# Re: CO2-Kh-PH table: Reference Wanted

```>Date: Sun, 13 Jul 1997 18:12:23 -0400
>From: Sanjay Joshi <sjoshi at psu_edu>
>
>I have seen the CO2-Kh-pH tables in several places. I am looking for
>a published reference that explains the chemistry behind these
>tables, and also provides the formulas and equations used to derive
>the tables.

I can *almost* explain the derivation of the tables.  Karla helped me
with the derivation and calculations once upon a time.  I still have
all the data but I am missing a key conversion that allows me to
perform the calculations.  I need to convince Karla to help once
again.  Or perhaps one of the chemistry gurus can fill in the missing
pieces.

First some background. This was written quite a while ago and I'm a
bit unsure of some of the details.  Please feel free to correct me.

=====  Begin: from an old posting  ======================================

Karla worked up some water chemistry last night to share with the net.
This is detailed information about the carbonate buffering system in an
aquarium and how CO2 and KH and pH are related.

Buffering
---------

First of all, when strong acids like sulfuric or hydrochloric acid are
added to water, they completely ionize into hydrogen ions and the
corresponding salt (HCL -> H+ + Cl-).  However, when weak acids like
carbonic or phosphoric acid are added to water, they form "conjugate
acid-base pairs" (HA and A-) that are in equilibrium.  There is an
"equilibrium dissociation constant" (K) for a weak acid defined as:

[H+] [A-]   where [H+] = concentration of hydrogen ions
K = -----------        [A-] = concentration of the salt or conjugate base
[HA]            [HA] = concentration of the weak acid

This relationship is for the reaction:

HA <==> H+ + A-

Note that K will vary slightly with temperature (and other things?).

The acid-base equilibrium is described by the Henderson-Hasselbach equation
(derived from the "equilibrium dissociation constant" relationship):

[A-]
pH = pK + log ----      or   pK = pH  when [A-] = [HA]  {since log(1)=0}
[HA]

The point where pH = pK is the point at which the system has the most
buffering capacity to handle additions of acids or bases and can occur at
multiple pH values for different buffering systems such as carbonate and
phosphate.

For example, with the carbonate buffering system, this occurs at pH 6.37
and pH 10.25.  At pH 6.37, H2CO3 (carbonic acid) and HCO3- (bicarbonate)
are present in equal concentrations. At pH 10.25, HCO3- and CO3--
(carbonate) are present in equal concentrations.

H2CO3  <==>  H+ + HCO3-  <==>  H+ + CO3--

Another common example is the phosphate buffering system (I believe
products like pH-UP and pH-DOWN are based on this system).  This has
equilibrium points at pH 2.13, 7.21 and 12.32.

H3PO4  <==>  H+ + H2PO4-  <==>  H+ + HPO4--  <==>  H+ + PO4---

Conjugate Base to Weak Acid Ratio
---------------------------------

At any pH point, the Henderson-Hasselbach equation describes the ratio of
[A-] (conjugate base) to [HA] (weak acid).  For example, consider the
carbonate system at pH = 7.0:

[A-]                          [HCO3-]
pH = pK + log ----     ->  7.0 = 6.37 + log -------  ->
[HA]                          [H2CO3]

[HCO3-]   4.27
0.63 = log (4.27)      ->  ------- = ----  ->  77% / 23%
[H2CO3]   1.00

[I'm afraid I can't figure out where 77%/23% came from! George]

Therefore at pH 7.0, the carbonate system is 23% H2CO3 and 77% HC03-.  If
you make a graph of pH versus the relative base/acid concentrations, you
will get a "S" curve due to the logarithmic nature of the equation.  An
important observation is that at the pK point the slope of the curve is
nearly vertical, i.e., a large change in relative concentration produces
only a small change in pH.

100% |             ___       ___
|            /         /
|           /         /
salt       |          |         |
concen.    |          |         |
50% |   H2CO3  +  HCO3-  +  CO3--
|          |         |
|          |         |
|         /         /
|    ___ /     ___ /
0% |___________________________
6.4      10.3
pH

(A graph of this appears on page 32 of the _Aquarium_Atlas_ (Baensch, 1987).

