 # Off the pH scale?

I recently read in my chemistry textbook that a solution of 6 M H[sub]2[/sub]SO[sub]4[/sub] would have pH around 0 to -1, while the pH of a 3 M KOH solution would be approx. 14.5. How is this possible? My book does no explaining of this and an internet search did help any.

Remember that pH is just a convenient way of expressing the hydrogen ion concentration.

[H+]=10^(-pH)

If the pH=-1, [H+] (i.e. the concentration of hydrogen ions) is anout 10, very very high indeed. For a strong lye, pH=15 means a very low ion concentration of 10^(-15).

Possible since pH is just a mathematical convenience.

I would also note that there are some acids that do not contain hydrogen at all (although, obviously, sulfuric acid is not one of them).

It’s been a while[sup]1[/sup] since I took a chemistry class, but for non-hydrogen-containing or very strong acids, there is probably an “equivalent pH” where it is titrated to a known pH with a standard solution of some base.

[sup]1[/sup][sub]Contrary to rumor, I was not one of Avogadro’s students. I was one of his classmates [/sub]

There’s an aquifer in Northern California, near Redding, called Iron Mountian. It has a pH of -3.5. Needless to say, it is under a Superfund site.

Not concentration, but activity, which “can be regarded as an effective thermodynamic concentration that takes into account deviations from ideal behavior.” (Peter Atkins, 1995, Concepts in Physical Chemistry)

In very dilute solutions, activity is linearly related to concentration, but at higher concentrations the relationship becomes much more complicated. So, activity is always partially a function of concentration (and pressure and temperatue and the concentration of everything else in the solution), but they really aren’t the same thing. Activity is also partially a function of temperature, which is why pH is temperature-dependent and must be adjusted (better ones are self-correcting) so that all readings are standardized to the same temperature/pH scale (25°C, I think).

As an aside, your pH meter is unique in that it is the only piece of lab equipment that can directly measure activity.

When adjusted for temperature, pH can only be between 0 and 14. Here’s why:

By definition,

pH + pOH = pKw

where pKw is the autoprotolysis constant of water. Autoprotolysis is the transfer of protons between like molecules. For water, this is expressed as:

2H[sub]2[/sub]O(l) = H[sub]3[/sub]O[sup]+/sup + OH[sup]-/sup

The equilibrium coefficient for this reaction (Kw) is

Kw = [H[sub]3[/sub]O][OH]/[H[sub]2[/sub]O]

(quantities in brackets refer to activities; quantities in parenthesis refer to concentrations). The activity of water = 1.0.

Kw is a constant. At STP (~1 atm, 25°C), Kw = 10^-14.01.
pKw is defined as:

pKw = -log Kw

which means that at 25°C, pKw = 14.01. So, going back to the first equation,

pH + pOH = 14.01
pH = 14.01 - pOH

Activity cannot be negative, although it can be zero. Therefore, pH must lie between 0 and 14.01 at 25°C. Kw will vary as a function of temperature, since

ln Kw -Gw°/RT

where Gw° is the ideal (standard-state, sort of) Gibbs free energy of the autoprotolytic reaction given above.

But, for pH to be meaningful, it needs to be always adjusted to 25°C so that 7 always means neutral, 0 always means strong acid, and 14 always means strong base. That’s why we do it…

and that’s why “true” pH is always between 0 and 14.

I was under the impression that pH is simply a representation of the hydrogen ion concentration in aqueous solution, and so a non-hydrogen-containing acid (presumably some Lewis acid?) would not have a defined pH.

Also, I fail to see why pH cannot go outside the bounds of 0-14. If I have concentrated 12 M HCl, won’t the pH simply be -1.08? (I have to admit, I’m a bit weak on “activity”, though…)

Also, I don’t believe pH can be as low as -3.5. This would correspond to a hydogen ion concentration of over 3000 M! The most concentrated strong acid I am aware of is sulfuric acid, which can only be concentrated to about 18 M.

(Speaking of the specific example of the aquifer, I would bet anything that the pH is 3.5, not -3.5, which is still pretty low for an aquifer.)

Lastly, I thought pH had to be temperature-corrected because pKw is temperature dependent. (It is only equal to 14.0 at 25.0 degrees C.)

Activity is NOT concentration.

Activity (a) is related to concentration (X) by:

a = yX,

where y (“gamma”) is the activity coefficient, which is itself a very complex function of concentration of the species in question, the ionic strength of the solution, and temperature. In an ideal (Raoult’s Law) solution, y = 1.0 and a = X.

First, note that the unit for X is molecular proportion (mol. prop) and can range from 0.0 to 1.0 (which is to say, 0% to 100%).

