Off the pH scale?

[mongrel_8]

I recently read in my chemistry textbook that a solution of 6 M H2SO4 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.

[Dr_Paprika]

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.

[Akatsukami]

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 while1 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.

1Contrary to rumor, I was not one of Avogadro's students. I was one of his classmates :)

[Great Dave]

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.

[Yossarian]

Originally posted by Dr_Paprika
Remember that pH is just a convenient way of expressing the hydrogen ion concentration.


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.

[Yossarian]

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:

2H2O(l) = H3O+(aq) + OH-(aq)

The equilibrium coefficient for this reaction (Kw) is

Kw = [H3O][OH]/[H2O]

(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.

[robby]

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.)

[Yossarian]

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+ as possible--in other words, X = 1.0. Therefore, the activity of H+ is also 1.0. What's the pH?

pH = -log [H+] = -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!

[ZenBeam]

Pantellerite writes
we've got as much H+ as possible--in other words, X = 1.0.

The OP was asking about H2SO4 in particular. For H2SO4 my chemistry book lists two stages of ionization:

H2SO4 --> H+ + HSO4- (100% ionized)
HSO4- <==> H+ + SO42- (K = 1.2E-2)

Doesn't this mean the concentration of H+ can be greater than 1, and therefore the pH can be negative?

[Yossarian]

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 H2SO4 are strong--but not as strong--because they don't completely dissociate readily: it's relatively easy to dissociate the first H+ ion, but it takes a bit more energy for the next. A weak acid, like acetic (CH3COOH) hardly gives up any H+ at all (CH3COOH = CH3COO- + H+). It's the same story for strong and weak bases.

[erislover]

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.

[CalMeacham]

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).

[ZenBeam]

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

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

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%).

When you say "molecular proportion", doesn't that mean proportion of H+ relative to molecules of something? If virtually all the H2SO4 give up at least one H+, and some give up a second H+, then the proportion of H+ to molecules of the original H2SO4 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 H2SO4, per molecule of H2O, and each H2SO4 gave up one H+, then the proportion of H+ to H2O 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?

[Yossarian]

Originally posted by ZenBeam
Well, 100% is 100%, so X can't be >1.0 because you can't have >100% of something.

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


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.


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%).

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

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...

[ZenBeam]

Proportion of the other aqueous ionic species in the system.

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 -log10(0.5) = 0.3 would be the highest pH possible, not 0.

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

9990 molecules HSO4-
10 molecules SO42-
1010 H+ (maybe this should this be 1010 molecules of H3O+?)
2 molecules H2O (just for fun)

What's the pH here?

[Yossarian]

Originally posted by ZenBeam
Proportion of the other aqueous ionic species in the system.

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 -log10(0.5) = 0.3 would be the highest pH possible, not 0.


True, IF concentration was the same as activity, which it isn't. If pH = -log XH, where XH was the concentration of H+ ions, you'd be right. But instead pH = -log aH, 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) mizi2

where mi is the molar concentration of each ion i and zi is the charge of each ion (+1 for H+). (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 = (Az2I0.5)/(1 + aBI0.5)

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+, 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 = Az2I0.5

For brines (I up to 0.5),

-log y = Az2[(I0.5/(1+I0.5)) - 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+ 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.

[ZenBeam]

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.

[Dr_Paprika]

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.

[jk1245]

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[/url] is a good link explaining the whole thing in much more detail than I could hope to do

[jk1245]

Dammit, my links never work.

http://chemistry.miningco.com/science/chemistry/library/weekly/aa050100a.htm


just cut and paste.

[ZenBeam]

jk1245, what is [H3O+]? I know it's the "concentration", but what precisely is it? An illustrative example would help, (maybe the case I posted two of my messages ago).

[Yossarian]

Great link! Thanks, jk1245. It's the answer I was hoping someone would post way back in this thread when I said...

Originally posted by Pantellerite

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.

So, I guess the answer to the OP is: "not normally, but follow this link!" Activity can be >1.0. jk1245's point that 0-14 pH is only valid for dilute solutions is the important distinction (note that the Debye-Huckel theory equations only work for I <0.5).

ZenBeam, sometimes pH is defined as -log [H3O+]:

H2O = H+ + OH-

can also be expressed as:

2 H2O = H3O+ + OH-

I don't think that there is any real difference between the "hydrogen ion" and the "hydronium ion"--just that it's a matter of convention.

