking it’s 1.0x10-7M).  Since there is no H2CO3 before reaching the equilibrium state, it is also 0M.

After reacting with water, one mole of HCO3 will form one mole of OH1 and one mole of H2CO3.  So, given that, let’s make a table to chart these changes:


HCO3
H2CO3
OH-
Pre-Equilibrium
0.315 M
0
0
Change
-x
+x
+x
Equilibrium
(0.315-x)M
xM
xM

Now we get to use the equilibrium equation:



Now we have something we can work with!  We know that the Kb of HCO3 is 2.25x10-8 from earlier.


It looks like here we have to use the quadratic equation to solve, but it turns out that since x is likely to be much, much smaller than 0.315, you can drop the x from the denominator without affecting the outcome too badly (you can double-check that this is a valid approximation later on--just put it back into the equation for Kb above and make sure it all checks out).  So:



Since “x” stands for the molarity of OH-, we can now use this value to solve for the pH of our solution!



So you can see that 1 tablespoon in 2 cups of water will have a pH of 9.9.  This is very high, and exactly what the other blogger demonstrated with universal indicator paper.

You can also see why adding more water doesn’t really do much to the pH of the solution.  Adding more water will only change the initial M-value that we used.  Let’s look at this equation with 20 cups (= 4.73 L) of water.






Almost no different!  So, what value of M value do you need in order to have a pH of 7?  Well, let’s work backwards.  If our pH is 7, so is the pOH.  Then, we can use the definition of pOH to find that we need x=10-7 to get a neutral solution.  Now our equation looks like this:




This is a tiny, tiny number!  Let’s keep going to see how many liters of water you need to add to a tablespoon of baking soda to get this result:




–TL;DR–

1 tablespoon of baking soda in 2 cups of water has a pH of 9.9.
1 tablespoon in 20 cups has a pH of 9.4.
You need 1418439 cups of water to make a solution with a pH close to 7.

Anyway, the key point of all of this is, because of the nature of the reaction that NaHCO3 has with H20, it takes an enormous amount of water to dilute even one tablespoon of baking soda down to a near-neutral pH.

No-Poo People who use baking soda (or those who are considering it) may want to take this fact into consideration when constructing their hair care regime.  It’s certainly true that pH is not THE ONLY important factor when it comes to hair health, but it’s also a fact that the scalp has an acid mantle that will be disrupted by having a high-pH solution onto it, and there are a number of people who say that hair itself is strongest when slightly acidic.

Furthermore, the pH of 1 tablespoon : 2 cups solution is starting to approach 10, which is the range where the cuticle of hair can becaused to lift.  This is very undesirable.  As the linked page says, the hair cuticle is not hinged, it cannot open and close freely!  Repeatedly lifting and smoothing it will damage it.

Those of us striving for a scientifically-sound hair care regime would do well to leave the baking soda in the kitchen.

If you'd like to know more about alternatives to baking soda, I'll be testing out common choices for a month at a time. So far: rye flour.

5 comments:

  1. Thank you for the awesome chemistry explanation! I was looking for something like this :)

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    1. I'm glad it was interesting! Thanks for reading (ˆ-ˆ)

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  2. Hello, thank you for taking the time and effort to make all these calculations that were triggered by my findings on baking soda: http://blog.kanelstrand.com/2014/01/baking-soda-destroyed-my-hair.html I was wondering if you would allow me to repost this post as an update to my original article, with a link back to your blog of course.

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  3. I wonder if you can do equations like this for soap..

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    1. I'd actually really love to post the equations for soap as well, but they're a little bit more complicated. Soaps are made of salts of fatty acids, and there can be different types and compositions, whereas baking soda is pretty simple. I've also been kind of distracted with my Game of Thrones costume analysis project, but I'd like to find time to work out a (simplified) version of the soap calculations as well. I can say based on testing with Universal Indicator paper, though, that the pH of soap also doesn't dilute much.

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