NATIONAL BUREAU OF STANDARDS REPORT

10 072

Progress Report

on SOLUBILITY OF CALCIUM HYDROGEN PHOSPHATE

AT 5, 15, 25 AND 37 °C

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

31 1.05-1 1-31 1056 1 June 30, 1969 10 072

Progress Report

on SOLUBILITY OF CALCIUM HYDROGEN PHOSPHATE

AT 5, 15, 25 AND 37 °C

by H. McDowell,* J. R. Sutter^ and W. E. Brown*

* Research Associates from the American Dental Association

in the Dental Research Section, National Bureau of Standards,

Washington, D. C. 20234 +

Assoc. Professor of Chemistry, Howard University, Washington, D. C. This investigation was supported in part by Research Grants

D E —00 57 2 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 Council on Dental Research of the American Dental Association; the Dental Research Division of the U. S. Army Medical Research and

Development Command; the Dental Sciences Division of the

School of Aerospace Medicine, USAF; and the Veterans

Administration.

IMPORTANT NOTICE NATIONAL BUREAU OF ST

for use within the Government, and review. For this reason, thi whole or in part, is not author, Bureau of Standards, Washingtc

the Report has been specifically

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 201 5.

ss accounting documents intended

subjected to additional evaluation

listing of this Report, either in

i Office of the Director, National

y the Government agency for which

opies for its own use. <NBS>

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

Solubility of Calcium Hydrogen Phosphate at

5, 15, 25 and 37°C

ABSTRACT

The solubility of CaHPQ4 has been determined in

the ternary system, Ca (OH) 2 -H3 P04 -H2 0, at 5, 15, 25

and 37° C. The same results are obtained when equilibrium

is approached from super- and under-saturation. The solubility product constant, K = (Ca2+

) (HP04

2 ), varies

sp

with pH unless formation of ion pairs CaHP04

° and

CaH2 P04

+ is taken into account. Solubility product constants and standard errors for CaHP04 are 1.97 ± 0.03 x

10~ 7

, 1.64 ± 0.01 x 10” 7, 1.25 ± 0.01 x 10~ 7 and

0.94 ± 0.01 x

10*' 7 at 5, 15, 25 and 37°C, respectively.

Association constants and standard errors are 243 ± 32,

187 ± 24, 377 ± 37 and 355 ± 62 for CaHP04

° and 5.2 ± 0.7,

9.6 ± 0.6, 9.6 ± 0.7 and 8.1 ± 0.6 for CaH2 P04 + at the

same respective temperatures. Thermodynamic functions

for the dissolution reaction for CaHP04 and the association reactions for CaHP04

° and CaH2 P04

+ are also

presented.
'
SOLUBILITY OF CALCIUM HYDROGEN PHOSPHATE AT

(1) 5, 15, 25 AND 37°C (1)

This paper is based on a dissertation submitted by H.

McDowell to the faculty of Howard University in partial

fulfillment of the requirements for the Ph. D. degree

by Hershel McDowell, Walter E. Brown,

Research Associates, American Dental Association, National

Bureau of Standards, Washington, D. C. 20234

and John R. Sutter,

Department of Chemistry, Howard University, Washington, D 0 C.

20001 .

INTRODUCTION

Anhydrous dicalcium phosphate, CaHPO* , is a stable phase

in the system, Ca (OH) 2 -H3 P04 -H2 0. Accurate information about

the solubility of CaHP04 is of great value in establishing

its stability range, in determining its thermodynamic

properties, and in understanding interrelations among the

calcium phosphates that precipitate from aqueous solutions.

Two values are given in the literature for the apparent

solubility product of CaHP04 at 25°C(2, 3) but these are not

(2) T. D. Farr, Tenn. Valley Authority, Chem. Eng. Rept.

No. 8, Wilson Dam, Ala., (1950).

(3) K. L. Elmore, and T. D. Farr, Ind. Eng. Chem. _3_2, 580

(1940)

.
- 2 -

in good agreement. It is now known that the apparent solubility products for calcium phosphates vary with the equilibrium pH if they are calculated without taking into account

ion-pair formation (4)

.

The neglect of ion-pair formation at rela-

(4) E. C. Moreno, T. M. Gregory and W. E. Brown, J. Res. NBS

70A (Phys. and Chem.) No. 6, 545 (1966)

.

tively high concentrations probably accounts for much of the discrepancy in the reported values

.

In this paper, we report the solubilities of CaHP04 at

four temperatures in the dilute range of the phase diagram for

this system. Three different experimental conditions were used

to minimize effects due to metastability of CaHP04 with respect

to other calcium phosphates, and to ascertain that equilibrium

was reached. Solubility products and standard heat and entropy

of the dissolution reaction for CaHP04 and association constants and standard heats and entropies of association for

the ion pairs, CaHP04

° and CaH2 P04

+

, are reported for 5, 15, 25

and 37°C. Equilibrations were made in the ternary system to

avoid complications in the interpretation of data which might

result from the presence of other ions.
3

EXPERIMENTAL SECTION

Chemicals . The CaHP04 was prepared by dehydration of

CaHP04 «2H2 Q. The CaHP04 *2H3 0 was prepared in pure, well

crystallized form by ammoniating an aqueous solution, initially saturated with Ca (H2 P04 ) 3 »H2 0 , in an ice bath. The

dehydration was accomplished by boiling 400 g of CaHP04 -2H 2 0

in 41 of 0.07M phosphoric acid for 72 hours. The solid was

collected by filtration and washed with small volumes of

distilled water until the pH of the wash solution reached 6.0.

