NATIONAL BUREAU OF STANDARDS REPORT

10 599

Progress Report

on SOLUBILITIES OF PHOSPHATES AND OTHER

SPARINGLY SOLUBLE COMPOUNDS

<nbs£>

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

311.05-11-3110561 May 28, 1971 10 599

Progress Report

on SOLUBILITIES OF PHOSPHATES AND OTHER

SPARINGLY SOLUBLE COMPOUNDS

Walter E. Brown, Director

American Dental Association Research Unit

National Bureau of Standards

Washington, D. C. 20234

This investigation was supported in part by research grant

DE00572 to the American Dental Association from the National

Institute of Dental Research and is part of the dental

research program conducted by the National Bureau of Standards

in cooperation with the American Dental Association; the

United States Army Medical Research and Development Command;

the Dental Sciences Division of the School of Aerospace

Medicine, USAF; the National Institute of Dental Research;

and the Veterans Administration.

NATIONAL BUREAU OF SI

for use within the Government, and review. For this reason, th whole or in part, is not autho Bureau of Standards, Washingt

the Report has been specifically

IMPORTANT NOTICE

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST)

on October 9, 2015.

ss accounting documents intended

subjected to additional evaluation

listing of this Report, either in

3 Office of the Director, National

iy the Government agency for which

opies for its own use. <NBS>

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
II I 1*1
1 . INTRODUCTION

Rapid increases in phosphate levels in lakes and rivers

near heavily populated areas have made the understanding of

factors that control these concentrations a matter of great

social importance. Phosphate is thought to be the nutrient

which is the limiting factor in the growth of algae in most

waters (1); it is possible that carbonate, under some conditions of heavy pollution, may also be a limiting substance.

Solubilities of salts of these two anions have been proposed

as having considerable control over their concentrations in

lake waters. The same considerations do not apply to nitrogen,

however, because all of its salts are quite soluble. Even

though metabolic processes, such as growth and decay of

organisms, are probably of primary importance in establishing

the concentrations of dissolved orthophosphate and carbonate

ion at any instant, particularly during warm periods, it

appears reasonable that physicochemical processes such as

dissolution of suspended particles and sediments, leaching

of rocks and soil, and precipitation of insoluble salts have

major roles in establishing over-all, long term levels.

The observation that continual entry of phosphate into a

lake does not increase the phosphate level in proportion to

the amount that has entered (1) clearly indicates that at least

some of the phosphate is being removed by a precipitation
process. The high levels of calcium frequently found in

lake waters suggest that a calcium phosphate may be one of

the forms by which it is removed. This alone is justification

for the treatment given here on the solubilities of calcium

phosphates. The emphasis here is on thermodynamic and operational points of view and on making the treatment sufficiently

general so that it can be applied to sparingly soluble phosphates

of cations other than calcium and to salts of other weak acids.

Much of the information described here was developed in

studies ( 2-8 ) of calcium phosphates for application to systems

containing tooth, bone and dental calculus; the method of

approach, however, is equally applicable to other complex,

multicomponent systems.

Solubilities of sparingly soluble salts is a subject

that has been basic to chemistry over the years, and more recently

has been treated extensively as applied specifically to

minerals and limnetic solutions (9,10). One might wonder what

more can be said on the subject. However, the treatment given

here differs from others in the emphasis that is placed on

how to interpret data for a multicomponent system in terms of

a four component system. This should be useful in the interpretation of data collected under natural conditions. Also,

stress is placed on considerations which are vital for making

measurements on solubilities of other salts, such as those of

iron which are thought to limit the phosphate concentrations

under acidic conditions. There is urgent need, on the one hand

2
for more solubility and thermodynamic data on a fairly large

variety of salts and minerals. On the other hand, it should

be recognized that the data must be quite accurate if one is

to avoid serious misinterpretations. For example, a

relatively small error in the solubility product of one

compound may have a dramatic effect because it may change

the identity of the compound which is thought to be most

stable for some solution compositions.

The application of solubility principles even to the simple

ternary system, Ca (OH) 2 -H 3 P04 -H2 0, is far more complex than is

usually thought to be the case. Much of this complexity

goes back to the fact that orthophosphoric acid is both a weak

acid and a polybasic acid; but there are other complicating

factors, including those caused by the presence of ions other

than calcium and phosphate, by the presence of more than one

solid phase, and by the formation of ion pairs and complexes.

It is possible to treat each of these factors provided it is

adequately understood, but we have found it desirable, both

conceptually and experimentally, to start out with the simplest

system first and then to add complicating factors. Some of

these complicating factors are described next as a preliminary

to examining the system in greater detail.

When the solubility of calcium phosphates is thought to

be the factor that limits the concentration of a solution,

attention is usually fixed on hydroxyapatite, Ca5 (P04 ) s 0H,

as the saturating solid phase because it is the least soluble

3
of the several calcium phosphates under the situations

encountered most frequently. Actually, directing one's attention solely to this salt is an over simplification because

hydroxyapatite is only one of several salts with which such

a solution may be supersaturated; under these circumstances

it becomes difficult to predict which will precipitate. The

other sparingly soluble calcium phosphates include CaHP04 "2H2 0,

CaHP0 4 , Ca8 H2 (P0 4 ) 6

• 5H2 0 and 0-Ca 3 (PO 4 ) 2 ; two others, tetracalcium phosphate, Ca 4 0(P0 4 ) 2 , and alpha-tricalcium phosphate,

a-Ca 3 (P0 4 ) 2 , do not persist when in contact with aqueous

solutions. Experience has shown that the two hydrated salts,

CaHP0 4 -2H 2 0 and Ca8 H 2 (P0 4 ) 6

• 5H2 0, are the ones that precipitate

most easily, particularly at ambient temperatures, provided

the degree of supersaturation with respect to hydroxyapatite

is sufficiently high so that their solubilities also have

been exceeded. Subsequently, these salts tend to convert to

the more stable forms, sometimes going through a sequence of

phases of increasing stability, but this can be a slow process.

Under circumstances where one of the other salts has formed,

it may not be possible to describe the composition of the solution from solubility considerations unless that salt is

taken into account. This is true even though the salt is metastable with respect to hydroxyapatite and one or more of the

other calcium phosphates provided the rate of its dissolution

is great compared to the rates of precipitation of the more

stable salts).

4
Ions in the solution other than calcium or phosphate

affect the solubilities of the solid phases in several ways:

(1) The most important factor is probably the acidity

or basicity of these components. This comes about because

the concentration of the P0 4

3 “ ion is strongly repressed

by relatively low concentrations of hydrogen ions. Thus,

as described later, the calcium and phosphate concentrations

in a solution saturated with respect to hydroxyapatite at a

pH of about 4 could be as high as 10”* 1 M. As is shown later,

when several other components are present it is the net

acidity (i.e., surplus of acid over base) or the net basicity

of these components that is one of the dominant factors in

setting the solubility of any calcium phosphate in a given

solution.

(2) Both calcium and phosphate ions enter into the

formation of ion pairs or complexes, between themselves as

well as with other possible components. As noted later,

uncertainties in the theory of ionic solutions make it difficult

to be certain about the presence or absence of specific calciumphosphate ion pairs. This uncertainty comes about because

the concentrations of these ion pairs are usually quite low

(less than 10 to 20% of the total calcium or orthophosphate

in solution) and because it is always necessary to make extrathermodynamic calculations to arrive at ionic activities.

However, we can now have some confidence that these ions do

5
form pairs because extensive studies (4-10) of the ternary

system, Ca (OH) 2 -H 3 P0 4 -H 2 0, at 5°, 15°, 25°, and 37 °C have

shown that the solubilities of four of the above calcium

phosphates can be described accurately if it is assumed that

the ion pairs, CaHP0 4

° and CaH2 P0 4

+ are present in solution.

In the pH range above 7 or 8, the ion pair CaP0 4 may be

present in significant concentrations (11) . Stable complexes are known to form between phosphate ions and a variety

of multivalent cations found in natural waters (e.g., iron

and aluminum) ; calcium ions form complexes with a rather

limited number of anions found in natural bodies of water,

primarily organic ions, and now, possibly, polyphosphate ions.

Carbonate ions, because of their ubiquitous

presence in all natural waters, must be given special consideration in this respect because there is still some

uncertainty about the stability of calcium-carbonate ion pairs

(12,13)

.

(3) The presence of other ions also increases the ionic

strength of a solution, thereby affecting the solubilities,

but usually in a less spectacular way that the first two effects.

A high ionic strength has two effects on the equilibria: (i) the

solubility is enhanced by decreases in the ionic activity

coefficients, and (ii) the distribution of phosphate ions

among the various species is altered because the activity

coefficient of a multivalent ion such as P0 4

3 ~ should be more

6
affected than that of, say, H2 P0 4 . Evidence of the latter

effect can be sought in the way the ionic strength affects

relative solubility of an acid calcium phosphate (e.g., CaHP0 4 ) as compared to a more basic calcium phosphate (e.g., Ca5 (P0 3)0H)

and in how it affects the apparent electroneutrality unbalance

(see equation (12) , later) . Neither of these effects is large

and, thus, would require very accurate data.

