August 2002

# Education

## What do you mean pH = -1? Super Acids

### JWF Muwanga-zake, Centre for the Advancement of Science & Mathematics Education (CASME), University of Natal

##### Introduction

In the CASME vacation school (funded by Carnegie), one group of educators came up with a pH < 0 after a titration, which challenged our conceptual understanding of pH. The author was one of the tutors in the vacation school, and was called upon to confirm whether there were indeed pH negative values. Most worrying was the apparent resignation to 'believing' rather than critically interrogating what was printed in textbooks.

The author has since then tried to explore the concept of pH, focusing on the way it is dealt with in South African classrooms. Literature shows that the concept of pH has some problems, especially outside the 0-14 range. Some suggestions on what negative pH values could mean are made.

This discussion, limited to some aspects of pH taught at school, is intended for opening up a debate with educators who would further investigate pH and how best to teach it. It does NOT provide an authoritative understanding of pH.

##### The Meaning of K

A constant of a reaction (K) defines the ratio of the concentrations of the products (e.g. dissociated ions) and the reactants (e.g. un-dissociated molecules).

For example, the dissociation constant K of a molecule HA is given as
K = [H+][A-]/[HA]. Square brackets denote concentration. Apparently this originated from Arrhenius's theory of electrolytic dissociation (among others): 'the molecule of an electrolyte can give rise to two or more electrically charged atoms or ions], and from the Law of Mass Action (by two Norwegians, Gulberg & Waage). If the dissociation happens in a solvent, at saturation, we can call the constant, solubility constant. The dissociation process might be endothermic or exothermic.

Let us look at a common dissociation: the autoprotolysis of pure water
The following equation describes the reaction of water with itself (called autoprotolysis):
H2O + H2O <==> H3O+ + OH¯. The equilibrium constant for this reaction is written as follows: K = [H3O+][OH-]/[H2O][H2O]. However, in pure liquid water, [H2O] is a constant, and together with K produces Kc.
Thus Kc = [H3O+] [OH¯]. This constant Kc, also known as Kw, is called the water autoprotolysis constant or water autoionization constant. It is assumed to be obtained in dilute solutions, and may not strictly apply to concentrated solutions. Many factors such as temperature and the nature of solute would affect this equilibrium. For example, while Kwhas been shown to be 1.011 x 10-14 at 25 °C (generally, a value of 1.00 x 10-14 is used), Kw = 10-12.3 at 100 °C.

The equation above shows that concentrations of H3O+ and OH¯ are in the molar ratio of one-to-one, and so [H3O+] = [OH¯]. Therefore, [H3O+] and [OH¯] = Kw, which is 10¯7 M in pure water at 250C. This leads to important models of acids and bases, not least the pH scale.

##### Acids and bases

According to Park (1998), the word acid comes from the Latin word acere, which means 'sour'. All acids taste sour. Well known are vinegar, sour milk, lemon juice, Aspirin (acetosalicylic acid). The word alkaline could be from the Arabic al-qily, which means, 'to roast in a pan" or "the calcinated ashes of plants'. The word "base" might come from bassus, which is Latin for low. There are numerous definitions of acids and bases, most of which relate with solutions containing water.

Arrhenius
A Swedish chemist, Svante August Arrhenius (1859-1927), defined an acid as a substance, which releases hydrogen ions in solution, and a base as a substance, which forms hydroxyl ions. Wilhelm Ostwald followed up this idea and calculated the dissociation constants Ka, which is a measure of an acid's strength. In 1904, H. Friedenthal recommended that the hydrogen and hydroxyl ion concentration be used to characterize solutions, since the [OH¯] = 10¯14 ÷ [H3O+]. The idea of a pH scale could have germinated then.

Now, it is widely argued that hydrogen ions cannot exist in water because hydrogen ions are actually protons, and so have a high electric field density. Furthermore, bases such as NH3 that do not have the OH- group are excluded from Arrhenius's definition.

