Calculating Molarity (mole/L)

As a preface to this post, I want to make doubly clear my stance on vaccines. There is no good scientific evidence to support the notion that vaccination is in any way an unsafe practice or that it is responsible for any manner of health problem above and beyond the diseases that vaccines protect against. Vaccination is the single most powerful health intervention created in the last 150 years of medicine. There is, in my opinion, some potential for this post to be used to damage the credibility of a person who I believe to be a necessary positive force in the Healthcare scene and I want to make it clear that this was not the intention of my writing here. Orac is a tireless advocate for science and for clear, skeptical thought in general and I respect him quite deeply for the time he puts in and for putting up with the static he puts up with.

That said, I believe that science advocacy is a double edged sword: if you didn’t get it right, it can come back to bite you.

I love Respectful Insolence, but I’ve got to ding Orac for failing to calculate molarity correctly. He is profoundly educated, but I think he’s a surgeon and not a physicist. We all have our weak points! (Thank heaven above I’m not ever in the operating room with the knife!)

In this post, which he may now have edited for correctness (and it seems he has), he makes the following statement:

More importantly, look at the numbers of precipitates found per sample. It ranges from two to 1,821.

O.M.G.! 1,821 particles! Holy crap! That’s horrible! The antivaxers are right that vaccines are hopelessly contaminated!

No. They. Are. Not.

Look at it this way. This is what was found in 20 μl (that’s microliters) of liquid. That’s 0.00002 liters. That means, in a theoretical liter of the vaccine, the most that one would find is 91,050,000 (9.105 x 107) particles! Holy hell! That’s a lot. We should be scared, shouldn’t we? well, no. Let’s go back to our homeopathy knowledge and look at Avogadro’s number. One mole of particles = 6.023 x 1023. So divide 91,050,000 by Avogadro’s number, and you’ll get the molarity of a solution of 91,050,000 particle in a liter, as a 1 M solution would contain 6.023 x 1023 particles. So what’s the concentration:

1.512 x 10-16 M. that’s 0.15 femtomolar (fM) (or 150 altomolar), an incredibly low concentration. And that’s the highest amount the investigators found.

Anybody see the mistake? Let’s start here: Avogadro’s number is a scaling constant for a linear relationship and it has a unit! The units on this number are atoms(or molecules) per mole. It converts a number of atoms or molecules into a number of moles.

‘Moles’ is a convenient person-sized number that is standardized around ‘molecular weight,’ which is a weight unit that arbitrarily says that a single carbon atom has a weight of ’12’ and results in atomic hydrogen having a weight of ‘1.’ That’s atomic mass units (or AMU), which is usually very convenient for calculating relative weights of molecules by adding up all the AMU of their atomic constituents. To use molarity, we usually need a molecular weight in the form of Daltons, or grams/mole. Grams per mole says that it takes this many grams in mass of a substance for that substance to contain a single mole’s worth of molecules (or atoms) where it is then implicit that the number of molecules or atoms is Avogadro’s number.

‘Mole’ is extremely special. It refers to a collection of objects that are atomically identical! If you have a mole of a kind of protein, it means that you have 6.02 x 10^23 number of this kind of identical object. If you make a comparison between two proteins, the same molar number of each with a different molecular weight is a different overall mass. Consider Insulin (5808 g/mole) compared to the 70S Ribosome (2,500,000 g/mole)… one mole of Insulin would weigh 5.8 kg while one mole of 70S Ribosome would weigh 2.5 metric tons!!! If they have roughly the average density of proteins, what would be the volume of 1 mole of 70S ribosome as compared to 1 mole of Insulin? It would be 430 times greater for the Ribosome; 2900 L for 70S Ribosome while Insulin is about 6 L!

Notice something here: an object with a big molecular weight occupies a bigger volume than the same object of a smaller molecular weight… regardless of the fact that they are at the same molarity. Molarity as a number depends strongly on the molecular weight of the substance in question in order to mean anything at all. For the Ribosome, the same molar concentration as for Insulin means a solution containing a much larger amount of solute.

In the post in question on Respectful Insolence, Orac is talking about a paper which observes particulate matter derived from vaccine specimens in an SEM. It is clear from the authorship and publication of the paper that the intent is to find fault in vaccines based upon the contents of materials examined by this probing… from what little I know about the paper, it does not seem to be producing any information that is truly that informative. But, you can’t fault a paper on a point that may not actually be as flawed as an initial interpretation would imply. The paper reports number of particles observed per 20 uL of a solvent. They find as many as 1,821 particles per 20 uL. We are not told for certain what these particles are composed of except that the investigators aren’t sure and shot an overpower EDS at everything and reported even the spurious results. Orac scales up this number to 1L to get 90.1 x 10^7 particles and then divides by Avogadro’s number to find what proportion this is of one mole of these particles, never mind that we don’t know how big the particles are in terms of molecular weight or how dense in volume per mass. He declares it to be a tenth of a femtomole and runs on with how tiny the concentration is. As I initially wrote this, I focused on the gleeful way in which Orac does his deconstruction in large part because it really isn’t a valid thing to laugh at when the deconstruction is not properly done.

Here is how someone of my background approaches the same series of observations. I can see from the micrograph in the blog post that the scale bar is something like 2 mm (2000 microns)… the objects in question are maybe tens to hundreds of microns in size. Let’s make a physicist supposition here and think about it: pulling this out of my ass, I’ll claim these are 1,821 approximately spherical identical particles of sodium chloride, each of 40 microns diameter. That gives a volume of 4/3*Pi*20^3 um^3 or 1.9 x 10^-12 m^3 per particle and 3.5 x 10^-9 m^3 for the whole collection of particles. Now, density usually is given in terms of g/cm^3 or g/mL… there are 100 cm per meter and you must convert three times to cube it, so 3.5 x 10^-9 x 100^3 = 3.5 x 10^-3 cm^3. Wait a minute, we’re now at a volume of 3.5 uL!!! Did you see that? A cubic centimeter is a mL and 0.0035 mL is 3.5 uL, or 17% of the original 20 uL sample volume! What molarity is this? The density of sodium chloride is 2.16 g/mL or 2.16 mg/uL… which is 7.56 mg. That’s 7.56 mg of salt dissolved in 20 uL. The molecular weight of sodium chloride is 58.44 g/mole or 58.44 mg/mmole, which gives .129 mmole. From this .129 mmole in .02 mL is 6.47 mmole/mL.

That’s 6.47 mole/L……. 6.47 M!!!!

Let’s pause for a second. Is that femtomolar?

Orac missed the science here! I initially wrote that he should be apologizing for it, but I’ve revised this so that my respect for his work is more apparent. The volume of these particles and their composition is everything. A single particle with a molecular weight in the gigadaltons or teradaltons range is suddenly a very substantial mass in low particle number. If these particles are as I specified and composed of simple salt, they are at a molarity that is abruptly appreciable. If we make these into tiny balls of Ricin, that’s unquestionably a fatally toxic quantity!

As with all things, dose makes the poison and there’s no Ricin in evidence, but this argument Orac has made about concentration, in this particular case is catastrophically wrong. A femtomole of a big particle that can be dissolved could be a large dose!

I forgive him and I love his blog, but let this be a lesson… you don’t just divide by Avogadro’s number in order to get meaningful concentrations!


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