Now, since we’re talking about pH, let’s consider the absolute best-case scenario: it’s an ideal solution (ergo, a = X) and we’ve got as much H[sup]+[/sup] as possible–in other words, X = 1.0. Therefore, the activity of H[sup]+[/sup] is also 1.0. What’s the pH?

pH = -log [H[sup]+[/sup]] = -log 1.0

since 0 = log 1, then pH = 0.

So, at the highest possible activity of Hydrogen under ideal conditions, pH = 0.

In real solutions, though y <> 1.0. In fact, except when the chemical specie is very dilute, y isn’t even constant! (Henry’s Law).

Activity does not equal concentration in a solution because either:

1. The presence of other ions in the solution “interfere” with that ion’s ability to interact, or
2. The presence of too many ions in the solution make the solution too “crowded” for the ion to interact as much as it would like.

By analogy:

1. is like going to a party with a girl that you’re “just friends with”. Other girls that you may’ve interacted with stay away because she’s hovering around you.
2. is like trying to find a friend in a crowded airport. You want to interact, but the crowd makes it difficult.

So, X can’t be better than a, which can’t be better than 1.0 and thus pH can’t be lower than 0.

BUT… as temperature changes and Kw changes we can play around with the upper limit of the apparent pH, but for reporting purposes, we always correct it to 25°C where the upper range is stuck at 14.01.

I’m not a chemist… but I am a geochemist!

Pantellerite writes

The OP was asking about H[sub]2[/sub]SO[sub]4[/sub] in particular. For H[sub]2[/sub]SO[sub]4[/sub] my chemistry book lists two stages of ionization:

H[sub]2[/sub]SO[sub]4[/sub] --> H[sup]+[/sup] + HSO[sub]4[/sub][sup]-[/sup] (100% ionized)
HSO[sub]4[/sub][sup]-[/sup] <==> H[sup]+[/sup] + SO[sub]4[/sub][sup]2-[/sup] (K = 1.2E-2)

Doesn’t this mean the concentration of H[sup]+[/sup] can be greater than 1, and therefore the pH can be negative?

Well, 100% is 100%, so X can’t be >1.0 because you can’t have >100% of something.

Unless you’re a coach, politician, or in some bizarro-physiochemical situation that I’m unaware of. (Being in the natural sciences, I’m only aware of natural conditions.)

But are there bizarre conditions in which–although X can be no greater than 1.0–a specie may be “hyper-active” and have a > 1.0? None that I know of, and I beg for a genuine physical chemist to show up and say for sure. But, in nature at least, I’ve never heard of it.

What you’ve brought up is the concept of strong and weak acids (and bases, for that matter). Very strong acids, like HCl, completely dissociate (split into ionic aqueous species) in water. More H, higher activity, lower (to 0) pH. Acids like H[sub]2[/sub]SO[sub]4[/sub] are strong–but not as strong–because they don’t completely dissociate readily: it’s relatively easy to dissociate the first H[sup]+[/sup] ion, but it takes a bit more energy for the next. A weak acid, like acetic (CH[sub]3[/sub]COOH) hardly gives up any H[sub]+[/sub] at all (CH[sub]3[/sub]COOH = CH[sub]3[/sub]COO[sup]-[/sup] + H[sup]+[/sup]). It’s the same story for strong and weak bases.

pantellerite, you need to buy a titrator for anything? We got your pKas, you intrinsic solubility instruments…

Showed up to this one a little late.

It’s been a long time since I’ve done chem lab, and I’m only a poor dumb physicist, but I have to chime in and say that Pantellerite’s arguments are what I was always taught. You can’t have more than 100%, and pH really can’t go outside the range of 1 to 14 (ignoring the “.01” in 14.01).

Sure you can. If something costs \$10 and I have \$15, I have 150% of the cost of the item.

When you say “molecular proportion”, doesn’t that mean proportion of H[sup]+[/sup] relative to molecules of something? If virtually all the H[sub]2[/sub]SO[sub]4[/sub] give up at least one H[sup]+[/sup], and some give up a second H[sup]+[/sup], then the proportion of H[sup]+[/sup] to molecules of the original H[sub]2[/sub]SO[sub]4[/sub] is greater than 1. Looking through my chemistry book, it appears that the proportion is relative to the solvent, so if you had 2 molecules of H[sub]2[/sub]SO[sub]4[/sub], per molecule of H[sub]2[/sub]O, and each H[sub]2[/sub]SO[sub]4[/sub] gave up one H[sup]+[/sup], then the proportion of H[sup]+[/sup] to H[sub]2[/sub]O is 2.

So I’ll ask, what precisely does X (molecular proportion) mean in the context of determining pH? i.e., proportion of what to what?

Poor analogy. The sum of all things in your system/solution equals 100%. If you add 0.5L of water to 1.0L of water, you may have 50% more water, but that doesn’t mean that you now sum the components in the system to 150%. It’s still 100%. Molecular proportion (which is just saying “how many percent of this solution is something” in terms of 0.0 to 1.0 instead of 0% to 100%) has a maximum value of 1.0.