[ZenBeam]

I'm not asking what is H3O+, I'm asking what exactly is [H3O+], i.e. the concentration, i.e. the number, including an example.

[jk1245]

ZenBeam,

[X] is just a short hand way to write "the concentration of X"

For example,

[NaCl] = 0.1M
[alcohol] = 20%

This is just a habit I picked up in undergrad school and never really let go of. It's not that useful since "100mM NaCl" also implies the same thing.

you could also (in informal communication at least) use it without a number.

".. in order to increase the reaction rate, we need to lower [PO4]..."

Again, I wouldn't submit a paper to Nature like that, but for notebooks and such it's fine.

[ZenBeam]

thanks jk1245, but I know it's the concentration. My question is, how exactly is that defined.

My reason for trying to pin this down precisely is that Pantellerite tells me the concentration can't be larger than 1. My chemistry book, on the other hand, tells me concentration is given by the molarity, M, which is "the number of moles of solute per exactly one liter of solution." This could easily be greater than 1. The first example even finds the molarity of 18.23 grams of HCl in 0.250 liters of solution to be 2.00 M.

Is my chemistry book's definition correct? Since there are 55.6 moles of water in a liter, why is it so hard to imagine a concentration of HCl or H2SO4 substantially greater than 1? Consequently, why should anyone even expect a hard boundary at 0 pH?

So is this the answer (finally. You still here, mongrel_8?) to the OP?: Concentration is measured in moles of H+ (or H3O+) per liter of solution, and there is no reason you can't have more than 1 mole of H+, or fewer than 10-14 moles of H+, in a solution.

By the way, my book talks a little about Debye-Huckel theory, but in the context of salt. It sounds like this would complicate matters for high concentration acids or bases, but there's no reason this would cause a hard boundary at 0 and 14.

[Yossarian]

Originally posted by ZenBeam
My reason for trying to pin this down precisely is that Pantellerite tells me the concentration can't be larger than 1. My chemistry book, on the other hand, tells me concentration is given by the molarity, M, which is "the number of moles of solute per exactly one liter of solution." This could easily be greater than 1. The first example even finds the molarity of 18.23 grams of HCl in 0.250 liters of solution to be 2.00 M.
[/B]

When we're dealing with activity (which is what's really important here), and trying to determine it from concentration, we recast concentration in terms of 100%, or 1.0. That's what I'm talking about here. Yes, you could have a solution with a concentration of 2 mol/L of something. So, there's 2 mol/L of that something and 2 mol/L TOTAL in the system, which means that X = 1.0, or 100% of the total stuff in the system is that something. So, whether the total amount of substance in a system is 0.0000001 moles or 1000 moles, it's always just 100% of what it is.

We do that to make concentration an intensive parameter (ie., a parameter independent of the size of the system) instead of an extensive parameter.

[ZenBeam]

Way back on the 22nd, you wrote:

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+ as possible--in other words, X = 1.0. Therefore, the activity of H+ is also 1.0. What's the pH?

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

since 0 = log 1, then pH = 0.

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

My chemistry book also has pH = -log [H+]. But my book uses the notation [H+] to mean moles per liter of H+ ions. Now, if you are using [H+] to mean X, i.e. proportion of H+ to solution, then your equation disagrees with my book, both notationally and numerically. Neglecting the difference in densities between water and the solution, you essentially have

pH = -log(X) = -log([H+] / 55.6)

where now [H+] is in moles/liter, since there are 55.6 moles of solution in 1 liter. You should be able to easily verify this, since if you 1 mole/liter of H+ in 55.6 moles (= 1 liter) of solution, X = 1/55.6.

If, in your quoted lines above, you are using the notation [H+] to mean moles per liter of H+, then 1) you are switching definitions without telling us and 2) your conclusion is wrong because [H+] = 1 is not the largest that [H+] can get.

[mongrel_8]

I just wanted to say that yes I am still here and following closely what is being said, even though I am having a bit of trouble comprehending it all. I still have yet to see an answer other than that you can get pH values above 14 or below 1 at temperatures other than 25 C. Correct me if I am wrong.

[Yossarian]

Alright... once more...

