A basic approach to body pH

The physical distance between Europe and North America is perhaps a few thousand kilometers but from a acid base point of view, we are talking light years. It is sad that acid-base (which should be relatively easily understood, now that we have modern computers to do the dirty work for us) is plagued by unnecessary argument. Particularly vituperative comments have come from Severinghaus and Siggaard-Andersen, criticising the work of Stewart and the "strong ion difference" concept. We believe that although Siggaard-Andersen has made a massive contribution to the understanding of acid-base, Stewart has provided a sound mathematical approach which is of great use in understanding acid-base. This article briefly examines Peter Stewart's approach to acid-base. It's best read carefully, but you can jump straight to the calculator, a Java applet that implements his model!

Basis

This web page makes few assumptions about your knowledge. It should be easily understood by anyone with a high-school education and a bit of a background in clinical medicine and physiology. We try to keep things simple - if you feel that our tone is patronising, then you are probably too advanced to be reading such simple stuff! Note that a bit of spadework is still required to work through the maths - pen and paper remain a wonderful medium for this! Because solutions for the more complex equations require the use of computers, we use a tiny little bit of "computerspeak" - as little as we can get away with! We do however feel obliged to use a star (*) to indicate multiplication and a slash (/) for division. This page does not assume that you can speak a computer language.

Remember from physiology that if you have two substances A and B reversibly combining to form substance C, that when the reaction is at equilibrium the equation governing the equilibrium is:

[A] * [B] = K * [C]

where K is the rate constant for the reaction.

Throughout the body pH is vitally important. Anyone who has a blood pH of below 7.0 is likely to die soon if this isn't corrected. Likewise, a pH of over 7.6 is frequently associated with serious illness and death. Remember that pH is the log of the reciprocal of the hydrogen ion concentration. Why use such a convoluted measure, and not just talk about the concentration of hydrogen ions? The answer is mainly historical but also perhaps reflects the ongoing inability of humans to cope with concepts that cannot be counted on fingers (with perhaps a little help from the toes)! Rather talk about pH 7 than 1 * 10-7 mol/litre, or (god forbid!) 100 nanomol/l of hydrogen ion.

Unfortunately, such terminology has only served to complicate things, especially when we start plotting how pH changes when we change another ion concentration - we have to continually remember that we are dealing with non-linear axes on any graph we might draw. We will try and clear up some of the confusion. It goes without saying that in any solution mass is conserved (the amount of each component remains constant unless some of that substance is added or removed, either physically or by participating in a chemical reaction), and that any aqueous solution is electrically neutral (the number of positive ions equals the number of negative ones). But let's review a few other fundamental definitions:

What is an acidic solution?

We define an acidic solution as one that has a higher concentration of hydrogen ions than of hydroxyl ions. A basic solution is the reverse of this, and a solution is acid-base neutral in the rare circumstance where the concentrations are exactly the same! Simplistically, we then say that an acid is something that makes a solution more acidic, while a base does the opposite! Note that already the Scandinavian hackles rise, as they fashionably follow Bronsted with his definition of an acid as a "proton donor" and so on - but we have done nothing wicked - we simply use a convenient definition that will have great utility in the clinical arena. We are not pooh-poohing Bronsted, we are simply applying a useful label!

What is a strong electrolyte?

Another convenience is to categorise substances that dissociate when put into water (electrolytes) into two groups. If they dissociate almost totally in the pH range we are interested in, we call them strong electrolytes - if only partially dissociated, they are weak. Note that we will also refer to ion concentrations in units of moles of charge rather than moles of substance, as we are after all more interested in the ionic (charge) properties of the substance than actual concentrations. For example, if each ion has a charge of +2 (say Calcium) then there are twice as many moles of charge as there are moles of calcium.

A convenient shorthand is use of the term normal, for example if we have 1 mole of Sodium Sulphate in a litre of water, then the solution is 2 normal in the sodium ion (there are two mol of Na+ per litre), as well as being 2 normal in sulphate (there is only one mole of SO4-- but it has a charge of 2-). Also note that technically we should be talking about "effective concentration" (or "activity") of an ion rather than concentration, as the two may differ. But for most ions in physiological solutions, the difference is not significant.