Finally, non-thermodynamic factors, such as rates of

dissolution and precipitation, barriers to nucleation or

crystal growth, ionic transport rates, and conversion of metastable solid phases into more stable solid phases, serve to

make these systems difficult to handle experimentally and

necessitate caution in the interpretation of results.

Recitation of these difficulties is made not to discourage

the reader, but instead to stress the need for use of a valid

thermodynamic treatment, a chemically and physically appropriate

model, and a systematic approach which will help to clarify

the concepts and avoid some of the pitfalls. This is the major

objective of this chapter.

2.0. SOLID PHASES

As noted earlier there are five solid calcium phosphates

that are of importance in terms of being sparingly soluble salts

that are relatively stable in aqueous systems. It is generally

considered that hydroxyapatite, Ca5 (P0 4 ) S 0H, is -the least soluble

of these salts. The other salts in order of increasing solubility

under normal conditions are: whitlockite or p-tricalcium phosphate.

7
p-Ca 3 (P0 4 ) 2

; octacalcium phosphate, Ca8 H2 (P0 4 ) e

• 5H2 0; anhydrous

dicalcium phosphate or monetite, CaHP0 4 ; dicalcium phosphate

dihydrate or brushite, CaHP0 4 *2H 2 0. Some investigators have

tended, more or less, to disregard the fact that each of these

salts has individual characteristics and that each, therefore,

would have its own solubility for a given solution composition,

temperature, and pressure. Thus, although Ca5 (P0 4 y s 0H is the

most stable salt under most conditions found in nature, it becomes less stable (i.e., more soluble) than CaHP04 and

CaHP0 4 °2H 2 0 if the solution is sufficiently acid. As is described

below, the pH at which the relative stability of two salts

is reversed (i.e., the singular point) depends on the concentrations of other components. The apparent solubility behavior

of Ca5 (P0 4 ) S 0H in a solution more acid than the singular point,

or which at some time had been more acid than the singular

point, may be quite different from what one would expect for

Ca5 (P0 4 ) s OH; the formation of the more acid calcium phosphates,

especially if formed as surface coatings, can lead to apparent

high solubilities. This effect is accentuated by the facts

that (i) the solutions tend to be so dilute that a large

volume of aqueous phase is required to dissolve a relatively

small amount of the more acid calcium phosphate phase, and

(ii) hydroxyapatite crystal growth appears to be a slow

process so that the elimination of the acid calcium phosphate

by conversion to hydroxyapatite may not occur to a significant

extent.

8
These facts have not always been taken adequately into

account; the resulting apparent anomalous behavior may have

led some investigators to the conclusion that the properties

of calcium phosphates in general, and Ca5 (P0 4 ) 3 0H in particular,

are rather variable (14-17) . Actually when working with well

purified materials, it was possible to show from solubility

measurements (4-10) that each behaves in a quite normal fashion

in terms of its solubility and thermodynamic properties.

The order of increasing solubilities of these salts listed

above is true only for a limited range of pH. This is shown in

Figure 1 where the negative logarithm of the calcium concentration, pCa, at saturation is plotted against pH for the five

salts mentioned above (4-9, 18) . A similar diagram is obtained when

the negative logarithm of the total orthophosphate concentration, pP, is plotted against pH. It is important to keep

in mind that these plots apply only to the ternary system

(i.e., solutions which contain only calcium, phosphate and

water ions). The relative solubilities (at 25 °C) of the

salts at a given pH are easily seen from this graph. At pH 7,

the order of increasing solubility is Ca5 (P0 4 ) S 0H, $-Ca s (P0 4 ) 2 , Ca8 H2 (P04 ) 6

" 5H2 0, CaHP04 and CaHP04 *2H 2 0; below pH 4.8 the most

stable salt is CaHP0 4 , and even CaHP0 4 -2H 2 0 is more stable than

Ca5 (P0 4 ) 3 OH below pH 4.3. The pH values for various singular

points for the ternary system are defined by the broken vertical

lines. If one were to extend this plot to high concentrations,

four other singular points would be found where the solubilities

9
of the two monocalcium phosphates, Ca(H 2 P04 ) 2 and Ca (H2 P04 ) 2 *H g 0,

which are found only under very acid conditions, become less

than those of CaHP0 4 and CaHP0 4 *2Hg 0 (19, 20).

2.1. HYDROXYAPATITE, Ca5 (P0 4 ) 3 0H

The compound of greatest interest in most physiological

and other natural circumstances is hydroxyapatite because this

compound is a prototype of the principal crystalline material

in tooth, bone and many minerals. It is the ultimate alteration

product of fertilizers in soils, and undoubtedly is a factor

in the precipitation of phosphates under limnetic circumstances

and in sewage and waste treatment facilities. Because of the

relationship between its structure and its chemical properties,

the crystal structure of hydroxyapatite (21-24) is shown projected on the o face in Figure 2 and in a stereoscopic

illustration in Figure 3. Some of the atoms in Figure 2 are

shaded to show their relationship to Cas Hs (P0 4 ) 6

* 5H2 0, described

later. The stereoscopic illustration has the advantage of

showing the groups of ions in three-dimensional perspective.

The ellipsoid representing each atom depicts its apparent

thermal motions. The unit cell contains two formula weights

and consists of the following groups of ions: (i) two OH ions

lying on the hexagonal axis at the center of the diagram. The

oxygen of the OH ion lies about 0.3 A above or below the plane

formed by three nearby calcium ions, and the hydrogen attached

to this oxygen has a strong librational motion and is even

10
farther away from the calcium triangle; (ii) six calciums in

the form of two triangles separated by half a unit cell translation plus a 60° rotation, (iii) six P0 4 groups in two

triangles similar to those of the calciums; and (iv) four

calciums made up of pairs located on the trigonal axes near

the corners of the cell depicted in Figures 2 and 3. By keeping

in mind these four sets of atoms, one can easily visualize the

structure of hydroxyapatite. A closely related mineral of

importance to these considerations is f luorapatite, Ca5 (P04 ) s F.

It differs from Ca5 (P0 4 ) 3 0H in that the OH” ions have been

replaced by F ions situated at the centers (i.e., in the planes)

of the calcium triangles instead of being 0.3 A away. The displacement of OH ions from the centers of the calcium triangles in Ca5 (P04 ) 3 0H introduces the possibility of disorder

in the three-dimensional arrangement of these ions. This

disorder probably affects the solubility of hydroxyapatite

through its effects on entropy and internal energy. The disorder

may be enhanced by partial replacement of OH by F or 0^~ ions

or by vacancies.

Normally one might expect all the OH groups in a given

column to point in one direction because this gives maximum

separation of protons that are on adjacent OH groups. On the

other hand, an F~ or 0

2 ~ ion located in the plane of the Ca

triangle can act as a center of inversion by being hydrogen

bonded to the OH

- groups above and below it. This arrangement

could decrease the enthalpy and increase the entropy over what

11
might be expected from a simple solid solution, thereby decreasing

the solubility significantly. Several studies on the solubilities

of laboratory preparations of hydroxyapatite have shown that

it is possible to describe its solubility in a straightforward manner. However, evidence is accumulating showing

that enamel, which is the most crystalline of the apatites

produced in vivo, has a variable solubility even though fluoride

is present in minor amounts. This may be caused by the

presence of crystal imperfections and impurities (such as

carbonate) , by the presence of other calcium phosphates, or

by particle-size effects. Furthermore, the effects of F~ ion

on the solubility of f luorapatite-hydroxyapatite mixtures are

poorly understood beyond being able to say that such mixtures

tend to form in the presence of F ion and are, therefore,

more stable than hydroxyapatite alone. These uncertainties

create major problems in the application of solubility principles

to limnetic solutions, as they do for tooth and bone chemistry.

2.2. OCTACALCIUM PHOSPHATE, Ca8 H 2 (P0 4 ) 6

• 5H 2 0

The structure of this salt (25) bears a special relationship to that of hydroxyapatite and is shown in projection in

Figure 4 and as a stereoscopic illustration in Figure 5. The

unit cell contains two formula weights. The shaded atoms

in Figure 4 correspond in position very closely to those of

hydroxyapatite which are shaded in

12
Figure 2. The presence of this layer in Ca8 H2 (P0 4 ) 6

• 5H2 0

permits an epitaxial relationship with hydroxyapatite (26, 27)

.

The resultant formation of surface layers and intracrystalline

mixtures is believed to be a major factor that has led to confusion about the composition and properties „ of hydroxyapatite.