The sum of the proportion of all of the elements (in this case, ions) in any system is 100%, which is to say 1.0. A system may consist of H, OH, and SO4 (as an example). No matter what the actual amount of these components are in terms of grams, moles, or whatever, their proportions sum to 1.0 (100%): if the system is half H, then (H) = 0.5, if the system is one-eighth OH, then (OH) = 0.125, and since everything else (not including the solvent) is everything else, (SO4) must equal 0.375.

When we want to know pH, we need to know activity. Once again, activity is related to concentration by:

a = yX,

where the units for X are molecular proportion, not moles per liter, not milligrams per kilogram, not drams per ounce--these units and many like them can be >1, but molecular proportion can NOT.

In ideal conditions, y = 1.0 and therefore a = X. Therefore, in ideal conditions, the maximum value for a = 1.0. And, thus, the minimum possible pH is 0.

In real conditions, y <> 0.0. In dilute systems (like natural waters), Henry's Law is obeyed and y is a constant (which can be calculated using Debye-Huckel), activity and concentration are linearly related, and the lower constraint for pH is still 0, but is more realistically (like you calculated way back when) more like 0.3.

In some real conditions, like the extreme cases that jk1245 linked us to, a > 1.0 and "negative pH" happens... but X is still in units of molecular proportion. But pH, like most concepts in physical chemistry, are rooted in ideal conditions, under which pH always has a range of 0 to 14. Which is why, like the link taught us, they cease using the pH scale in those situations.

[ZenBeam]

I still have yet to see an answer other than that you can get pH values above 14 or below 1 at temperatures other than 25 C. Correct me if I am wrong.

Well, you're half wrong. My previous answer was that, even at 25 C, and for ideal conditions (y = 1.0 in pantellerite's post),

pH = -log[H+]

where [H+] is the concentration given in moles / liter. Since 55.6 moles = 1 liter of water, you might expect a maximum for [H+] of about 55.6, for a minimum pH of about -1.75, neglecting differences in density. For the cases in your OP, this agrees with what you quoted. For H2SO4, pH = -log(6) = -0.78. For 3 M KOH, pOH = -log(3) = -0.48, so pH = 14 - pOH = 14.48, also in agreement.

Now, this is all according to my college chemistry book (surprisingly enough, titled College Chemistry. I don't have it here, so I can't give you the authors), and it all hinges on how the concentration [H+] is defined. pantellerite is using seemingly the same equation for his ideal case

pH = -log[H+]

but he has the concentration [H+] given as "molecular proportion, not moles per liter". This leads to roughly a factor of 55.6 difference in what those two numbers are, which leads to a log(55.6) = 1.75 difference in the pH quoted using his formula and mine. Under pantellerite's formula, pH is limited to between 0 and 14, except in exceptional cases. Under my formula, the minimum would be more like -1.75.

I don't believe this is a difference in semantics, or that two different definitions are actually in use. I believe one of these two definitions for concentration is incorrect in the context of pH. And maybe I'm the one who's wrong, but I'll have to see a link somewhere stating that concentration used is supposed to be proportion, not molarity. It seems clear from reading my book that they mean molarity, so I'm not just going to take someone's word for it.

And just for laughs, jk1245 is no help. He writes

[X] is just a short hand way to write "the concentration of X"

For example,

[NaCl] = 0.1M
[alcohol] = 20%


The first definition agrees with me, the second with pantellerite. :rolleyes:

Anyway, it's late. I'm gone.

[ZenBeam]

OK, I did some more searching. A site called Acids, Bases and pH (http://web.jjay.cuny.edu/~acarpi/NSC/7-ph.htm) is a site which says

In a sample of pure water, the concentration of hydronium ions is equal to 1 x 10-7 moles per liter (0.0000001 moles per liter).

Since pure water has a pH of 7, this site agrees with me, and contradicts Pantellerite. The concentration of hydronium ions in terms of molecular proportion would be 1.8 x 10-9. so -log(molecular proportion) for pure water gives about 8.75.

Another site, called Understanding pH (http://www.acid-base.com/ph.html), says it more clearly (bolding mine).

The pH is the negative logarithm of the [H+]. Logarithm has to be to the base ten and concentration must be measured as activity in moles per liter.

There's a neat little applet which lets you select pH or [H+], and see how the other changes.