Here is a list of common strong ions (in biological solutions):

• Na+, K+, Ca++, Mg++
• Cl-, SO4--
• lactate-
Weak electrolytes (at physiological pH) include the carbon dioxide / bicarbonate system, and reactions involving the proteins found in some body fluids.

What body compartments are there?

In all biology, approximations are necessary. Although each of the perhaps 1014 cells in the body is unique, we can define groups of cells and treat them as single entities. Likewise, we have identifiable subsets of body fluids that we can define and analyse. Important components are

• Intracellular fluid
• Interstitial (tissue) fluids including lymph but excluding blood plasma
• Blood plasma
• Gastric secretions
• Pancreatic secretions

..but of course there is also..

• Bile
• Saliva
• Fluid within the small bowel
• fluid within the large bowel
• Cerebrospinal fluid
• Synovial fluid (and fluid within bursas)
• The aqueous humour
• .. and so on!
We will concentrate on the first five 'compartments'.

1. A simple case - Pure Water

Water is weird. This weirdness is the basis of life. Among its most peculiar characteristics are:

• Strongly ionic substances dissociate when placed in water
• Water itself dissociates, but only a little
• Water contains a lot of water!

The last statement deserves a bit of explanation. Because water molecules are small (molecular weight of just 16+1+1 = 18) the molar concentration of water is enormous - about 55.3 mol/litre at 37 degrees C. There are a billion times fewer hydrogen ions than there are water molecules in a glass of water. Remember that even the hydrogen ion is a convenient fiction - it's just a simple symbol (H+) for a statistically very complex association between a proton and a whole lot of surrounding water molecules that the proton finds very attractive. The life of the common proton is far more complex than even the most Byzantine modern TV soap opera!

Water dissociates as follows:

H2O <<<====> H+ + OH-

This is a very rapid reaction indeed, and we will always assume that equilibrium is reached instantaneously in biological solutions.

At equilibrium, the following equation must hold:

[H+] * [OH-] = KW * [H2O]

As we've already implied by saying that water doesn't dissociate very much, the dissociation constant KW is rather small, in fact, about 4.3 * 10-16Eq/l at 37o Celsius. This is really tiny. Also note that KW varies rather remarkably with temperature - for example at 25 C it is about 1.8 * 10-16Eq/l ! If you're smart, you'll immediately realise that the pH of a glass of water changes dramatically as its temperature changes! Further contemplation will also assure you that we can assume a new constant

KW' = KW * [H2O]
because the tiny changes in water concentration will be totally insignificant compared with the total amount of water present.
KW' is just a convenient label for the product of the two constants, KW and the molar concentration of water.

As an aside, you can approximate the value of KW' by:       KW' = 8.754 * 10-10 * e(-1.01*106) / T2)
where temperature T is expressed in degrees Kelvin

Okay, so how do we find the pH of pure water?

From the above we know that

[H+] * [OH-] = KW'
But it is clear that even if we know KW', we still need to find one of the other variables, the hydroxyl ion concentration, if we are to determine the hydrogen ion concentration. Fortunately, in pure water the only ions present are hydrogen ion and hydroxyl ion, so if the water is to be
electrically neutral then:
[H+] - [OH-] = 0
We now have two simultaneous equations, and it is easy to see that [H+] = [OH-], so [H+] is the square root of KW'. We can also see the practical significance of our definition of an acid-base neutral solution - if the hydrogen and hydroxyl ion concentrations are the same, then
[H+] * [H+] = KW'
which is another way of saying that a solution is acid-base neutral if the hydrogen ion concentration is equal to the square root of the KW'. The definitions of acid (acidic) and alkaline (basic) solutions follow accordingly:
A solution is acidic if [H+] > ROOT (KW')

A solution is basic if [H+] < ROOT (KW')

where "ROOT" is just our way of saying that we are taking a square root.

2. Strong ions in Water

Take some water. Add strong electrolytes, such as NaOH and HCl, which we know will almost completely dissociate. We now have a heady mix of water, Na+, Cl-, H+ and OH- ions. What will happen to the hydrogen ion concentration in this mix?