The unshaded atoms in Figure 4 have essentially no relationship

to the structure of hydroxyapatite. The water molecule (0(43)

in reference 25) is of chemical interest in that this site may

be only partially occupied, thus introducing possible

variability into the chemical properties of octacalcium phosphate .As a result of the similarity in the two structures, it

appears that Ca8 H2 (P0 4 ) s

# 5H 2 0 may participate in the nucleation

and growth as a precursor to hydroxyapatite; it may act, also,

as a transition layer between the hydroxyapatite crystals

and the aqueous phase. It is thought (28) that a hydrated

structure such as octacalcium phosphate will have a lower

inter facial energy as compared to that of an anhydrous ionic

structure such as hydroxyapatite. Thus, octacalcium phosphate

may have an important bearing on other properties of hydroxyapatite such as (i) influencing the morphology of the crystallite,

(ii) participating in the reactions with fluoride ions, and

(iii) possibly stabilizing hydroxyapatite so that it can exist

as very fine crystallites. The existance of hydroxyapatite

in this finely divided form in most of its occurrences

is one of its most important properties, giving it the

characteristics of a colloid.

13
Additional insight into the manner in which octacalcium

phosphate may complicate the chemistry of hydroxyapatite can

be gained if we consider the different processes by which

hydroxyapatite can be formed by precipitation from an aqueous

solution. It was proposed (28) that hydroxyapatites may be

formed through three mechanisms.

(1) One of these is easily carried out in the laboratory.

Octacalcium phosphate is precipitated first (e.g Q , by hydrolysis

of CaHP0 4 *2H2 0 or by titration of a calcium solution with a

phosphate solution at room temperature and approximately

neutral pH) . Once formed, the octacalcium phosphate can be

hydrolyzed to hydroxyapatite by boiling or by treating with

an alkaline solution. In this process the larger crystals

tend to become coated so that complete hydrolysis is difficult,

and the gross composition tends to be intermediate to the two

salts. An important aspect of this process is that it tends to

occur in situ so that the resulting hydroxyapatite crystals

have the platy or blade- like morphology of the original

octacalcium phosphate. This is an important indication of

the previous history of such crystals.

(2) A second and the most obvious process would be the

growth of hydroxyapatite directly without an intermediary

phase such as octacalcium phosphate. Under many conditions,

this route is very slow or ineffectual. Many possible constituents of lakes and rivers, such as carbonate, magnesium

and polyphosphate ions, seem to inhibit precipitation of

hydroxyapatite.

14
(3) A third process might be a combination of these two

and would involve the successive steps of precipitation of,

say, a unit-cell thick layer of octacalcium phosphate on the

(100) face of hydroxyapatite followed by its hydrolysis to

hydroxyapatite; this process would then be repeated, each

time converting a unit-cell thickness of octacalcium phosphate

to a double unit-cell thickness of hydroxyapatite. Such a

mechanism was proposed (28) because of the structural similarity

described above and because direct precipitation of hydroxyapatite

seems to be very difficult.

Fluoride ions would play an important role in the first

and third of these processes in that they tend to accelerate the

hydrolysis of octacalcium phosphate to an apatite. This process

was proposed (28) as a partial explanation of why extremely small

amounts of fluoride in tooth mineral can have such profound effects

in reducing caries. Limnetic conditions are such that they, too,

could facilitate the formation of octacalcium phosphate as

an intermediary in the formation of hydroxyapatite since the

growth of octacalcium phosphates seems to be preferred at

lower temperatures. This mechanism and the proposed role of

F~ ions could, therefore, be important factors in the kinetics

of precipitation in natural bodies of water.

2.3. WHITLOCKITE, p-Ca 3 (P0 4 ) 2

Like hydroxyapatite, j3-Ca 3 (P0 4 ) 2 is a mineral (29, 30) . It is somewhat unusual in that cations smaller than calcium

(Fe 2+ and Mg 2+

, in particular) seem either to make it more

stable or to facilitate its formation in aqueous media.

15
For a long time the composition of this salt was in doubt

because of a discrepancy between the unit cell constants and

the apparent symmetry (31) . Recent determinations of the

structure are resolving both the uncertainty in the structure

and the reason for the stabilizing effect of small cations.

Recent solubility measurements on the pure calcium salt (9)

have shown that above about pH 6 (see Figure 1) this salt is

second only to hydroxyapatite in its insolubility. Therefore,

under suitable circumstances one might expect that it would

form at least as an intermediary phase to hydroxyapatite.

Special considerations should be given to this compound in

those modifications of sewage or industrial-waste treatment

processes in which the presence of Fe 2+or Mg s+ ions may

facilitate formation of p-Ca s (P0 4 ) 2

2.4. BRUSHITE, CaHP0 4 -2H 2 0

Although under most circumstances, this salt is the most

soluble of the five that we are discussing here, it deserves

special consideration; because of crystal-growth kinetics it

appears to form very readily. Therefore, it is found under

many circumstances where it is metastable with respect to the

other four salts. Another feature that seems to be important

to both this compound and octacalcium phosphate is the presence

of waters of hydration. It seems that the presence of these

waters may reduce the interfacial energy of these salts when

they are in aqueous systems. This would enhance the relative

16
stability of the hydrated salts when the particles are extremely

small, that is, during the nucleation and initial growth

stages. Thus, it is possible that one of these salts is actually more stable than hydroxyapatite during the incipient

stage of formation. In any event, it is a common experience

in the laboratory to find that brushite (or octacalcium phosphate) forms initially and the more stable anhydrous salts

form subsequently, either by hydrolysis in situ, as in the

case of transition from octacalcium phosphate to hydroxyapatite,

or through a dissolution step followed by a precipitation step,

as seems to be the case usually in the conversion of brushite

into one of the other more insoluble forms.

2.5. MONETITE, CaHP04

Although this salt is always more stable than brushite,

except possibly at very high pressures and temperatures below

about 5°C, it is encountered less frequently than brushite

under most circumstances. This is in part due to the fact

that the salt is somewhat difficult to detect by X-ray methods.

Petrographic microscopy should be used when its presence is

suspected because its birefringence is relatively high (33)

.

It has a relatively low probability of occurrence under limnetic

conditions for two reasons: (i) the pH range in which it is

the most stable salt (about 5 to 1) is rarely encountered in lakes

and rivers, and (ii) its metastability with respect to several

17
other calcium phosphates is not compensated by a rapid rate

of crystal growth as is the case with brushite.

2.6. AMORPHOUS CALCIUM PHOSPHATE

In recent years considerable emphasis has been placed on

the observation that precipitated calcium phosphates frequently

display no crystalline characteristics. These products are

also thought (34-36) to have a composition closely approximating that

of Ca 3 (P0 4 ) 2 except that they contain considerable water. These

materials cannot be treated according to the solubility product principles described later even though they might have a

fairly fixed composition because it is unlikely that they fulfill

the necessary condition of having a fixed standard free energy

of dissolution per mole. There is no reason to believe that a

material as ill-defined as amorphous calcium phosphate would

have fixed thermodynamic properties. Whether such materials

ever have significant effects on the compositions of limnetic

solutions is difficult to say. It seems probable that they

would nearly always be metastable with respect to all five of

the sparingly soluble calcium phosphates described above, and

can therefore be formed only when (i) the solution is so highly

supersaturated that the reactions are too rapid for crystal growth

of any of the well-defined crystalline phases or (ii) the solution

is only moderately supersaturated, but crystal growth poisons

are present that prevent the formation of the other phases.

2.7. CALCIUM CARBONATES

Calcite and aragonite are the two most commonly occurring anhydrous calcium carbonates. Several other

18
anhydrous forms, including vaterite, several high pressure

modifications of calcite (37, 38) and a multiply twinned type

of aragonite structure, should be kept in mind as having an

outside chance of being involved in some way in growth from

water solutions. The two hydrated salts, CaC0 3 *H 2 0 (39-41)

and CaC0 3 *6H 2 0 (42, 43)

,

are more likely participants in

nucleation in crystal growth; they would tend to be favored

by (i) properties such as interfacial energy and kinetics

of crystal growth, and (ii) conditions such as low temperatures,

high pressures, and growth poisons (Mg 2+ and polyphosphates)

which poison the growth of the anhydrous salts more than that

of the hydrated salts.

3.0. THE LIQUID PHASE

3.1. DEFINITION OF COMPONENTS

For the sake of convenience in defining the activities

or chemical potentials of the components, we have chosen to

describe the ternary system with the components Ca(0H) 2 , H 3 P0 4

and H2 0. For many years the components that were chosen to

describe this system were CaO, P 2 05 and H2 0. However, in

dilute solutions, the activities and chemical potentials of

the oxides cannot be visualized or expressed as easily as those

of Ca (OH) 2 and H 3 P04 , which can be defined as follows:

^Ca (OH) s (Ca2+

) (OH ) s

, ( 1 )

19
•4H s P04 = (H+

) 3 (P0 4

3 -)

,

(2)

Vo = (H+

) (OH")

,

(3)

^Ca (OH) s = Va(OH) 2

+ RTta(Ca2+

) (OH") 2

, (4)

tJH3 P04 = U°H po4

+ RTXn(H+

) 3 (P0 4

3"), (5)

Vo = |a° + RT j!n (

H

+

) (OH”) . n 2 U

(6)

In this chapter, parentheses denote activities of ions, and

square brackets denote concentrations.