How this system relates to buffering can be seen from an example.  Consider
the process of protein ammonification (from "Water Chemistry in Closed
System Aquariums", A.J.Gianoscol, 1987).  One of the byproducts of this
process is phosphoric acid (H3PO4).  At a pH of around 7, phosphoric acid
will rapidly dissociate completely to H+ and dihydrogen orthophosphate
(H2PO4-) and dihydrogen orthophosphate will dissociate partially to H+ and
monohydrogen orthophosphate (HPO4--) (note that monohydrogen orthophosphate
won't dissociate to H+ and phosphate (PO4---) until the pH gets up around
the third equilibrium point at pH 12.32).  The free hydrogen ions (H+) can
then combine with bicarbonate (HCO3-) to form carbonic acid (H2CO3),
shifting the acid/salt balance slightly downward.  Since the reaction takes
place near the pK point, the slight shift in concentration does not
measurably affect pH.

To put it simply, the system is "buffered" since any free H+ ions can
combine with bicarbonate without altering the pH much.  Naturally, as
more and more "buffering capacity" is used up, the pH will be able to
shift more and more.  Also, salts from the weak acids build up in the
water.  Both these consequences point to the need for occasional water
changes to remove the salts and replenish the buffer.

Since the phosphate system has an equilibrium point at pH 7.2, you
would think it would be preferred to the carbonate system for
freshwater aquariums.  This is not true for three reasons.  First,
aquariums naturally produce organic acid compounds (metabolism
by-products) so you are most concerned about buffering acids.  If you
keep your pH at around 7.0, you are already below the phosphate pK
point and will keep getting further away as acids are buffered,
reducing the amount of buffering potential.  Using the carbonate
system, pH 7.0 is above the pK so that any acid buffering will move
the pH even closer to the best buffering point.  Second, plants can
use the carbon compunds which are part of the carbonate system.
Third, phosphates tend to grow algae, which is not desired.

Using the Henderson-Hasselbach equation and some chemical
calculations, you can create a chart showing the relationship of pH to
KH to CO2 (wink, wink, nudge, nudge).

=====  End: from an old posting  ======================================

Sounds easy, right?  OK, here are some scribbles from my old work
sheet -- the results of which created a fine pH/KH/CO2 table.

First, the "equation" that Karla gave me:

[A-]
pH = pK + log ----
[HA]
-or-
HCO3-      CO2
pH = 6.37 + log [ ------ = ------- ]
H2CO3   1.64CaCO3
-or-
HCO3-      CO2
pH - 6.37 = log [ ------ = ------- ]
H2CO3   1.64CaCO3

I don't remember how to interpret this.  Is HCO3-/H2CO3 "the same as"
CO2/1.64CaCO3?  Or is the relation part of the equation?  What's the
deal with "1.64"?  HCO3- is the KH which is 17.8 mg/l CaCO3.  Where
does this fit in?  Are the concentrations in millequivalents (I
suspect it is but I am meq impaired).

Next, the results. I calculated two points for each of many pH values.
The points are the CO2 concentration for that pH at KH=1 degree and
KH=8 degrees.  This makes a nice linear chart with CO2 on the vertical
axis, KH on the horizontal axis and showing lines of constant pH.

Here's some sample results (CO2 in mg/l):

CO2@    CO2@
pH   log[A-/HA]  [A-/HA]     KH=1    KH=8
---  ----------  -------    -----   -----
6.4     0.03       1.07     12.01   95.98

6.7     0.33       2.14      6.00   47.99

7.0     0.63       4.27      3.01   24.05

7.7     1.33      21.38      0.60    4.80

8.0     1.63      42.66      0.30    2.41

Notice the nice "power of 10" relationship between pH and CO2 values.
(pH=6.7, CO2=6.0) and (pH=7.7, CO2=0.6).

I made a notation at the top of the table showing a factor of "0.292"
for KH=1 and "2.334" for KH=8 (2.334 = 8 * 0.292).  I don't remember
what this factor meant.

Also note that [A-/HA] * CO2 is a constant value (with rounding):

1.07 * 12.01 = 12.8       1.07 * 95.98 = 102.7
4.27 *  3.01 = 12.8       4.27 * 24.05 = 102.7
21.38 *  0.60 = 12.8      21.38 *  4.80 = 102.6

So, if someone could figure out how to calculate the [A-/HA] values
for CO2 and KH, the equation is simple to rearrange to solve for
any one of the variables given the other two.

I feel sooooo stupid.

I'll forward this to Karla.  Perhaps she will be inspired to help out
once again.

George
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