[jk1245]

And just for laughs, jk1245 is no help. He writes

[X] is just a short hand way to write "the concentration of X"

For example,

[NaCl] = 0.1M
[alcohol] = 20%


The first definition agrees with me, the second with pantellerite. :rolleyes:



Hey! Last time I try to help. Lousy ingrates :)

I think both definitions are "correct" for most cases. That is, the differance is insignificant. From what I recall, the "true" definition is moles/L though.

Ultimately though, you may want to take my suggestions with a grain of NaCl; I'm a biochemist so I only work at physiological pH (6-8). Makes it a LOT easier. :D

[Sysop]

I'd just like to say: "You guys have WAY too much time on your hands!" :)

[jk1245]

Originally posted by Sysop
I'd just like to say: "You guys have WAY too much time on your hands!" :)


Yeah, we could be dabating which "Star trek" episode is better. :D

[ZenBeam]

Yeah, we could be dabating which "Star trek" episode is better.

But we don't, because everyone in this thread agrees it's the one with the space hippies, where when they finally get to their paradise planet, all the plants are full of acid! :eek:

[Yossarian]

Originally posted by ZenBeam
Yeah, we could be dabating which "Star trek" episode is better.

But we don't, because everyone in this thread agrees it's the one with the space hippies, where when they finally get to their paradise planet, all the plants are full of acid! :eek:

Hear, hear!

Okay, ZenBeam you linked me your cite. Here's mine... unfortunately, being net-un-savy and bookish, mine comes from my handy desk reference Concepts in Physical Chemistry by Peter Atkins (1995, WH Freeman and company). If you grab a copy (or hell, I'll FAX you a copy of the page!), it's on page 6.

Bearing in mind that pH = -log aH,


Thermodynamic expressions stated in terms of activities are exact, but to make them of practical value it is normally necessary to relate them to measurable properties, such as the molar concentration or the molality of the species. To this end, activites are expressed in terms of mole fractions, X, by writing a = yX where y is called the activity coefficient.

Emphasis, obviously, mine. Why mole fractions (0.0 to 1.0, another way of saying mole proportion)? Because...


...to ensure that the activity is dimensionless.


pH units are dimensionless; activity is dimensionless; concentration is not dimensionless. If you wanna see the proof that activity is dimensionless, just say so.

But there's my cite. While I was researching this (instead of working, I might add), I found yet another definition of pH. I'll hold back... for now.

New sig line idea: "Honk if you passed P-chem!"

[LazarusLong42]

Before I start, I want to say that my copy of Atkins' Physical Chemistry is five years older than Pantellerite's (he has the fifth edition; I have the fourth edition sitting here).

On page 227, pH is defined as:

pH = -log a(H3O+)

a in this case is defined as the activity of hydronium (or protons) in the solution of interest.

This activity is defined as:

a = ym/m0

where m0 is a hypothetical standard state of molality 1 mol/kg. (In these equations, molality rather than molarity is used. Molality measures moles of solute per kilogram of solvent; Molarity measures moles of solute per liter of solution.)

So, while Pantellerite is correct that, basically, mole fractions are to be used in this equation (because m/m0 will be a dimensionless "fraction"), he is incorrect in that this mole fraction will have a size essentially equal to the concentration of the solution.

Thus, for a 2.0 molal solution of HCl, m/m0 = 2.0 (dimensionless)

Now, onto the question of negative pH.

Before we move on, don't forget that y (really a gamma) in these equations. That's the activity coefficient. For very dilute solutions, it can be calculated from the Debye-Hückel Limiting Law:

y = 10^(-|z-z+|AI0.5)

where

I = 0.5(m+z++m-z-)

If this law continued to hold true for solutions to 1 molal (it doesn't), the activity coefficient would be seen to drop tremendously, to about 0.3 for 1 molal HCl (or similar solutes with two singly-charged ions).

However, this law only holds up to, at best, about 0.1 molal. It's shaky even there. There's a refinement of it that holds around 0.1 molal and gets shaky as you approach 1 molal. According to my text, no current theory is reliable at or above 1 mol/kg.

That leaves us is a bit of a bind, theoretically speaking. At 1 molal, we can be pretty sure the pH will not be 0 yet, because y will be (extrapolating from experimental results in Atkins) between 0.4 and 0.7 for HCl (lower for solutions with more ions or higher charges). For this, then,

pH = -log ym(H+) = -log (~-0.5)(1) = ~0.3

(to approximate)

As the molality increases, the activity coefficient will decrease, causing ym/m to move ever more slowly toward 1 as m increases above 1. (For the pH to be less than zero, the term ym/m must be greater than 1.)