The very useful fact that the chemical potential of any

component, whether it is a weak or a strong electrolyte, can

be defined in terms of an ionic product is frequently overlooked. As a consequence of this definition, the chemical

potential of an acidic or basic component rises or falls

as the pH is changed even though the concentration of the

component is held constant. A graphic illustration of this

effect is described later where the chemical potential of

Ca(0H) 2 is shown to decrease along an isotherm even though

the concentration of calcium ion is increased over a thousandfold. For this reason it is important to distinguish clearly

between the concentration of a component and its chemical

potential. Although these effects are generally recognized

in the treatment of inorganic processes, they should be given

20
more consideration than presently in treating metabolic processes

where the driving forces for diffusion and chemical reactions also

involve differences in chemical potentials.

In the ternary system, the chemical potentials are

relatively easily definable quantities because the calcium

and phosphorus concentrations and pH are measured directly.

One then needs only to know the ionic strength to be able

to use an empirical relationship such as the Debye-Huckel

or the Davies equation (4) to calculate the activities of the

individual ions. In these calculations, one must also use

equilibrium constants for the dissociation of H s P0 4 and for

the calcium-phosphate ion pairs that may be present.

The application of the equilibrium constants obtained

from the study of the ternary system to those that contain

additional components is described in a later section. It

is noted here, however, that in natural waters there are

exceptional difficulties in defining the concentrations of the

components. Phosphate, for example, occurs in various other forms

such as (i) organically bound phosphate, (ii) particulate

material, (iii) adsorbed on particulate material, (iv) complexed ions, and (v) possibly as polyphosphates or metaphosphates. It is only the "free" orthophosphate and the orthophosphate

ion-paired to calcium (which can usually be calculated) that

should be counted as affecting the solubility of a calcium

phosphate salt.

The task of defining the components to be used in

describing a system should not be taken lightly. Although

21
neutral salts and non-ionizing components tend to have

secondary effects on solubility and frequently can be ignored,

some components may introduce new solid phases even when present in very low concentrations, which would then dominate

the nature of the equilibrium. HF is an example of such a

component because it forms highly insoluble fluorapatite

(or fluor apatite-hydroxyapatite solid solutions) . In general,

it is advantageous to choose all other components as acids

or bases because it is the acidity or basicity that is usually

the primary factor affecting the solubility of a phosphate

or carbonate salt because of its effect on the degree of

dissociation of H 3 P0 4 or H2 C0 s . The exception to this rule

would be a constituent that forms soluble complexes or insoluble

precipitates. As is shown later, the selection of the

additional components as acids and bases makes it easier to

treat their effects by combining them into a single variable

which has been called the "electroneutrality unbalance".

3.2. SOLUBILITY ISOTHERMS

We have already mentioned some of the complications

that are introduced into understanding the solubilities of

the calcium phosphates because of the fact that phosphoric

acid is a weak polybasic acid. For this reason, we approach

the subject from a basic position. The Gibbs phase rule,

when applied to a system containing three components and in

which a solid phase and a liquid phase are in equilibrium at

22
fixed temperature and pressure, allows only one degree of freedom.

This fact is easily seen from Figure 6, a logarithmic plot

of the concentration of calcium hydroxide against that of

phosphoric acid. The compositions of the solutions that are

saturated with respect to hydroxyapatite are given by ihe

line designated Ca5 (P0 4 ) S 0H. The fact that the equilibrium

compositions are described by a line (the "isotherm" ) is a

consequence of the single degree of freedom. Thus, if the

[Ca(0H) 2 ], [H 3 P04 I or pH of a saturated solution is specified,

then the composition of the solution is completely fixed by

the point where that composition or pH line crosses the isotherm. Along this isotherm the chemical potential of hydroxyapatite is constant even though the concentration of the

components Ca(OH) 2 and H 3 P04 vary over three orders of

magnitude. This is true because saturation requires that

^HA + 9 ^H 2 0

= 5 ^Ca (OH) 2

+ 3 mH 3 P04

^

where AF°^ is the standard free energy of dissolution of

hydroxyapatite. This relationship is a direct consequence

of the assumption that the solutionis saturated with respect

to hydroxyapatite and, therefore, the reaction

Ca5 (P0 4 ) 3 0H (s) + 9H2 0

(

i) = 5Ca(0H) 2 (aq) + 3H S P04 (aq) (8)

takes place without a change in free energy. In the dilute

solutions under consideration, c^emi cal potential

of water, is essentially constant.

23
One can now visualize a hypothetical process in which

the composition is changed toward the region of high concentrations and low pH' s (toward the upper right along the isotherm)

,

at all times maintaining equilibrium. In this process, five

times the decrease in chemical potential of Ca(0H) 2 is exactly

compensated by three times the increase in chemical potential

of H s P 04

. The chemical potential of Ca(0H) 2 is decreased

during this process because the pH has a greater effect on the

chemical potential of Ca (OH) 2 than does the concentration.

The isotherm is the line of demarcation between the

region of supersaturation with respect to hydroxyapatite

(above and to the left of the isotherm in Figure 6) and the

region of undersaturation. In the latter region, the reaction

defined by equation (8) would be accompanied by a decrease in

free energy. It must be kept in mind that equation (8) is not

the only reaction that can be written. Isotherms are shown in

Figure 6 for other calcium phosphates which define the solution compositions where there would be no change in free

energy for the dissolution-precipitation reaction for each of

these salts. The fact that these isotherms have regions of

positive slope in the range of concentrations of primary interest

to us is an important characteristic of phosphatic systems and

is a consequence of phosphoric acid being a weak electrolyte.

The same situation applies to the solubilities of calcium

carbonates. It accounts for the well known phenomenon that

calcium carbonate dissolves when the partial pressure of C0 2

24
is increased. On the other hand, the slope of an isotherm

for a salt formed from a strong acid and a strong base should

be monotonically negative over the entire range (in the

absence of complex formation) . The slope of the isotherm

for hydroxyapatite becomes negative when the pH reaches a value

in the vicinity of pK3 of phosphoric acid, but the phosphate

concentrations are then so low and the pH's so high that

this is not a range of significance except possibly when waste

water is being treated with lime.

With the exception of the isotherm for octacalcium

phosphate, which was calculated by use of a tentative solubility constant, the other isotherms given in Figure 6

represent experimental measurements. The isotherms for all

the salts shown in Figure 6 are nearly superimposed in the

range of higher concentrations and lower pH's. This is why

it is necessary to use a plot of the type shown in Figure 1

to separate the isotherms for the various salts. The fact

that these isotherms are nearly superimposed in Figure 6

relates to a property of phosphoric acid which can be used

to give additional insight into these systems, as described

in the section on the "electroneutrality surface". Next,

however, we consider some of the consequences of having two

solid phases in equilibrium with a liquid phase.

25
3.3 . SINGULAR POINTS

As noted above, when a single equilibrating solid phase

is present, application of the phase rule shows that the

ternary system has one degree of freedom; the presence of a

second equilibrating solid phase reduces the degrees of freedom by one so that the system becomes invariant (at a given

pressure and temperature) . This is represented in Figure 1

as the point of intersection where the isotherms of two

solids intersect; the composition at the point of intersection

is known as the "singular point". A total of six singular

points are shown in Figure 1. The singular points, although

present, cannot be seen in Figure 6 because the isotherms

are so nearly superimposed. One can approximate three

additional singular points by extrapolating the isotherms of

Ca5 (PO 4) 3 0H, Ca8 H2 (P0 4 ) 6

• 5H 2 0 and g-Ca 3 (P0 4 ) 2

, Figure 1, to the points

where they intersect. However, the metastability of these

compositions with respect to CaHP04 *2H 2 0 and CaHP0 4 (and

the ease with which CaHP04 ’2H 2 0 precipitates) prevents them

from having more than theoretical significance. The invariance

of the calcium and phosphorus concentrations and pH at the

singular points is destroyed when the system contains more

than three components. It is shown next that the variability

in the concentrations of these components does not signify that

their chemical potentials are variable at the singular point.

26
At the singular point, saturation with respect to two

solids requires that the solubility products for both are

satisfied. Thus at the singular point of Ca5 (P0 4 ) 3 0H and

CaHP04 , for example, we have

(Ca3+

) B (P0 4

3-

) 3 (OH) = (9)

and

(Ca2+

) (HP04

3 ") = K

m

( 10 )

or

(Ca2+

) (H+) (P04

3-

) = K^K S/ (11)

and are the solubility products for hydroxyapatite

and monetite, respectively, and K 3 is the third dissociation

constant for phosphoric acid. Simply by dividing equation (9)

by the third power of equation (11) and substituting for

(H* ) (OH

-

), we have

(Cas+

) (OH

-

) 2 (12)

and by dividing by the fifth power of equation (10)

,

(H+

) 3 (P04

3_

) = V*1 ’

we have

(13)

These equations reveal that the chemical potentials of Ca(0H) 2 and H 3 P04 are fixed at tKe singular point at a given temperature

27
or pressure because the addition of other components will not

change the quantities on the right hand sides of (12) and (13)

.