We already know that water dissociation constrains us to:

[H+] * [OH-] = KW'       .. Equation #0
and we can readily deduce that electrical neutrality ensures that:
[H+] - [OH-] + [Na+] - [Cl-] = 0       .. Equation #1
We can therefore substitute KW'/[H+] for [OH-] in Equation #1, and determine [H+]. We get an equation:
[H+] - KW'/[H+] + [Na+] - [Cl-] = 0
which we can readily tweak by multiplying throughout by [H+]:
[H+]2 + [H+] ( [Na+] - [Cl-] ) - KW' = 0
At last! A use for quadratic equations!
The above is a standard quadratic equation of the form:
a*x2 + b*x + c = 0
We know from our high school days that the solution of such an equation is given by: 1
x = - b/2a +- ROOT ( (b/2a)2 - c/a )
Substituting in for a=1 b= [Na+] - [Cl-] and c = - KW' this gives us the formula:
[H+] = - ([Na+] - [Cl-])/2 + ROOT ( ([Na+] - [Cl-])2/4 + KW' )
If we know the amount of sodium and chloride ion in solution, we can readily determine the hydrogen ion concentration. Even more elegantly, we can determine the result for any solution containing only strong ions! We simply fill in the concentrations of those ions in solution, where we had Na and Cl! The perceptive reader will realise that it is only the difference in ionic concentrations that matter - we can abbreviate the above equation to:
[H+] = ROOT ( KW' + SID2/4) - SID/2       .. Equation #2
where SID is our abbreviation for the difference between the concentrations of the strong base cations (eg Na+) and the strong acid anions (eg Cl-). SID is what Stewart calls the Strong Ion Difference .. an extremely useful concept. For a detailed consideration of this elegant solution, see Stewart's book. Similarly,
[OH-] = ROOT ( KW' + SID2/4) + SID/2       .. Equation #3
Note that if SID is negative, then the hydrogen ion concentration is always greater than the hydroxyl ion concentration. The converse holds for a positive SID. In these solutions it is clear that if the hydrogen ion concentration changes the SID must have changed.

If SID is positive and bigger than about 10-6 Eq/l you can see that KW becomes insignificant, and Equation #3 becomes very nearly the same as:

[OH-] = [SID]
We can use this and Equation #0 to derive the hydrogen ion concentration:
[H+] = KW' / [SID]

If the SID is negative and bigger than about 10-6 Eq/l then Equation #2 simplifies out to: [H+] = -[SID] but such solutions are not commonly encountered in biological systems.

"Okay, this is all very well" you say "but what is the practical significance of all this maths?" Easy. The above tells us that in a solution containing strong ions, if you want to calculate the pH, you must:

1. Know the concentrations of the strong ions, and
2. Plug these values into the equations;
before you can work anything out. And if you add basic or acidic substances, you cannot just say "We added so much sodium hydroxide so the pH will change by so much". You have to work things out using the above equations. Conversely, if you know the values, they are sufficient to work out the pH.

You can also see that if we have (for example) an acidic solution and we progressively add base, there will be a sudden, rapid rise in pH as we approach the point where [SID] is zero, and then add just a tiny bit more base. Consider the following diagram (our applet) showing how pH changes with SID - Press the button!

If you can read this message, then your browser is almost certainly not Java enabled. To view the acid-base calculator, get a Java-enabled browser!

pH is easily measured so now we finally understand the concept of "titratable acidity" of for example urine (we add base until there is a sudden rise in pH and this is the point where "titration is complete" - that is, the SID of the solution is zero). But remember that this doesn't tell us what titratable acidity means, and also note that the "sudden change in pH" is an artefact induced because pH is logarithmic: nothing that dramatic happens if we look at actual hydrogen ion concentration! Try this - in the above applet, change the Y axis parameter from pH to [H+] and press on the Calculate button again!

It is instructive to read Stewart's book where he looks at titration of interstitial pH using HCl - the counter-intuitive nature of pH is clearly seen.

Gaps & Gamblegrams

Electrical neutrality in solution demands that Equation #1 is satisfied - the sum of [negative] and [positive] ions is always zero. This is conveniently represented by two adjacent bar plots as shown in the following illustration (after Stewart, p43). The solution is a simple one containing [H+], [OH-], [Na+] and [Cl-]. The bar plot (Gamblegram) clearly shows the disposition of ions. The minuscule amount of H+ ion is not shown. Such plots can be used for far more complex solutions, often with unmeasured ions (where the term "gap" is often used, as in "anion gap"). It is crystal clear from the plot that the SID is equal to the amount of hydroxyl ion. In alkaline solutions the SID will be positive, in acidic, negative.