Lowry and Brønsted
In 1923, Johannes Nicolaus Brønsted (Denmark) and Thomas Martin Lowry (England) published essentially the same theory about how acids and bases behave. Since they came to their conclusions independently of each other, both names have been used for the theory name.
Cited in Hawkes (1992)
Brønsted: ". . . acids and bases are substances that are capable of splitting off or taking up hydrogen ions, respectively." Or an acid-base reaction consists of the transfer of a proton from an acid to a base. Here is a more recent way to say the same thing:
· An acid is a substance from which a proton can be obtained.
· A base is a substance that can remove a proton from an acid.

T. M. Lowry, in his paper "The Uniqueness of Hydrogen" Chemistry, and Industry 42 (19 January 1923) pp. 43-47, argued:
"It is a remarkable fact that strong acidity is apparently developed only in mixtures and never in pure compounds. Even hydrogen chloride only becomes an acid when mixed with water. This can be explained by the extreme reluctance of a hydrogen nucleus to lead an isolated existence.... The effect of mixing hydrogen chloride with water is probably to provide an acceptor for the hydrogen nucleus so that the ionisation of the acid only involves the transfer of a proton from one octet to another."

Note:
1. The difference between Lowry and Brønsted; - Brønsted used H+, but Lowry, used the H3O+ that is commonly used today.
2. Because there are a variety of solvents in which a transfer of protons can happen, an acid could be taken as one that increases the cationic species (such as hydrogen ions) natural to the solvent. (for enrichment, consider dissolving acetic acid in water, and then in sulphuric acid, and in alcohol). A substance acts as an acid or base in relation to a solvent used. With regard to Lowry - Brønsted, acidity is really a measure of how strong the water molecule is, compared to the base in the acid molecule (H-Base).

It should be emphasised that the water molecule must exert some force to break the bond between the proton and the base in the acid (energy is required for this), and then to bond with that loose proton (energy is probably released from this). It is commonly argued that the balance between these energies would determine whether the acid loses its proton, whether the acid is strong or weak, and whether the dissolution process would be endothermic or exothermic.

##### Protons are not often "donated"

It is unlikely that acids simply 'donate' protons since those protons have to be broken off a bond. In the Lowry-Brønsted acid, the hydrogen ion is bonded to a base. The formation of acids is exothermic, and so breaking acids up requires energy. When a base approaches an acid, it will (if it is strong enough) take the proton off the acid molecule. Some bases are stronger than others in fighting for protons. Therefore, the acid molecule does not "give" or "donate" as stated for example in Brink & Jones (1979; 160), and Toon & Ellis (1973: 475). Bases that have a greater desire for protons take possession of the proton by force, except in some rare cases where complex ions 'repulse' protons.

##### Problems with the Lowry- Brønsted theory

This theory assumes a solvent that easily participates in hydrogen ion transfers such as water, but fails to explain acid base behaviour in aprotic solvents (solvents that do easily participate in transferring hydrogen ions such as benzene). A more general theory, such as the Lewis Theory of Acids and Bases explains how such solvents might work.

##### Lewis Theory of Acids and Bases

You will note that the Arrhenius, as well as the Lowry- Brønsted definitions limit acids - bases reactions to a transfer of hydrogen ions (protons) - hence acid-base reactions are known to be protic reactions. However, there are chemical reactions that act as acid-base reactions, which are not directly protic (protolytic, protonic). According to Gilbert Newton Lewis (1875-1946), the American chemist, acids, and bases are not tied to water or restricted to protons. The crucial things being moved around are charges: Well, it is true that a proton is charge (a positive charge). However, Lewis includes electrons: an acid is an electron-pair donor, and a base is an electron-pair acceptor. (Note that Lowry mentioned a transfer of H+ between octets). This would seem to be similar to a definition of a reducing agent and an oxidising agent respectively. However, the difference is that Lewis's definition applies to an un-shared pair of electrons in the outer energy level. You may also look up the Lux-Flood definition of acids and bases. These definitions are out of the South African Matric syllabus and so will not be explored further.