The singular points are important features of the system

because they represent compositions with relatively high

probability. This is a direct result of the phenomenon of

metastability. An example illustrates this point. Consider

a solution which is saturated with respect to hydroxyapatite

and which has a pH less than that of its singular point with

brushite (i.e., to the left of pH = 4.3 in Figure 1) . The

solution will be supersaturated with respect to brushite, which,

therefore, can precipitate, making the solution under saturated

with respect to hydroxyapatite. Dissolution of the basic salt,

hydroxyapatite, and precipitation of the acid salt, brushite,

will cause the pH of the solution to rise. This process would

continue until the pH and solution composition reached the

singular point. Conversely, brushite in a solution with a pH

above that of the singular point would tend to cause the

precipitation of hydroxyapatite; subsequent dissolution of

brushite would drive the pH down; again, these processes

would continue until the singular point was reached.

Relationships similar to (12) and (13) are obtainable

at the singular point for any pair of calcium phosphates. The

chemical potential of Ca(0H) 2 or H 3 P0 4 would again be constant

although, for each pair it would have a different value.

According to equation (12), the activity of the calcium ion

at the singular point would increase a hundredfold with a

tenfold increase in hydrogen ion activity. These increases
can be brought about only by the presence of other components

(e.g., HC1 , H 2 C0 s , H 2 S04 ) in the solution? concomitantly, the

P0 4

3 “ ion activity would be decreased by the presence of

these other anions. One can see these relationships better

by keeping in mind that the charge of the foreign anion is

balanced by a greater calcium concentration. The consequent

greater calcium ion activity results in a lower P04

3 " ion activity

because of equations (9) and (11) . Similarly, according to

equation (13), the activity of P04

3 " would be increased, and

the activities of the calcium and hydrogen ions would be

decreased, by the presence of components such as NaOH, KOH

and Mg(0H) 2

. Since the composition and pH of the singular

point solution is affected by the presence of other ions,

equations (12) and (13) are especially useful for estimating

the pH of the singular point of a solution in which the

phosphate concentration is greatly exceeded by those of

other anions such as carbonate and sulfate, as is the case

in many natural waters. Note the direction of these Effects?

an excess of added anion increases the calcium and hydrogen

ion activities, and an excess of added cation increases the

P0 4

3 ~ activity and the pH.

Since Ca(0H) 2 is a legitimate solid in the ternary system, it

could co-exist with hydroxyapatite to form a singular point which

could be of great interest in the treatment of sewage with

lime. However, this region of the system has not been

studied adequately because the phosphate concentrations are

too low to measure conveniently.

29
3.4. THE "ELECTRONEUTRALITY SURFACE"

Knowledge of the dissociation constants of phosphoric

acidand the stability constants for calcium-phosphate ion

pairs permits one to calculate the pH of any dilute solution

for which the concentrations [Ca(0H) 2 ] and [H 3 P04 ] are

given. Furthermore, one can define a surface in threedimensional space by plotting -log [Ca(0H) 2 ] against -log

[ H 3 P04 ] in the plane of the paper, as was done in Figure 6,

and the pH in the third direction. This surface can be

depicted as in Figure 7 by lines of constant pH which define

the contours of the surface. Although this surface is more

properly called a "pH surface", if was first called (44) the

electroneutrality surface for the reason that it is necessary

to introduce the condition of electroneutrality,

[H 2 P04 “] + 2 [ HP04

2-

] + 3[P04

s “] + [0H]~ - [ H+

] - 2 [ Ca2 +

] - [ CaH2 P04

+

] = 0 (14)

to be able to calculate the pH at any given point. More

details about this surface are given elsewhere (44)

.

The important feature about this surface is the presence

of two steep faces where the pH contours fall very close to

i one another in the range of pH from about 4 to 10. The i existence of these steep faces is the reason why the isotherms

in Figure 6 are nearly superimposed over much of their lengths.

These isotherms, like those of all other electrically neutral composition

30
must lie in the electroneutrality surface. In those regions

where the surface is very steep, and when the surface is

viewed from above as in Figure 6, the isotherms will appear

nearly superimposed. When the isotherms are projected into

a face that contains the pH axis, as is the case in Figure 1,

the isotherms appear well separated. Diagrams similar to

Figure 6 may be calculated for fixed compositions of a

fourth component and for other temperatures and pressures

through use of the solubility and ion-pair formation constants

which have been derived from studies of the ternary system.

The two steep faces, in the pH ranges 4 to 6 and 8 to 9,

are separated by a slightly flattened region at pH 7. Since

pK2 for phosphoric acid is 7.20, it is apparent that the

flattening of the surface in this region corresponds to

the pH where the two ions H2 P04 and HP04

2 have equal activities

and the system has a maximum in its buffer capacity. The flat

surfaces below pH of 4 and above 10 are regions that are buffered

by the combinations H3 P04/H 2 P04 and HP04

2 /P0 4

3

, respectively.

Thus the steep faces occur at pH values that are intermediate

to the pK values for the acid. Accordingly, the system Ca(0H) 2 - H 2 C0 3 -H s 0 should have a steep face at a pH of about 8.3.

The presence of the steep faces in Figure 7 is very

valuable for interpreting experimental data or for designing

experiments. For example, it is difficult to tell whether a

solution is saturated with respect to any specific calcium

phosphate from a knowledge of the calcium and phosphate concentrations alone? it is necessary to know the pH as well.

31
In other words, in the steep faces the calcium and phosphate

concentrations are slowly changing variables, whereas the

pH is the most sensitive variable, and it is the one that

should be chosen if the approach to equilibrium is to be

monitored. This is not true, of course, in a multicomponent

system that contains a high concentration of buffer.

Another use for the type of diagram shown in Figure 7

comes from the fact that the solutions, regardless of their

state of saturation with respect to the various calcium

phosphate salts, are restricted in their compositions to a

rather narrow region. Most solutions of interest fall in the

pH range 5 to 9. Thus this is a region of fairly high

probability from the standpoint of finding solution compositions,

and one can use it for purposes of approximation in much the

same way as one would use an isotherm. The presence of other

components will displace the steep surface in a manner that will

be described in a later section. However, before doing this it

is necessary to consider the effects of additional components in

a more general way.

3.5. MULTICOMPONENT SYSTEMS

Isotherms describing the solubility of hydroxyapatite

in the presence of constant amounts of a fourth component

are shown in Figure 8. For reasons given later, the amount

of the fourth component is given in terms of the negative

logarithm of the concentration, U(±), of this component, where

U (-) represents an acid and U (+), a base. The isotherm labelled

pU (0) is the same as the one given for hydroxyapatite in

the ternary phase diagram in Figure 6. In accord with the

32
phase rule, the addition of a fourth component introduces a

second degree of freedom, but by restricting the concentration of that component to a given value, the degrees of

freedom are again reduced to one. Thus, all solutions containing 10" 4 mol/

H

of, say, HC1 would have compositions

defined by the line pU (-) = 4 when saturated with respect

to hydroxyapatite; solutions containing 10" 3 mol/£ of HC1 would

correspond to the isotherm labelled pU(-) = 3. These

isotherms lie in the region which, in the absence of HC1,

would be supersaturated with respect to hydroxyapatite,

thus showing that the effect of the acid, as one would expect,

is to increase the solubility of hydroxyapatite. The latter

two isotherms were calculated on the basis of equilibrium

constants obtained from the ternary system. The position of

the isotherm should not be shifted very much if, say, HC1 were

replaced by HNO s or by any other strong acid of the same

normality that did not have specific interactions with

calcium or phosphate ions. A similar set of isotherms,

shown below that of pU (0) in Figure 8, is obtained for a

series of solutions with constant concentrations of a base

such as NaOH or KOH. Examples with pU (+) = 3 and 4 are shown

in Figure 8. In the range of higher pH's and lower calcium

and phosphate concentrations, the curves with positive values

of pU(±) acquire negative slopes as do the curves with negative values of U(±). However, this portion of the phase

diagram is in the range of such low phosphate concentrations

33
that it does not have much practical significance.

According to the view presented here, it is the acidity

or basicity of the other components that is important as long

as they do not form significant concentrations of ion pairs

and complexes. Furthermore, in the presence of several of

such components, it is the excess of base over acid, or vice

versa, that determines the position of the isotherm. This

leads to the definition of the quantity U(±),

U(±) = V v,c, , (15)

i where vj is the valence and C t is the concentration of the

tTi

i ion, and the sum is taken over all species of ions other

than calcium, phosphate, and water ions, and calciumphosphate ion pairs. Because the solution is electrically

neutral, an alternative definition of U(±) is given by

U(±) =-Y VjC, , (16)

4 where the sum is taken over calcium, phosphate and water

ions (i.e., the ions present in the ternary system). This

is why the quantity U(±) has been called (44) the "electroneutrality unbalance". In essence, U(±) is the measure of

the excess of basic over acidic components but it takes into

account only the dissociated portions of these acids and

bases. Ion pairs and complexes are counted as acids or bases

depending on whether their valence charges are negative or

positive, respectively o For these reasons, the quantity U(±)

34
can be considered to be the fourth component, which, when

positive, is a base and, when negative, is an acid. Its use partially

ignores the effects of ionic strength on activity coefficients,

but for the dilute solutions under consideration here, this

would not be a major source of error. Ion pairs and complexes (other than the calcium-phosphate ion pairs) must be

taken into account explicitly, and the concentrations of

calcium or phosphate in these forms must be subtracted

from the quantities [Ca(0H) 2 ] and [H 3 P04 ] used as the coordinates in Figure 8.