3. A more complex setup - Adding a weak electrolyte

Adding a weak electrolyte (one that only partially dissociates in the pH range we are considering) complicates things rather a lot. In body fluids such as plasma, the most important weak electrolyte is albumin, but the principles hold for all weak electrolytes.

A weak acid

Let's look at the case of a weak acid, HA (such as albumin) that dissociates to form H+ and A-, as follows:

As above, we can work things out if we have the correct numbers, and plug them into the relevant equations. Two of the equations we already know, those governing the dissociation of water and the requirement for electrical neutrality (slightly modified to include the dissociated anion A-, derived from the acid):

[H+] * [OH-] = KW'       .. Equation #0

[H+] + [OH-] + [SID] + [A-] = 0       .. Equation #1A

The other two equations are not too difficult, as they are based on the dissociation of the acid, and the necessity for conservation of the total amount of acid, which we abbreviate to ATOT. These are (in turn):
[H+] * [A-] = KA * [HA]       .. Equation #4

[HA] + [A-] = [ATOT]       .. Equation #5

Unfortunately for people such as myself who are used to counting on their fingers, the solution of the above four equations is based on cubic equations. Fortunately, when we use digital computers to solve such equations, the difficulty evaporates! We explore this below. When we actually go to the bother of solving the equations, we reach several initially rather counter-intuitive conclusions.

1. The weak acid is only really important in the range where SID is between zero and [ATOT]. Outside this range the hydrogen ion concentration changes in much the same way as it does without the presence of the weak acid. When [SID] is negative, then [A-] becomes tiny, and so [H+] approximates - [SID].
On the other hand, with [SID] > [ATOT], OH- have to fill the gap between the two (for [A-] can't of course exceed [ATOT]), so [OH-] = [SID] - [ATOT]. Consider the following:

Click on Calculate and then change the Albumin value to say 40, and click once more. The difference between the two curves is clearly seen - albumin only really has an effect between an SID of zero and an SID of about 0.02. (An albumen of 40 g/litre more-or-less corresponds to an ATOT of 0.016 Eq/l, perhaps a bit less)!

2. Inside this range of 0 .. [ATOT], hydrogen ion concentration changes more rapidly with [SID] than is the case where no weak acid is present! This destroys our conventional view that weak acid "buffers" lessen the impact of a pH change - the presence of a weak acid causes [H+] to change more rapidly with changing [SID] !

Click on the above graph to get the picture with an albumin of 40g/l, and then decrease the albumin to 0 and click again - notice how the new, red, albumin-free graph changes LESS steeply! Wham!

3. The point where [H+] = [OH-] is NOT necessarily where [SID] = [ATOT] !
It is best to play extensively with the actual equations in order to see how truly non-intuitive their effects are! Remember that in order to predict any result of altering the nature of a strong ion solution containing a weak acid, we need to know:
1. [SID]
2. [ATOT]
3. KA
4. K'W
Without even one of these four, you are stymied, and cannot for example predict pH. Note that in this example, (as in all the others on this page) we have dependent and independent variables. Things that we change by external maneuvers (such as [SID] or [ATOT]) are independent variables. Values that respond to such changes (e.g. pH) are dependent variables.
 WE MUST KNOW ALL THE INDEPENDENT VARIABLES IN ORDER TO CALCULATE THE DEPENDENT ONES!

A weak base

Calculations for a weak base are similar to the case of the weak acid above. In an analogous way, added weak base is most effective in the SID range zero to -[BTOT]. The difference is that buffering here does occur: the base flattens the [H+] versus [SID] curve very effectively!

Solving equations using computers

If you've successfully come this far, the following section should not be beyond your capabilities, even if computers scare you witless! You may however skip the following computerese. If you choose to read it, then also get a piece of paper and write down the equations, and work things out. We take the equations from above:

[H+] * [OH-] = KW'       .. Equation #0

[H+] + [OH-] + [SID] + [A-] = 0       .. Equation #1A

[H+] * [A-] = KA * [HA]       .. Equation #4

[HA] + [A-] = [ATOT]       .. Equation #5

Our approach:
A program to do this

Many people are terrified of computers - an unnecessary state of mind, as all computers do is give us a way of implementing an algorithm like the one above in a fast and accurate way! In order to implement our algorithm we need to express it in an unambiguous way. We use statements written in a computer language to do so.