##### Strong acids and bases

The key point is that strong means 100% ionised. This happens when water molecules are much stronger than the base in the acid. (Strength is not the same as concentration, which is often for acids expressed as the number of moles per unit decimetre). It is assumed that all the acid dissociates, and all its protons are in solution. No acid molecules are left. Examples of strong acids in schools include;

Hydrochloric acid (HCl + H2O ==> H3O+ + Cl-; i.e., [H3O+] = [HCl]),

Sulphuric acid (H2SO4 + 2H2O ==> 2H3O+ + SO4-2; i.e., [H3O+] = 2[H2SO4]), and

Nitric acids (HNO3 + H2O  ==> H3O+ + NO3-; i.e., [H3O+] = [HNO3].

Similarly, there are almost no un-dissociated base molecules in solutions of strong bases. It is important to remember that K is proportionately high in comparison to concentrations, and that we do not talk of equilibrium, in the case of strong acids and bases, especially in their dilute solutions.

##### Weak acids and bases

Many of the acids are weak and partially dissociate to produce H3O+ and their bases or anions (A-). This is because water cannot grab all the protons from such acids. Consequently, a measurement of [H+] does not give the total hydrogen ion concentration because some of the hydrogen remains undissociated. However, if the acid is titrated against a dilute base, the undissociated molecules progressively dissociate and eventually the total hydrogen ion concentration can be found. This is why the "titratable acid" is distinguished from the actual acidity (pH). With strong acids, the titratable acid and acidity are assumed the same.

With weak acids and bases, we can talk of equilibrium because the reactants are not finished. It should also be noted that an equilibrium sign does not apply to irreversible reactions, such as where a gas is liberated and lost to the environment. Second, it is wrong to represent equilibrium by two unequal arrows pointing in opposite directions, as often seen in some textbooks. This mistake, often wrongly applied to show which direction of the equilibrium is favoured, should be avoided. We have to differentiate between the quantities yielded (perhaps as a result of a change in conditions surrounding the equilibrium - i.e. Le Chatelier's principle), and the state of dynamic equilibrium.

At equilibrium, the number of products being formed is exactly equal to the number being broken down to re-form reactants. That is, the RATE of the forward reaction is exactly equal to the rate of the reverse reaction. Therefore, the arrows must be equal to indicate that the forward rate equals the reverse rate. For example, a weak acid (HA) dissociates reversibly in water:

HA + H2O  <---> H3O+  + A-. Rearranging the equation to isolate the hydrogen ion concentration:
[H3O+] = K[HA][H2O]/[A]  ------- >  [H] = K[HA]/[A]

Assuming that [H2O] is in excess and therefore constant; remembering that [OH-] [H3O+] = 10-14  Therefore, log {H+] = log K + log {HA] – log [A-], and -log [H+] = log [A-] - log K - log {HA]

##### Acid-base reactions tend towards weaker acids and bases

Take the example: HCl + H2O <===> H3O+ + Cl-. In general, in acid-base reactions such as this, equilibrium favours the production of the weaker acid (e.g. H3O+ ) and base (e.g. Cl-). Put simply, the acid produced has a stronger base than that in the acid, which lost the proton (i.e. H2O base is stronger than Cl-). We may in fact consider acid-base reactions as a competition in which bases fight for protons.

##### What is pH?

The concept of pH was introduced in 1909 by the Danish chemist Søren Sørensen as a convenient way of expressing acidity. The reasons for the introduction of pH include:
· The slow colour-change tests, and a need for electrical methods of determining acidity or basicity;
· It is easier to write and compare pH than actual concentrations; e.g., "the pH varies from 2 to 13", instead of [H3O+] varies from about 0.01 M to 0.0000000000001 M. Please note that comparisons of pH (or even of acidity) are limited to the use of the same conditions.
· The pH scale based on Lowry- Brønsted model is minimised to 0 to 14 because it assumes the presence of excess water.

The letters pH are an abbreviate for "pondus hydrogenii" (translated as potential hydrogen) meaning hydrogen power as acidity is caused by a predominance of hydrogen ions (H+). Sørensen used to write pH as PH. The Compact Oxford English Dictionary states that the modern notation 'pH' was first adopted in 1920 by W. M. Clark (inventor of the Clark oxygen electrode) for typographical convenience, and claims that the "p" stands for the German word for "power", potenz, so pH is an abbreviation for "power of hydrogen". "p-Functions" have also been adopted for other concentrations. For example, "pCa = 5.0" means a concentration of calcium ions equal to 10-5 M.