The isotherms for the various values of pU(±) can be

viewed as forming a three-dimensional surface in a way

somewhat similar to that of Figure 7 except that the coordinate

in the vertical direction would be pU(±) instead of pH. Since

each point on an isotherm has a specific pH, one can define

lines of constant pH on this surface. These are shown, in

projection, in Figure 9, which is a duplicate of Figure 8

except that it shows lines of constant pH.

Figure 9 displays at a glance considerable information

about how solubility of hydroxyapatite is affected by other

components over the normal range of limnetic solutions.

Compositions with pH's less than 4 are not shown because

hydroxyapatite is no longer the most stable phase; above pH = 9,

the orthophosphate concentrations are too low to be of much

significance compared to other forms of phosphate found in

natural waters. Figure 9 shows that a wide range of calcium

35
and phosphate concentrations can be saturated with respect to

hydroxyapatite at a given pH. The breadth of this range

(i.e., the fanning out of the isotherms) is much greater

in the dilute region of the diagram. Figure 9 also shows

that the calcium and phosphate concentrations of saturated

solutions are quite high in the acid region. As noted

previously, the phosphate concentration at a pH of 4 is

remarkably high, being of the order 0.1 M over a considerable

range of values for U(±).

The lines of constant pH can be used to determine whether

or not the solution is saturated even when one does not

know the value of U(+) for that solution. Thus, the lines

of constant pH can be used as "isotherms 1 ' when U(±) is not

known or is a variable. For example, a solution undersaturated with respect to hydroxyapatite would have a pH

below that shown in Figure 9 for a point defined by its

calcium and phosphorus concentrations. A disadvantage of

this method (as with any other method) is that either the

concentrations of ion pairs and complexes must be negligible

or they must be taken into account explicity in the manner

noted previously.

3.6. EFFECTS OF DILUTION

The positive slopes of the isotherms in Figures 7 through

9 have an interesting effect on what happens when a saturated

solution is diluted, say, by rain water which is relatively

36
free of solutes. The slope of an isotherm over much of its

length is so nearly one that even after a severalfold dilution

the composition point is not far from the isotherm. The

degree of undersaturation of the new solution, as measured

by the ion activity product, would probably be quite low;

however, as long as the composition lies on a "steep face"

of the electroneutrality surface (i.e., in the pH range 4 to

10) it is likely to be near the isotherm. This is true even

when the value of U(±) of the initial solution is quite large

because the new composition and the newly applicable isotherm

both lie in the " steep face" for solutions with the new

value of U(±). This phenomenon can be explained on a physical

basis. Dilution of the solution causes a decrease in the

calcium concentration, but this is largely compensated by an

increase in the OH and P0 4

3 ~ ion concentrations so that the

ion-activity product is not reduced as much as one might

expect. On the other hand, dilution of a saturated solution

with one that has a positive value of U(±) is likely to lead

to a final solution that is supersaturated with respect to

the salt. The reason for this is that the isotherm applicable

for the final solution will have been shifted farther to the

right then would be the case if U(±) of the diluent were

zero or negative, and, the composition of the mixture will tend

to lie to the left and above the applicable isotherm (i.e.,

in the region of supersaturation) . This type of consideration

exemplifies the use of the "generalized four- component phase

37
diagram" . Detailed calculations, based on the initial and

final compositions of the solutions would also permit one

to analyze the effects of dilution for any given situation,

but this would be much more tedious than the use of the phase

diagram, and it would be nearly impossible except when the

initial and final pH's were known. These pH's would have

to be known in any case where the solutions contain weak

electrolytes because their contributions to U(±) vary with

pH and dilution. Carbonate, of course, is such an electrolyte; sulfate and chloride, the other principal anions

contributing to U(±) in most waters, are not affected by pH

in the ranges found in most lake waters.

3.7. APPLICATION TO NATURAL WATERS

No effort is made here to test the application of Figure 9

to specific examples of lake water because this would require

knowledge that the solution was saturated with respect to

hydroxyapatite. However, it should be noted that the calcium

concentration usually exceeds that of phosphate by one to two

orders of magnitude. This is consistent with a large negative value of U(±) which would be supplied by the carbonate,

chloride, and sulfate ions. The latter provide most of the

negative charge needed to counterbalance the positive charge

of the calcium ions. Large negative values of U(±) lead

to high calcium and low phosphate concentrations as seen in

Figure 9 when the pH is above about 6.

38
3.8. OTHER CALCIUM PHOSPHATES

A diagram similar to Figure 9 can be calculated for

each of the other four calcium phosphates for which solubility

constants are available. Because of the way the isotherms

tend to superimpose, the positions of the isotherms in these

diagrams would be quite similar to those in Figures 8 and 9.

They would differ considerably, however, in the locations

of the pH lines in Figure 9. The greater the relative

solubility of a given salt, the more its lines of constant

pH would be shifted toward the upper right in the diagram.

Increasing the ionic strength with neutral salts, which tend

to increase the solubility of the salt, would shift the pH

lines in the same direction. Simi]arly, factors that enhance

solubility of a given salt, such as small crystallite size,

crystal impurities or structural defects, would shift the

positions of the pH lines, but they would shift the positions

of the isotherms only slightly. The effects of these structural

factors on the solubility of hydroxyapatite is poorly understood,

but there is reason to believe that they may be quite significant (46) .

3.9. OTHER ELECTRONEUTRALITY SURFACES

The electroneutrality surface shown in Figure 7 for the

system Ca (OH) 2 -H 3 P04 -H 2 0 was calculated taking into account

the ion pairs of CaHP0 4

° and CaH2P0 4 +

, but these are of relatively

39
minor importance over most of the achievable range of concentrations. A system such as Fe (OH) S -H 3 P04 -H2 0, where complexing

between the cation and the phosphate is strong, would produce

a grossly different surface. The shape of this surface can

be calculated provided the stability constants for the

complexes are known. The electroneutrality surface for such

a system may also have steep faces, but these are likely to

be displaced toward lower pH's. The reason for this displacement would relate to the facts that (i) the complexing

enhances the dissociation of the protons on the phosphate

groups, and (ii) the inflections in the steep faces correspond approximately to the pH values half way between the

first and second and the second and third dissociation constants of the acid. As noted above, the system Ca(OH) s - H 3 C0 3 -H2 0 is also likely to have an electroneutrality surface

with a steep face at a pH of about 8.3. The location and

slope of this face may be of considerable value in describing

calcium carbonate solubility. As with the phosphate system,

other ions will affect the shape of the surface for the

carbonate system. The presence of other ions might be

handled, as in Figures 8 and 9 for each of the calcium carbonates,

by defining a quantity similar to U(±) given here for the

phosphate system. On this diagram, the lines of constant

chemical potential of the component H 2 C0 s would correspond to

lines of constant partial pressure of C0 2

. It can be easily

shown that the chemical potential of Ca(0H) 2 is also a constant

along these lines.

40
3.10. CHEMICAL POTENTIAL DIAGRAMS

Figures 1 and 6 have considerable utility in that the

quantities that are plotted are directly measureable. However, the presence of other components destroys their

utility and it becomes necessary to use a phase diagram such

as Figure 9 where one can compare the directly measured

quantities (calcium and phosphate concentrations and pH)

with functions (isotherms and lines of constant pH) that have

been derived from theoretical considerations. There is

another type of derived diagram (Fig. 10), usually called a

"potential diagram", which is quite useful in multicomponent

systems. Equation (7)

,

which is valid regardless of the

number of components, can be written

Hra(OH) a = “5“ mh s po 4

+ ^ (af ha +

Substituting equations (4) and (5) into (17) gives

log (Ca 8+

) (OH

-

) 2 = -

-I log (H+

) 3 (P0 4

3-

) + K.

It can be seen from equation (18) that the chemical potential

of Ca (OH) 2 is linearly related to that of Hs P04 , and that

the negative inverse of the slope of the line, 5/3, is the

Ca/P ratio of hydroxyapatite, the equilibrating phase 0 Similar

relationships have been derived for the other calcium

phosphates (47) in a more general way. The slope and position

of the line for each solid is characterized by the Ca/P

ratio and its solubility product (i.e., its standard free

(17)

(18)

41
energy of dissolution). Brushite and monetite would both

have a line with a slope of one, but the line for brushite

would be displaced in the direction of higher chemical potentials

Potential diagrams (47) have proven quite useful in

attempting to identify the phase that may be saturating a

given solution or a series of solutions. In addition to

the usual difficulties related to uncertainties as to the

formation of ion pairs and complexes and whether equilibrium

has been attained, this type of plot has two shortcomings:

(i) it is strictly a comparison of derived quantities (involving

ion activities which can be calculated only through the use

of arbitrary conventions), and (ii) the concentrations are

not shown explicitly, so that two solutions differing greatly

in composition may fall on the same point in the potential

plot. In spite of these shortcomings, potential plots are

useful adjuncts to the other types of plots.