We will now 'formalise' the above algorithm. We won't yet write it in a specific computer language, we will just make things more precise, without (we hope) compromising readability. Our only assumptions are:

1. We have something (a function) called ROOT that somehow works out a square root
2. The function F(MY_GUESS) works out Equation #6
3. the function ABS works out the absolute value of a number. That is, ABS(-0.001) and ABS(0.001) give the same result, 0.001.

TEA BREAK!

If you were unwise enough to read the above in one sitting, we suggest that you take a tea break to marshal your resources!

Remember the fundamentals:

• There are complex and often non-intuitive relationships between the various components involved in acid-base.
• We must decide what are dependent variables and which ones are independent.
• If we alter an independent variable, then the dependent variables will change.
• If we observe a change in a dependent variable (such as [H+] or pH) then there must have been a change in at least one of the independent variables.

4. Strong ions with Carbon dioxide

People who are familiar with acid base as it is commonly taught, usually cut their teeth on the Henderson-Hasselbalch equation. There is little wrong with the H-H equation, other than the fact that it only represents part of the truth. Here we start to explore how carbon dioxide behaves, but in what we regard as the proper context. Much of the following may seem familiar, but be careful - don't lose sight of the big picture!

Take our familiar mixture of strong ions and water, and expose it to CO2. What happens? Four things can happen to CO2 gas when exposed to water - it can dissolve, react with water to form carbonic acid, or even form bicarbonate or carbonate ions. We will explore each of these in turn, but the two most significant reactions are the formation of carbonate and bicarbonate, as each has its own equilibrium constant. By now you will realise that these reactions with their equilibrium constants will have a profound influence on the whole system, and it is only in the context of the whole system that we can understand the role of carbon dioxide. Let's see:

1. CO2 can dissolve, as expressed by the equation:

CO2(gas) <=====> CO2 (dissolved)

Reverse Reaction

===> Depends on partial pressure of CO2 <=== Depends on concentration of dissolved CO2
Rate of forward reaction = Kf * PCO2 Rate of reverse reaction = Kr * [CO2 (dissolved)]
..AT EQUILIBRIUM..

[CO2(dissolved)] = SCO2 * PCO2        .. Equation #7A

For Kf/Kr we have substituted SCO2, otherwise called the Solubility of CO2. In other words, the amount of dissolved CO2 depends on the partial pressure of CO2 times a rate constant. SCO2 is dependent on temperature, and at 37 C it is about 3.0 * 10-5 Eq/litre/mmHg. Note that if we are examining a fluid that isn't in contact with gas, we still talk about a partial pressure  of that gas in solution "as if" it were exposed to, and in equilibrium with, a gas containing that gas at that partial pressure.  This is just another convenient way of representing the concentration of dissolved gas in solution.

2. CO2 can react with water to form carbonic acid:

CO2 + H2O <==> H2CO3

 Equilibrium is represented by: [CO2(dissolved)] * [h30] = K * [H2CO3]        .. Equation #7B If we treat [h30] as constant, and rearrange things a bit we get: [H2CO3] = KH * PCO2 The value of KH at 37 C is 9 * 10-8 Eq/litre - because of this, the H2CO3 concentration is far smaller than the amount of dissolved CO2.

The reaction of CO2 with water is SLOW , with a half time of about 30 seconds, fortunately speeded up to microseconds by the carbonic anhydrase abundantly present in most tissues.

3. H2CO3 thus formed can dissociate into bicarbonate and hydrogen ions:
H2CO3 <==> H+ + HCO3-

 Equilibrium is represented by: [H+] * [HCO3-] = K * [H2CO3] It follows that: [H+] * [HCO3-] = KC * PCO2        .. Equation #8 A good value for KC is 2.6 * 10-11 (Eq/l)2/mmHg.

4. Once formed, HCO3- may rapidly dissociate:

HCO3- <==> H+ + CO32-

 Equilibrium is represented by: [H+] * [CO32-] = K3 * [HCO3-]        .. Equation #9 A typical value for K3 is 6 * 10-11 Eq/litre.