From -log {H+] = log [A-] - log K - log [HA] above, substitute -log[H] for pH, and -log K for pK. pH is the familiar measure of acidity or alkalinity, and pK is a measure of dissociation in water (remember we removed [H2O] and [OH-] from that equation - they are part of pK). By definition, pH = -log [H3O+] (or in this case -log [H+]. So, the definition of pH inherently assumes the presence of water in excess, and the factor 10-14 within pK.

A definition that uses the hydrogen ion activity:
In other cases, pH has been defined as pH = -log aH+ where aH+ is the hydrogen ion activity. In solutions that contain other ions and under varied conditions, activity and concentration are not the same. The activity indicates the hydrogen ions that are active, rather than the true concentration; it accounts for the fact that other ions and conditions surrounding the hydrogen ions might shield them and affect their ability to participate in chemical reactions.

The IUPAC definition
IUPAC endorsed a pH scale based on comparison with a standard buffer of known pH using electrochemical measurements. IUPAC restricted the pH range to dilute aqueous solutions of less than 10-1 mol.dm-3: 2<pH<12.

Comparing pure water and pure acids
Pure water
Consider 1000 ml of water with a density of 1.00 g/ml - this 1.00 litre (1000 ml) would weigh 1000 grams. This mass divided by the molecular mass of water (18.0152 g/mol) gives 55.5 moles. The "molarity" of this water would then be 55.5 mol / litre or 55.5 M. It might also be interesting to note that [H3O+] = 10-7 in pure water implies that for every one H3O+, there are 10,000,000 (i.e. 107) un-dissociated water molecules. The amount of H3O+ increases by adding acidic compounds. Adding an alkaline compound increases OH-.

Something that appears to escape critical consideration is not that [H3O+][OH-] = 10-14 at 250C, but that the range of both [H3O+], and [OH-] is between 100 and 10-14 at 250C; both these factors must be considered in our thinking about pH. For example, 1020 x 10-34 = 10-14, but assumes that [H3O+], and [OH-] could be in the range of 1020 to 10-34. In this example, we should ask ourselves, and investigate whether it is possible to have a range of 1020 to 10-34 for [H3O+], and [OH-] at 250C. Until that is established, the limiting condition seems to be [H3O+][OH-] = 10-14 , in combination with a range of 100 and 10-14 for both [H3O+], and [OH-] at 250C. What these limits imply is that [H3O+], and [OH-] cannot go beyond the 100 and 10-14 range at 250C; and so the pH ranges between 0 and 14 at 250C.

Pure H2SO4 is weak

The Lowry- Brønsted model implies that pure H2SO4 is weak because it does not release its protons easily, although it has plenty of protons (hydrogen ions). Adding drops of water might liberate some protons but not all. (i.e. making a solution of water in H2SO4 – note that an acid that is >50% of the solution turns out to be the solvent by definition of a solvent = the greater in proportion in a solution). It seems that plenty of water is needed to enable the release of all protons and make H2SO4 a Lowry- Brønsted acid. This is probably why the Lowry- Brønsted seems to have problems with concentrated acids.

##### What do you mean pH = -1?

A pH = -1 implies that [H3O+] = 10 M or 101 moles/dm3. Indeed, any concentration of H3O+ (i.e. [H3O+], and not of the acid molecules) greater than 1M would translate into a pH that is < 0; i.e. would be negative. A 12M H2SO4 would have a pH of –1.08 and then –1.38, 11M HCl would have a pH of –1.04, etc. However, such pH values might be meaningless in the Lowry- Brønsted model.

What does this argument say about water?

Water acts as a base when an acid is introduced into it, and water has to be stronger than the base in the acid to pull the proton away from the acid, if a substance is to fit the definition of a Lowry- Brønsted acid. For example, the pull that water exerts upon the proton in HCl, is stronger than the bond between the Hydrogen ion (proton) and the Chloride ion (base). What is not clear though is how many water molecules surround or pull the Hydrogen ion.