It is apparent in Figure 10 that the point where two

lines intersect (the singular point) has constant values for

the chemical potentials of Ca (OH) 2 and H 3 P0 4 regardless of

what other components may be present. This is a direct

result of equations (12) and (13)

.

4.0. DESIGN OF SOLUBILITY EXPERIMENTS

Some rules for the design of valid solubility experiments

are given in this section. One needs to make only a cursory

review of the literature on the solubilities of sparingly

soluble calcium phosphates to see that the results of different

42
investigators can be in considerable disagreement. When

properly carried out, solubility measurements are tedious

and costly. It is important, therefore, to design the experiments carefully to avoid introducing a weakness that will

destroy the value of the results. The following principles

should help in this respect.

4.0.1. Phase rule considerations: The phase rule provides

minimum criteria which, although frequently ignored, must be

fulfilled before equilibrium experiments can be valid. The

first step in the design of the experiments should be to

analyze the number of degrees of freedom in terms of the

components to be used in the equilibrations. Our experience

has been that it is best to minimize the number of components

so that the number of degrees of freedom is limited to one. in

this way it is possible to plot the experimental data directly

in the forms given in Figures 1 and 6. Experimental errors

can frequently be detected as points that do not fall on a

smooth curve. Buffers are frequently used to control the

pH, but there is another way of controlling the pH range

(described later) which does not introduce additional components with the concomitant uncertainty of having specific

interactions with the calcium and phosphate ions. The addition

of a constant amount of another component to all of the

equilibrations in a given series does not, in effect, increase

the number of degrees of freedom, but it may prevent the use

of the electroneutrality balance as an internal check.

43
4.0.

2. Solid phases: The first requirement is that the

identity and pertinent properties of all the solid phases that

can form in the system should be known. X-ray diffraction

patterns and chemical analyses are necessary, but insufficient,

criteria for establishing the purity of a given preparation.

Both are insensitive to the presence of minor phases and

coatings which could markedly alter the apparent solubility

of the major phase. Examination with a petrographic microscope is essential for establishing the absence of impurities

and for ascertaining that the crystals are suitable for the

measurements

.

4.0.

3. Criteria for equilibrium: Strict criteria must be

used in ascertaining that equilibrium is achieved in the

solubility experiments. Whenever possible equilibrium should

be approached from both super saturation and undersaturation.

The approach from supersaturation can usually be achieved by

pre-equilibrating at a different temperature or by the use

of a salt that is more soluble than the one under study.

However, the approach to equilibrium from supers atur at ion

does not always go smoothly; intervening precipitation of

a metastable salt or a slow rate of crystal growth may preclude equilibration from supersaturation. Both of these

factors appear to apply to solutions that are supersaturated

with respect to hydroxyapatite. Knowledge of this type can

be of practical value, however, in understanding the behavior

of the system under natural conditions.

44
4.0.

4. Monitoring approach to equilibrium: One should use

the most sensitive variable to monitor the approach to

equilibrium. As noted previously, changes in pH can be detected

more readily than changes in calcium and phosphate concentration when the composition is on the "steep faces" of the

electroneutrality surface. The same would not be true in

the ranges of pH below about 4 and above about 10. In the

range 4 to 10, however, relatively large changes in pH have

been found to occur even when no change in calcium or phosphorus concentration could be detected. An electroneutrality

surface is so easily calculated by the use of computers that

it should be calculated prior to design of the experiments

so that one can decide on the most sensitive variable for

the purpose of monitoring. A disadvantage of the use of

pH for this purpose is that it cannot be measured continuously on the solution being equilibrated because of possible

contamination from the salt bridge. Continuous monitoring

can be achieved, however, through the use of conductivity (48)

.

4.0.

5. Equilibration times: Constancy in the monitored

quantity should be achieved over a large range of equilibration times. It is sometimes possible to get very short

equilibration times by the use of leaching columns. The

comparison of these data with those obtained from long-term

end-over-end rotations can provide confidence that equilibrium

has been achieved. It should be recognized, however, that an

45
approach to equilibrium that requires dissolution of one or

more solid phases and precipitation of another, particularly

when the liquid is in the dilute range, can be exceedingly

slow and can give steady-state conditions that mimic equilibrium.

Lack of equilibrium can sometimes be detected by plots of

the experimental data, such as Figures 1 and 6 above, or by

use of the adjustment procedure, described later, for treatment of the data.

4.0.

6. Condition of electroneutrality: Since this is a condition that must be fulfilled by any chemical solution, it

should be used to test the validity of the data. A nonzero value indicates either (i) an error in analysis, (ii)

presence of impurities, or (iii) an incorrect calculation

of the phosphate ion distribution because of the use of

inappropriate phosphoric acid dissociation constants, ionpair formation constants, or expressions for calculating

ion-activity coefficients.

4.0.

7. Solid-to-solution ratio: If the ratio, weight of the

solid to the volume of aqueous phase, is too small, the rate

of approach to equilibrium may be too slow for it to be achieved in a

reasonable length of time. This problem becomes especially

important if difficulties with preventing growth of microorganisms are likely to limit the equilibration time. On

the other hand, high solid-to-solution ratios are likely to

46
lead to two types of difficulties: (i) small amounts of

impurities — either as foreign ions or as secondary solid

phases — may exert a disproportionate effect on the

equilibrium composition, and (ii) the amounts of desorption of

calcium and phosphate ions from the surface may be large

compared to that obtained from dissolution of the crystalline portion of the sample. Although, in theory desorption

and resorption of ions from surfaces should not affect the

equilibrium constants derived from the experiment, these

processes do alter the apparent stoichiometry of the reaction.

These effects can be of relatively large magnitude when working with

insoluble salts in the dilute region of the phase diagram.

4.0.8. Stoichiometry of the reaction: Another internal

check on the validity of the experimental results is the

requirement that the solid should dissolve stoichiometrically

.

Desorption reactions, as noted above, can invalidate the use

of this as a check, but these can be minimized by a preliminary

equilibration with the solution to be used in the final

equilibration or by the use of the leaching-type technique (10)

.

Hydrolysis (i.e., conversion of a metastable salt under study

to one that is more stable) will alter the apparent stoichiometry

of the reaction. However, hydrolysis does not necessarily

vitiate the results. The composition of the solution may still

be adequately representative of equilibrium with respect to

the metastable phase provided the latter phase is present in

47
preponderant amounts so that its rate of dissolution exceeds

greatly that of precipitation of the more stable phase.

4.0.

9. Metastability: For the reason cited above, it is

possible to study metastable equilibria provided favorable

equilibration conditions are chosen and other precautions

cited here are taken into account to verify that the results

represent equilibria. In many phase-diagram

studies it is possible to mix the components together in

almost any form that will give the desired final composition

and then to wait long enough for final equilibrium to be

attained. Equilibria involving dilute solutions, however,

tend to be slow, and one must start with the solid phase

to be studied, whether it is the stable or the metastable

one, and use experimental techniques that facilitate rapid

equilibrium. Sufficiently large solid-to-solution ratios

and continuous stirring are particularly important.

4.0.

10. Range of concentrations studied: The range of compositions studied should be as large as possible within the

limitations provided by the accuracy in the analyses, on the

dilute side, and uncertainties in treating the data (i.e.,

calculation of activity coefficients) on the concentrated

side. Another possible limiting factor is metastability with

respect to other salts in the system. When the supersaturation

with respect to the more stable salt becomes too great, then

its rate of formation becomes excessive.

48
A broad range of concentrations is necessary if one is

to distinguish between the effects related to the solubility

product of the salt and those due to ion-pair formation.

This can be seen by considering a potential plot such as

that given in Figure 1. Unless ion-pair formation is properly

taken into account, the line would not be linear. This lack

of linearity would not be detected if the range of chemical

potentials that is covered by the experimental data is too

narrow. It is fortunate that ion pairs such as CaHP04

° and

CaH 2 P0 4 +

, which differ by the dissociation of a hydrogen

ion, will usually have their most significant concentrations

in different pH ranges of the phase diagram, thus making it

possible to detect their presence if the pH range is

sufficiently broad. In the system Ca (OH) 2 -H 3 P0 4 -H2 0 it is

possible to achieve a broad range by equilibrating the solid

with various initial concentrations of H 3 P0 4

. In this way,

concentration ranges of more than three orders of magnitude

can be covered (4, 6, 7). In a multicomponent, buffered

system, a similar range of chemical potentials can be

achieved by use of a large range of Ca/P ratios in the solutions, and in the system Ca (OH) 2 -H 2 C0 s-H 2 0, varying the partial

pressure of the C0 2 achieves the same purpose as varying

the initial concentration of H s P0 4

. The apparent concentration of the ion pair CaH s P0 4

+

is greatest in the low pH range where no difficulties are

encountered with metastability or measurement of the calcium

49
and phosphate concentrations, but the high ionic strengths

of the solutions make calculation of ionic activities

uncertain. The concentration of CaHP04

° in solutions

saturated with respect to CaHP0 4 -2H 2 0 or CaHP0 4 is assumed

to be constant throughout the range. However, its concentration becomes significant only in the higher part of the

pH range when the total calcium and phosphorus concentrations

are small because of the reduced solubility of the salt.