The big picture - finding the pH!

As usual, we need four simultaneous equations to work out all the dependent variables (given the independent ones). These are our old familiar equation #0, Equations #8 and #9, and the requirement for electrical neutrality:

[H+] * [OH-] = KW'       .. Equation #0

[H+] * [HCO3-] = KC * PCO2        .. Equation #8

[H+] * [CO32-] = K3 * [HCO3-]        .. Equation #9

[SID] + [H+] - [OH-] - [HCO3-] - [CO32-] = 0        .. Equation #10

It is important to distinguish between dependent and independent variables. Bicarbonate concentration and hydrogen ion concentration are dependent variables, but SID and pCO2 are independent!

We work out our equation for [H+] in the same way we did in Section 3 above, and solve it using a computer, again as above. On contemplating the results thus obtained, a whole host of interesting conclusions emerge. These include:

• Where the SID is negative, the excess negative strong ions must be balanced by positive ions (H+ being the only candidate) so OH-, HCO3- and HCO32- become rare and endangered species - their concentration becomes negligible.

• With a positive SID, [HCO3-] very accurately approximates SID. This equality is immensely useful, and also implies that [HCO3-] is independent of PCO2 in interstitial fluid! (at least in the physiological range).

• Totally counter-intuitive is that [CO32-] decreases with increasing PCO2!! Conversely, if we lower the PCO2 enough, carbonate levels, although tiny, may rise to a point where the solubility product for calcium carbonate (10-8) is exceeded. This may lower calcium levels, causing tetany! Click on the calculate button below, to see how carbonate concentration increases as PCO2 decreases:

and a Weak Acid

Following the pattern we established in previous sections, we identify the independent variables ( [SID], PCO2, ATOT which are respectively the strong ion difference, partial pressure of CO2 and the total amount of weak acid present). In addition, we need to know KW', KA, KC and K3. Given these, we can calculate any one of eight dependent variables:
HCO3-, A-, HA, CO2 (dissolved), CO32-, H2CO3, OH-, and H+. Note that dissolved CO2 and H2CO3 are easily determined from Equations #7A and #7B.

Exactly as we have done before, we derive six simultaneous equations, most of which are old friends:

[H+] * [OH-] = KW'       .. Equation #0

[H+] * [A-] = KA * [HA]       .. Equation #4

[HA] + [A-] = [ATOT]       .. Equation #5

[H+] * [HCO3-] = KC * PCO2        .. Equation #8

[H+] * [CO32-] = K3 * [HCO3-]        .. Equation #9

and finally, to maintain electrical neutrality:
[SID] + [H+] - [HCO3-] - [A-] - [CO32-] - [OH-] = 0

The equation that we derive in an analogous fashion to our method above where we discussed a computer solution is..

[SID] + [H+] - KC * PC / [H+] - KA * [ATOT] / (KA + [H+]) - K3 * KCPC / [H+]2 - KW' / [H+] = 0
.. and the computer approach is just the same.

"ionz" - an acid-base Calculator

Try experimenting further with our Java applet for blood plasma. This applet uses the above equation and iterative approach. First, simply press the "calculate" button. Then play - try seeing what happens to pH as you vary the SID and Albumin level. Then change the independent variable (on the X axis). Later you can see how other dependent variables (eg CO3--) change with variations in e.g. pCO2.

If you can read this message, then your browser is almost certainly not Java enabled. To view the acid-base calculator, get a Java-enabled browser!

ionz - Copyright (C) J van Schalkwyk, 1999

Afterword

I'd like to thank everyone for their feedback (mainly positive) about this page, particularly Rinaldo Bellomo and John Kellum, who have contributed enormously to spreading the news about the physicochemical approach, and also Jon Waters, and Philip Watson. A good website that explores both the traditional, and now the Stewart approach, is that of Kerry Brandis.

Thanks too to PJ Hilton, who pointed out a typo in Stewart's book, which we carried over in a mildly embarrassing fashion.

Recent developments are explored on yet another page on anaesthetist.com. Further constructive comment is of course still welcome. By the way the actual distance between Brown University (Stewart) and Copenhagen (Siggaard-Andersen) is 5958 km, just for the record.