‘Heat of Dilution of Acids’ tables show that the overall process of acid dilution is generally exothermic, but, apparently, as long as there are enough water molecules to pull those protons. Once the proton has been torn off the acid molecule, it has a very high electrical field intensity (because it is very small) that forces it into water molecules. The exothermic nature of acid dilution implies that excess energy is released, resulting in a higher temperature.

It is suggested that a number of water molecules might be needed to break the proton-base bond. Thus, the capacity of water to tear off and accommodate acidity (H3O+ ions) might be limited to the concentration of acid molecules in water at which acids act as Lowry - Brønsted acids (i.e. at a certain concentration of acid molecules, water cannot pull away any more protons from the acid). Beyond that concentration (let us call it a point of saturation), the extra hydrogen ions held by the acid are not available but are measurable as negative pH. According to the pH scale, that saturation point would occur when [H3O+] = 1M or 100 moles/dm3; that is when pH =0. This means that 1 mole of H3O+ is accommodated by 55.5 moles of water (or 1 molecule of H3O+ requires 55.5 molecules of water). If another mole of a monoprotic acid is introduced, that water would not have capacity to take any more protons. Thus, a negative pH becomes meaningless to the Lowry - Brønsted model, since it creates a false impression of extreme acidity, which is not the case, and since it deceives us into believing that [H3O+] can exceed 100, at 250C. The negative pH can be calculated, and can be determined by titration because the base used for titration introduces more water, as well as base ions, which liberates more H3O+ from the acid.

Hence, contrary to what some textbooks state, the pH scale is probably limited to between 0-14, beyond which it does not measure acidity meaningfully.

Other substances in solution, the nature of solvent, as well as physical conditions such as temperature and pressure, can influence the concentration of ‘free’ hydrogen ions, and therefore pH. These factors are likely to be even more effective in acids that can dissociate or evaporate to form gases. Two solutions with the same amount of hydrogen could have different amounts of hydrogen ions (protons) available, and therefore different pH values under different conditions. Similar hydrogen ion (proton) concentrations would register different pH values in different solvents because the neutral pH is different for each solvent. For example, the concentration of hydrogen ions in pure ethanol is about 1.58 × 10-10 M, so ethanol is neutral at pH 9.8. A solution with a pH of 8 would be considered acidic in ethanol, but basic in water! Think about a solution of water in an acid. Again, we see a limitation to the application of the pH scale based on the Lowry- Brønsted theory.

##### The concept of ‘Super Acids’

The concentration of hydrogen or acid can increase indefinitely since water mixes in all proportions with acids. Besides, with suitable equipment, we can prepare ‘pure’ acids with no water in them – most likely in conditions that are not standard. Acids whose pH < 0 are called super acids (Cotton, Wilkinson, & Gaus (1987: 221). Super acids are used to provide protons, and are believed even to force substances to accept protons. Similarly, there are super bases.

According to Cotton, Wilkinson, & Gaus (1987: 221), super acids are necessarily non aqueous (i.e., have little or no water) since the acidity of any aqueous system is limited by the fact that the strongest acid that can exist in the presence of water is H3O+. In that system, a stronger acid than H3O+ loses its proton to H2O to form more of H3O+. To measure this kind of acidity, one has to go beyond the normal pH scale.

The usual measure used for super acids is the Hammett acidity function:

H0 = pKBH+  - log  [BH+]/[B], where B is the base and BH+ its protonated form. pKBH+ is –log K for the dissociation of BH+. By using bases with very negative pKBH+ values, the H0 scale may be extended to negative values. The H0 scale becomes identical to the pH scale in dilute aqueous solution. On this scale, pure H2SO4 (1012 M) has a H0 value of –12, and for Oleum, about H0 = –15, etc. Take note that the Hammett acidity function clearly avoids water in its equation.

##### Implications for teaching/learning about pH

The problem with all definitions is how each can be demonstrated, and when or even whether they should be taught at all. The other problem with teaching about other scales of acidity is that we continue to look at them from the pH scale. One would need to derive another regime of testers and probably equipment to be able to work outside the pH scale. The limitations and constraints placed on the pH scale ought to be emphasised – that is: the pH scale is limited to the Lowry- Brønsted model; it has a limited range, and; it is applicable at room temperature (alternatively, the scale changes with conditions surrounding the equilibrium).

