# How to calculate the number of protons in a sphere given the volume and the pH?

The spherical radius of a nano-droplet of a solution ($$\mathrm{pH}$$ of which is $$6$$) is given as $$\pu{20nm}$$.

I can find the molar concentration of $$\ce{H+}$$ and subsequently the number of $$\ce{H+}$$ ions, but I am getting a fractional number ($$\approx 2 \times 10^{-2}$$). This is obviously wrong as there should be at least $$1$$ proton in a compartment which has an acidic $$\mathrm{pH}$$. What am I doing wrong?

• It's not wrong - it's just that there would be a 2% chance of presence of a single H3O+ in one of such droplets if you, say, sprayed such solution. Nov 10, 2019 at 21:39
• @Mithoron : But how can the pH of all such droplets be 6 then? Or is it the average pH which is 6? Nov 10, 2019 at 21:44
• 20 nm is so little that pH loses sense, it isn't even a phase, but some nanodroplet. One could say it's average pH if you had mist made of them. Nov 10, 2019 at 22:13
• Chemical equilibriums, similarly as temperature, are statistical thermodynamic properties of large scale systems. They are gradually losing sense for small enough systems. Imagine there is attribute of people with probability 0.01. If there is 10 people and none of them has such an attribute, does it mean it's probability is zero ? Of course not. Nov 11, 2019 at 3:19

As pointed out in the comments, the concept of pH as a thermodynamic property of nanodroplets does not quite make sense unless the nanodroplets are in equilibrium (stable). If they are in equilibrium you can still apply the equation $$\textrm{pH}=-\log_{10}(a_{\ce{H+}})$$ but now you have to be aware that the activity will be influenced by surface (and other) effects and the approximation $$\textrm{pH}=-\log_{10}(c_{\ce{H+}})$$ may not hold. But assume it does. Then as you point out, the average number of hydronium ions per nanodroplet is

\begin{align} n_{\ce{H+}} &=\pu{10^{-6} M}\cdot(\frac{4}{3}\pi\times\pu{20^3 nm^3}\times\pu{10^{-27} {(m/nm)^3}}\times\pu{10^3 L/m^3})\times \pu{6.022\times10^{23} mol^{-1}}\\&=0.020 \end{align}

Which is ok. That just means that on average only 1 out of 50 nanodroplets contains a solvated proton. Note by the way that one nanodroplet also contains about a million water molecules.

This is a real "rabbit hole" of a question, given all that is not specified, which is why I used sleight-of-hand and wrote that I am ignoring surface effects and assuming the particles are at equilibrium. It doesn't address how to prepare nanodroplets with the desired average proton concentration or what equilibrium means (are nanodroplets exchanging matter or are they isolated?). As noted in comments, composition (buffering molecules if any, salts and other ions including hydroxyl), droplet charge etc, are being ignored and you can expect statistical fluctuations in concentrations between different droplets. However if I stick to the assumption of equilibrium then I can say (for what it's worth) that ~2% of droplets have ~pH 4.3, and most of the rest are at neutral pH.

• "... on average only 1 out of 66 nanodroplets contains a solvated proton" at a given time. If the solution is buffered, the pH will still be 6 as a time-average concentration (or activity) of protons. If the solution is not buffered, a pH of 6 is kind of meaningless even for larger volumes.
– Karsten
Nov 11, 2019 at 3:32
• I'd flip that last sentence around. It means that $0.015%$ of the protons have a very low pH and the other $0.9985%$ are just plain water Nov 11, 2019 at 8:04
• @Poutnik I understand the argument, it is not simple. My point is that if you have a thermodynamicaly stable aerosol you can certainly define a temperature, as well as the average concentration of anything in the aerosol. You can have fluctuations between the droplets, of course, but you can (ignoring the strict definition of pH in term of activities) simply express the H+ concentration as pH (as an algebraic manipulation). Nov 11, 2019 at 9:39
• I would rather say pH, similarly as temperature or chemical equilibrium are statistical phenomena of large scale. For small enough scales, all of them do not have sense anymore.For pH, it turns into probability of H3O+ occurence. Nov 11, 2019 at 9:39
• I agree it seems absurd to define a concentration when you only have 1 of something in a volume. Note you have fluctuations within segments in a bulk phase, too. And there is the ergodic principle. The catch is that in the bulk you have material transfer between segments, and surface effects disappear. Nov 11, 2019 at 9:41