Proportion of the other aqueous ionic species in the system. But remember, X itself doesn’t matter: it’s the activity of X. X for H[+] may be a feble 0.5, but it may be more active than the other ions and have an activity closer to 1.0. Concentration is related to activity, which is really what pH is all about, but activity is also strongly controlled by temperature and the ionic strength of the solution (Debye-Huckel Law).

Lunch time for me! Later…

This still doesn’t seem correct. There will be as many negative ions as positve, so the proportion couldn’t get much above 50%, and nowhere near 100%. Under this definition, a pH of -log[sub]10/sub = 0.3 would be the highest pH possible, not 0.

Let’s try a concrete case, albeit with made up numbers.

9990 molecules HSO[sub]4[/sub][sup]-[/sup]
10 molecules SO[sub]4[/sub][sup]2-[/sup]
1010 H[sup]+[/sup] (maybe this should this be 1010 molecules of H[sub]3[/sub]O[sup]+[/sup]?)
2 molecules H[sub]2[/sub]O (just for fun)

What’s the pH here?

True, IF concentration was the same as activity, which it isn’t. If pH = -log X[sub]H[/sub], where X[sub]H[/sub] was the concentration of H[sup]+[/sup] ions, you’d be right. But instead pH = -log a[sub]H[/sub], and even if X = 0.5, a can still = 1.0.

ZenBeam, you’re obviously interested enough to need to know about the Debye-Huckel theory, which is how we can calculate all the activites for all the ions in an aqueous solution of known composition.

Remember: a = yX. We know concentration (X), we want to know activity (a). So, the D-H theory is gonna tell us how to solve for y.

First, the ionic strength (I) of a solution is:

I = ½ (Sum) m[sub]i[/sub]z[sub]i[/sub][sup]2[/sup]

where m[sub]i[/sub] is the molar concentration of each ion i and z[sub]i[/sub] is the charge of each ion (+1 for H[sup]+[/sup]). (Sum) is the sigma notation for summation that I can’t figure out how to do.

For not terribly concentrated aqueous solutions (I < 0.1),

-log y = (Az[sup]2[/sup]I[sup]0.5[/sup])/(1 + aBI[sup]0.5[/sup])

For each ion we calculate a y, where z is that ion’s charge, I is the ionic strength of the solution, a is the ion’s radius in angstroms (for H[sup]+[/sup], a = 9), and A and B are temperature-dependent constants (at 25°C, A = 0.5085 and B = 0.3281).

For really dilute solutions (I < 5e-3),

-log y = Az[sup]2[/sup]I[sup]0.5[/sup]

For brines (I up to 0.5),

-log y = Az[sup]2[/sup][(I[sup]0.5[/sup]/(1+I[sup]0.5[/sup])) - 0.2I]

The hypothetical solution you described with 2 water molecules would obviously have an ionic strength >0.5, so you’d have to think up something different.

So, ZenBeam, there’s the math: you figure it out! Of course, the thing is when we analyze aqueous solutions, we can not directly analyze the concentration of H[sup]+[/sup] ions for various reasons I won’t get into. So, how do we get H? We can’t! We don’t! Instead, we measure the activity of H… pH!

Sources:

G. Faure, 1991, Principles and Applications of Geochemistry, Prentice Hall.

R. Garrels and C. Christ, 1965, Solutions, Minerals, and Equilibria, Harper & Row.

Thanks, Pantellerite. It’s late and I’ll have to think about this for a bit. When I’ve got an answer for the pH, I’ll post back. Unless I get an imaginary pH. Well, maybe even then.

Been a while since I looked at this.

I’ll agree we learned about activity rather than hydrogen concentration in sophomore physical chemistry. But this doesn’t do one heck of a lot to answer the OP and is clearly outside the intent of the author of the (?high school) chemistry book. I have no doubt it defines pH as I did (that is, marginally incorrectly). I don’t remember questions in my high school physics book talking about wind resistance, pulleys with friction and Coriolis forces, either.

Yes, you can have acids with negative pH. pH is just a mathematical approximation of -log[H3O+]. Remember, though that pH is only useful in aqueous solutions or in somewhat dilute acids. Also,like a lot of math functions, the pH scale begins to break down as you reach the extremes (also, typical pH meters begin to act non-linear at this point). At this point you need to use the Hammett scale, which measures color change of certain indicators. Some flourine based acids (HF, HSO3F) are called superacids, meaning they are more acidic than 100% H2SO4.

[“url=http://chemistry.miningco.com/science/chemistry/library/weekly/aa050100a.htm”]here is a good link explaining the whole thing in much more detail than I could hope to do