Use of textbooks

Textbooks are not necessarily wrong, but the way some educators use them tends towards establishing belief systems rather than investigation. Printed matter seemed to be an end in its own right. It is time that teachers critically analyse what is printed, use it to further investigate concepts, and interrogate their own thinking rather than simply believing the print.

A need to investigate
We should develop a culture of consulting widely; textbooks do not always agree and rarely print similar approaches. Some of the claims can be put to the test in a school laboratory, taking care of the exact conditions under which these are tested. Let us not sit back to wait for some ‘authority’ to provide already processed knowledge. In this regard, it would be encouraging if educators attempt to investigate negative pH values.

What should we teach and how?
Teaching approaches are rarely static or rule-bound. Our conceptual understanding naturally takes different conceptual routes. No arguments and suggestions are written in stone. An educator can draw on classroom experiences to map out an approach. For example, traditionally, we tend to teach Arrhenius’s definition first. However, learners seem to easily confuse Arrhenius’s definitions with the more acceptable Brønsted-Lowry definitions. Hawkes (1992) argues

It is inherent in human nature that we accept what we are told first and relinquish or change it with difficult. The Brønsted-Lowry approach should be presented first, because it is simpler.

The following suggestions might add value to the way acids and bases are dealt with:

• Chemical bonding should be revised thoroughly before dealing with acids and bases.·

• The uniqueness of hydrogen bonds, especially in acids (remembering that H3O+ is an acid) ·

• Energy involved in the formation of acids and water· Dissociation constants of water, and some acids·

• The meaning of the pH scale, and why other scales might be necessary·

• Energy involved when an acid is added to water (Heat of Dilution)· The different acid-base models – strengths and weaknesses·

• Relative strengths of bases, especially other bases compared to water·

• At a higher level, the different acid-base models ought to be investigated and criticised

##### Conclusion

It seems clear that we need to renegotiate the meaning in the various definitions and scales of acidity, and to develop ways of teaching them. Educators have to critically debate issues that seem to be taken for granted and form networks through which they can share knowledge, especially drawing upon their experiences in class, and be ready to provide logical answers to their learners when they end up with funny values such as negative pH values.

It may be necessary to introduce the concept of ‘Super Acids’ to learners since they are exposed to acids whose molarity is greater than 1 at school. For the moment, it seems, every acid with a pH value of less than 0 should be included among ‘Super Acids’.

Contact JWF Muwanga-Zake by e-mail at E-mail: jmuwanga@lycos.com

References and Further Information:

Brink, B. P. & Jones, R. C. 1979. Physical Science 10. Juta & Co, Ltd. Cape Town.

Cotton, F. A., Wilkinson, G., & Gaus, P. L. 1987. Basic Inorganic Chemistry. John Wiley & Sons. New York.

CRC Handbook of Chemistry and Physics. 1st Student Edition. 1987

Ellis, H. Eds.1984. Revised Nuffield Advanced Science Book of Data. Longman Group (FE) Ltd. Hong Kong.

Hawkes, S. J., 1992. Arrhenius Confuses Students. Journal of Chemical Education, Vol. 69, No. 7; July 1992; pages 542-543.

IUPAC Online [Available] www.iupac.org [15th July 2002]
Park, J. L.,
1998. Acids and bases. Online. [Available] http://dbhs.wvusd.k12.ca.us/AcidBase/Acid-Base    [7th July 2002]
Toon, E. R. & Ellis, G. L.
1973. Foundations of Chemistry. Holt, Rinchart & Winston, Inc. New York.

·                 Bases feel slippery, sometimes people say soapy. This is because they dissolve the fatty acids and oils from your skin and this cuts down on the friction between your fingers as you rub them together. In essence, the base is making soap out of you. Yes, bases are involved in the production of soap! In the early years of soap making, the soaps were very harsh on the skin and clothes due to the high base content. Even today, people with very sensitive skin must sometimes use a nonsoap-based product for bathing.