In other words, I can't give you a theoretical answer. The only answer to be had lies in experimental evidence, or newer theory that I don't have here--and I think that the latter is unlikely, given that the Debye-Hückel law is something around 100 years old. But I can give you some answers:

1. The pH of a 1 molar solution of any strong acid is alomst guaranteed to be higher than 0. In fact, from my experience with actually putting the pH meter in 1 M acids, generally you're looking at 1-1.5 for a pH;

2. Accurate determination of pH does require use of dimensionless numbers, but not in the way Pantellerite was indicating;

3. There's no such thing as an ideal case, but because most students of general chemistry haven't had all that much math, they just teach the ideal, simplified case and leave it at that. (It's even worse in Chemical Engineering, where you're told "just assume it's the ideal case. That's close enough.);

4. Doing any science on the boards requires the use of way too many vB tags.

5. I really should haul my ass to work.

Thanks for listening :)

LL <--B.S. Chemistry, B.S. Chemical Engineering, M.S. Neuroscience

[LazarusLong42]

My bad. There's a capital A in my third equation down; it's 0.509 kg/mol at 25 °C. Sorry.

LL

[Yossarian]

So, where the fuck have you been???

Thanks for the clarification!

(And thank you everybody else in this thread for offering me the rare opportunity to "pad" my meager post count!)

[LazarusLong42]

Originally posted by Pantellerite
So, where the fuck have you been???

Watching from a distance as this nearly turned into a Great Debate. I felt I needed to actually look at my old P-Chem book because, well, I was curious as to the answer myself :)

Thanks for the clarification!

No problem!

LL

[robby]

Outstanding explanation, LL!

Looking around work, I came across the 6th edition of Atkins' Physical Chemistry. Unfortunately, like all p-chem books, it makes my eyes glaze over...

Now, what about these guys below, and their pH of -3.6 (at the Iron Mountain mine water)?

http://chemistry.miningco.com/science/chemistry/library/weekly/aa050100a.htm

Are they simply using a different scale?

(BTW, apologies to BigDaddyD for questioning the report of such a low pH earlier in this thread. It didn't seem possible to me, and frankly, still doesn't.) :)

[LazarusLong42]

Originally posted by robby
Now, what about these guys below, and their pH of -3.6 (at the Iron Mountain mine water)?

So, obviously, I should read all the links given in the entire thread before posting, even though what I posted wasn't untrue, just incomplete. :)

A word on accuracy of information in textbooks: My fourth edition Atkins was published in 1990. Likely, nothing reviewed in the literature after 1980, was included in the textbook, which means nothing discovered and published after 1975-ish was included... which includes this Pitzer theory (according to this 1999 paper about Iron Mountain water (http://pubs.acs.org/hotartcl/est/2000/research/es990646v%5Frev.html), the Pitzer theory was published in 1977.

(I mean, hell: that paper was published in 1999. The water samples in question were collected in September of 1990. You do the math :))

I'd bet the Pitzer theories aren't in the fifth edition Atkins; they might be in the sixth edition, depending on whether they're too complicated for P-Chem students. (Pitzer is listed in the references as having published a thermodynamics text in 1995 which, I bet, has this information in it.)

Anyway...

Look at Table 1 from that paper (choose the full text in HTML link, and scroll down to near the bottom). You'll see calculated activity coefficients ranging from near 1 up into the thousands. Obviously, activity coefficients that high would allow for negative pH--down to about -4 in this case.

I'm actually impressed with the precision of their calibration curves, as well, which tells me that this is actually real, and not somethign that's been fudged out of the data.

So, the answer to the OP is:

Yes, you can have a negative pH (and, presumably, a pH higher than 14). It's due to a very high activity coefficient caused by (AFAICT) intermolecular forces of some sort. However, these extremely low and high pH's are not the result of simply having 10^(-pH) M hydrogen ion concentration. Note especially that the solutions from the Iron Mountain mines were also extremely high in iron and other ions, which increases the ionic strength, which increases the activity coefficient, which decreases the pH.

Thanks for pointing out that link, robby.

LL