# How to approximate the pH of a box of water molecules with one proton?

I have a cubic box of length $$L = 12.42\ \mathrm{\overset{\circ}A}$$, containing $$60\ \ce{H2O}$$ molecules and $$1\ \ce{H+}$$ ion. How can I approximate the $$\mathrm{pH}$$ of the system?

Following the basic definition, $$\mathrm{pH}=-\log\left([\ce{H+}]\right)$$, I tried to calculate it like the following two ways:

\begin{align} \mathrm{pH} &= -\log\left(1/60\right) = 1.78 \tag1\\ \mathrm{pH} &= -\log\left( \frac{1/6.022\times 10^{23}}{12.42^3\times 10^{-30}\times 10^3} \right) = 0.0621 \tag2 \end{align}

However, I am not sure whether these approximations are valid or not in this regime.

• Chemistry is not about individual molecules at all. But if you insist, well, then your second way must be right. Feb 22 '16 at 11:21

For a rough, back of the envelope, sanity check sort of calculation, your second approach is more appropriate than the first. (Though check your value for Avogadro's number.) This is something that should jump out at you if you keep track of units in your calculation. The $\ce{[H+]}$ in the equation should be measured in moles of hydrogen ion per liter, so your calculation needs to result in an inserted value that's in moles of hydrogen ion per liter. You can also rule out the first equation by performing a thought experiment: if the size of the box doesn't factor into the equation, does it make sense that the pH wouldn't change if the box size changes? (No matter how much the size changes?)

However, the statement that pH is equal to the negative log of the hydrogen ion concentration is strictly speaking only true for dilute aqueous solutions near standard temperature and pressure. For concentrated acids/bases, solutions which are not primarily water, or for pressure and temperatures well outside standard, things get more contentious. There's a number of different interpretations of what "pH" means, and each have their advantages/disadvantages, depending on the situation.

One of the big considerations here is that "concentration" is not really really the thermodynamically relevant parameter when figuring out how chemical systems behave. Instead, it's the compound activity which is typically the more important quantity to keep track of. As it happens, the thermodynamic activity of a compound is normally proportional to the concentration in dilute solutions, so by convention we set unit activity to be equivalent to 1M solution.* This means the distinction between concentration and activity is normally ignored/glossed over in most introductory chemistry courses, but make no mistake, it's the activity of the compounds which should be used in things like the Henderson–Hasselbalch equation, not the concentration.

*(This normalization by 1 mol/L is the reason we can take the log and not run up against any strange results with dimensional analysis - it's technically a dimensionless value inside the log: the actual concentation, divided by the reference concentration of 1 M.)

The reason this matters for you is that you're dealing with a situation which isn't quite the same as a dilute aqueous solution. Nominally, it sounds like you're close. Your water density is around standard water density, and the amount of hydrogen ions are close to 1 mole per litre. But there are other considerations to take in to account. For example, what temperature and pressure are you using? Extreme values can change the hydrogen ion activity. Also, you didn't mention anything about the presence of a counter-ion. The unbalanced electrical charge might affect the predicted activity as well. The other issue is that just a single hydrogen ion will result in some pretty extensive fluctuations in the system as the system changes state. The "pH" (or hydrogen ion activity) of the system may actually exhibit time dependence, which might be important for the system you're modeling.

If you're truly interested in what the equivalent pH of the solution is, you should look into simulation techniques which attempt to actually gauge the chemical activity of the hydrogen ion under the conditions you're interested in. -- That is, figure out why you want to know the "pH" of the system, and look for more direct measures of that.

• Thank you for the answer. To be more be specific, I have been trying to see the activity of 1 $H_3O^+$/2 $H_3O^+$ ions in presence of oxyanions, $XO_4^{2-}$. The systems have been modelled at NTP having overall density of about 1.0-1.1. Can you please help me with some more insights? Feb 23 '16 at 12:33
• And thank you very much for pointing out the mistake in putting Avogadro's number, which now been corrected! Feb 23 '16 at 12:46
• @SangkhaBorah I've actually never done such simulations myself, so I'm a little short on practical help. What I would highly recommend is an extensive literature search for people which are doing similar things, (looking at hydrogen ion activity in simulations) and attempting to replicate the approaches they used.
– R.M.
Feb 23 '16 at 15:48