Analytical errors and metastability with respect to the more

basic calcium phosphates tend to be the limiting factors in

this range.

4.0.11. The equilibrium model: The model to be used for

treatment of the data entails selection of (i) dissociation

constants for the acid, (ii) stability constants for the

ion pairs or complexes, (iii) an expression for calculating

individual ion activities, and (iv) standard values for the

thermodynamic properties of ions and components and their

variations with temperature. Decisions have to be made as

to whether the reaction is taken to be stoichiometric, how

to weight the measured variables, and whether the condition

of electroneutrality is to be used as a condition function.

The greatest difficulty is encountered with the choice

of types of ion associations to be used in the model. The

following species have been included (4,6-7,11,49-52) in

descriptions of calcium-phosphate equilibria: CaHP0 4

°,

CaH2 P0 4

+

, CaP04 ”

, H5 (P0 4 ) 2 ", H4 (P0 4 ) 2

3 ~„ H 3 (P0 4 ) 23 ", CaH 3 (P04 ) 2

CaH s (P0 4 ) s s ~

,

CaH (P0 4 ) 2

3-

, Ca(P04 ) 2

4-

, and

50
Ca2 H^P0 4 ) 2 °. The relatively low concentrations of the ion pairs

and other ionic associations in this system make it difficult

to select the appropriate species to be used in the model.

The same appears to be true of the system Ca (OH) s -H2 C0 3 -H 2 0.

The situation can best be described by some examples. The

maximum ion-pair formation found (4) in the solutions saturated

with respect to CaHP0 4 *2H 2 0 in the pH range 3.6 to 7.7 was

about 25% of the calcium. Hydroxyapatite (5) being the most

insoluble salt over much of this pH range, provides an

example of the other extreme. Generally, formation of the

ion pairs CaHP04

° and CaH 2 P04

+ in solutions saturated with

respect to hydroxyapatite is not very significant, never

exceeding 3.5% of the calcium. There are two qualifications:

(i) In the low pH range (below 5) , the solubility of hydroxyapatite approaches that of CaHP04 *2H 2 0 and, therefore, these

solutions may contain significant concentrations of the CaHP0 4

+

pair . (ii) When the Ca/P ratio of the solution is high,

as is the case with many natural waters, the fraction of

the phosphate in the form of ion pairs may be quite large

even though their absolute concentrations are low.

Possible errors in the dissociation constants for

phosphoric acid and uncertainties in the use of expressions

for calculating ionic activity coefficients further confound

the problem of selecting ion pairs for the model because

omcan never be certain that some of the apparent formation

of ion pairs is not attributible to errors from these sources.

51
It should be kept in mind also, that equilibrium constants

such as those for dissociation of phosphoric acid and ion

pairs, were derived from systems which are likely to differ

grossly in composition from the one being studied, and a

totally different mathematical treatment may have been used

in deriving these constants. Although some consolation may

be taken from the fact that the constants are sometimes

"limiting law" values, frequently they are not, and at other

times the extrapolations are so great that specific ionic

interactions may limit their accuracy for use in other systems.

Consideration, therefore, should be given to refining the

values for such constants using the solubility data from

the system being studied and keeping in mind the empirical

character of the calculations.

In general, one should avoid the temptation provided

by the fact that the fit of experimental data to a mathematical

model can always be improved by including more equilibrium

species in the model. However, the number of such equilibria

should be kept to a minimum to be realistic. In spite of

the large number of species (cited earlier) which have been

suggested as having sensible concentrations in the dilute

range of the system Ca (OH) 2 -H s P0 4 -H 2 0, it is found (4-10)

that the pairs, CaHP0 4

° and CaH2 P0 4

+ sufficed to describe

the data in the pH range from about 4 to 7 . The pair CaP04 (11)

may have to be included in the higher pH range. Considerable

52
help in reaching the decision concerning the minimum number

of ion pairs was provided by the adjustment procedure for

treating data described below. There are specific reasons

for questioning the reliability of constants derived for

ion associations involving more than one calcium and one

phosphate.

Solubility studies and potentiometric (pH) titrations

are the two methods that are most commonly used to determine

stability constants for ion pairs and complex ions. Reference

is made later to the advantage given by having -the additional

constraints provided by the conditions of saturation and

stoichiometric dissolution. Titration studies (51) used in

studying calcium phosphate ion association have led to the

conclusion that multiple ion groups were present. However,

calculations show that some of the solutions used in these

studies were highly supersaturated. Before the stability

constants for multiple groupings such as CaH 3 (P0 4 ) 2 and

Ca 2 H 2 (P0 4 ) 2

° derived from these studies can be fully accepted,

it is necessary to show that the super saturation did not

lead to formation of a colloidal precipitate that went

undetected. Supersaturation in a titration experiment is

likely to give more difficulty than is given by the same

phenomenon in the study of the solubility of a metastable

solid; the presence of a large amount of equilibrating solid

tends to overcome the effects due to precipitation of small

53
amounts of the more stable phase, and the longer equilibration

times used in solubility measurements tend to reveal any

changes taking place in the solution. In selecting an

expression for activity coefficients, one should be aware

of the deficiencies in the two that are most often used. On

the one hand, there are thermodynamic strictures (53)

II against the use of the Debye-Huckel equation to calculate

activity coefficients of ions of different valencies; on

the other hand, there is an apparent physical unreality

in the Davies equation (54) which requires the use of the

same activity coefficient for all ions of a given valence.

As a result, one should keep in mind the more conservative

view that the primary objective is to treat the data in a

way such that the derived constants have transferral value

from a laboratory system to the one in which they are going

to be applied. That is, one should seek to achieve maximum

cancellation of errors through the use of a uniform treatment and compatible constants. A step has been taken in this

direction by assembling an internally consistent set of

thermodynamic values for the calcium phosphates which is

based on solubility measurements (55)

.

4.0.12. Treatment of data: Now that the use of computers permits

sophisticated treatment of large blocks of data, there is

little justification for point-by-point calculation of the

equilibrium constant, followed by calculating an average

value and a standard deviation of the average from the

54
individual constants. We have found most satisfactory (4, 56)

the use of a generalized least-squares procedure in which

the various equilibrium expressions, the electroneutrality

and the stoichiometry of the reaction (when possible) for

the model are introduced as constraints, the variations of

which are to be minimized. Estimates of the experimental

errors enter into the weighting scheme as part of the calculation of the errors of the derived constants. One of

the most useful features of the output is the list of

adjustments that would have to be made in the experimental

data to minimize the deviations in the constraints imposed

by the model. Consistently large adjustments reveal that

the model is inadequate or that systematic errors, possibly

due to lack of equilibrium, are present in the data.

One might anticipate intuitively that the more

restrictive an experimental model, the more reliable the

constants that one can derive from the data. It was possible

to test this hypothesis (4) through the use of the adjustment procedure. Incorporation of the condition functions

based on electroneutrality and congruent dissolution into

the mathematical model significantly decreased the standard

deviation of the adjustment and thus increased the reliability

of the derived constants over those obtained when these

condition functions were omitted. Similarly, the use of the

solubility data as compared to titration data, has the

advantage of introducing one or two extra constraints which should

increase the reliability of the dissociation or stability

55
constants derived from the data. The only constraints

applicable to titration data are the dissociation equations

and the electroneutrality expression.

A computer program has been reported (57) for handling

titration-type data for the calculation of ion associations.

This program was not designed to take into consideration

the condition functions related to saturation and stoichiometric dissolution , nor was it explicitly stated whether the

condition function for electroneutrality had been included.

4 .0.13 Reporting of results: Although it should not be necessary

to make this admonition, the fact is that two types of

deficiencies frequently appear in the reports on solubilities

of sparingly soluble salts. The primary data, concentrations

of all components and pH, are not always given along with

the derived equilibrium constants and/or graphs. The

absence of primary data implies a lack of esteem for the data,

and it prevents the re-evaluation of the data at a later

time when a more appropriate model may be available. Another questionable practice is to list the logarithms of

the constants for a given set of experiments rather than

the constants themselves. It is not always recognized

by the reader that constancy in pK is far less significant

than constancy in K. Finally, some estimate should be given

of the variation in the derived product that may be caused

by errors or by junction potentials in the pH measurements.

Very frequently the error in pH, which is a logarithmic

quantity, is the greatest source of variation in the

56
derived constants. In the adjustment procedure described

above, this error is taken into account automatically,

otherwise it may show up only in the actual variations of

the derived equilibrium constants.

57
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63

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