The solid was then washed thoroughly with acetone and dried

in vacuo (3 hours at 80°C,then 60 hours at 50°C) . Examination

with a petrographic microscope indicated that (a) refractive indices of this solid matched those reported for CaHP04 (2)

,

(b) the material was free of extraneous phases, and ( c) the

individual CaHPQ4 «2H2 0 crystals had converted into multicrystalline CaHP04 with an average domain size of about 20 urn.

The calcium and phosphorus contents and the mean errors were

I 29.49 ± 0.10% (theoretical, 29.46) and 22.71 1 0.06%

(theoretical, 22.71), respectively.
-4-

Doubly-crystallized phosphoric acid hemihydrate,

2H3 P04 *H2 0, was used to prepare a 1 M stock solution.

Other acid solutions were prepared by diluting the stock

solution with freshly-boiled distilled water and standardized by chemical analysis.

Analyses . Phosphorus was determined spectrophotometr ically

using the molybdovanadate reagent of Brabson, et_ a_l (5) .

(5) J. A. Brabson, R. L. Dunn, E. A. Epps, Jr., W. M.

Huffman, and K. D. Jacob, J. Assoc. Anal. Chem. 41 , 517 (1958) .

Standard phosphorus solutions were prepared from primary

standard KH2 P04 from a commercial source. Absorbance measurements were made with a double beam spectrophotometer. A

least squares equation relating the absorbances of the standard

solutions to their concentrations was used to compute the

concentration of phosphorus in the samples. The standard error in

this method is about ±1.0% of the phosphorus in the sample.

Calcium was determined with an

atomic absorption spectrophotometer equipped with a hollow

cathode calcium lamp and a single slot burner. Standard

calcium solutions containing a Ca/P weight ratio of 0.5 were

prepared from CaC03 and reagent grade KH2 P04 . The CaC03 was
-5-

prepared by reprecipitating reagent-grade CaCOs three times from

hot hydrochloric acid solution

/with ammonium carbonate solution. Strontium chloride solution was added to each standard to give a strontium concentration of 2.00 mg/ml to suppress interference by phosphate. The

samples were diluted to an appropriate calcium concentration

and the strontium concentration was made equal to that in the

standard solutions. Absorption measurements were made at 4227 &

in an air-acetylene flame. The calcium concentrations in the

samples were estimated using a standard curve 0 The standard

error in this method is about ± 1.5% of the calcium in the

sample. The pH measurements were made with a commercial pH

meter equipped with glass and saturated calomel electrodes

and standardized with certified National Bureau of Standards

buffers. The uncertainty in the pH measurements was estimated

to be about ± 0.008 pH units.

Equilibrations . At 25 and 37 °c# three types of equilibrations were used: (1) "leaching experiments" in which a

dilute phosphoric solution was passed through a column of

CaHP04 , (2) "batch experiments" in which CaHP04 was rotated

with dilute phosphoric acid, and (3) " super saturation experiments" in which a solution supersaturated with respect to

CaHP04 was passed through a column of the salt. At 5 and 15°

c

only leaching experiments were used.
- 6 -

The apparatus and technique used in the leaching experiments were essentially the same as those described by Moreno,

Gregory and Brown (4). An 8 cm layer of CaHP04 , supported

in the thermostated column by a medium porosity fritted glass

plug, was washed several times with dilute phosphoric acid

and was then leached with the same acid. The temperature

in the column was controlled within ± 0.01°G. A leaching rate

was selected so that doubling the flow rate would cause no

change in the pH of the effluent 0 Nitrogen was used to keep

the solutions free of C02

. At least three aliquots of each

effluent were taken for calcium and phosphorus analyses.

In batch experiments, 4.0 g of CaHP04 was added to each

of several glass stoppered bottles and washed several times

with small volumes of the acid solution with which the solid

was to be equilibrated. About 125 ml of the acid was then

added and the bottles were sealed with paraffin wax and rotated

end-over-end at 7 rpm in a thermostated water bath. In preliminary experiments, lasting up to seven days, it was shown

that equilibrium was attained within 24 hours. Thereafter,

solutions equilibrated for about 24 hours were filtered using

the leaching column, and their pH and calcium and phosphorus

concentrations were determined.
— 7 —

To attain equilibrium from supersaturation , solutions

supersaturated with respect to CaHP04 were passed through

a column of this salt in the leaching apparatus. These solutions were prepared by leaching either a column of CaHP04

• 2H3 0

at room temperature or a column of CaHP04 at a temperature 10°

C

lower than that at which equilibration experiments were made.

The pH and calcium and phosphorus concentrations of the effluent were determined before and after it passed through the column.

Calculations . Initially, "apparent" solubility product

constants, Kgp, were calculated by the procedure described

by Moreno, et. _al (4) , except that the Davies equation (6)

(6) C. W. Davies, "Ion Association", Butterworths Inc.,

Washington, D. C. 1962, p. 41.

was used to calculate activity coefficients. In the calculation

of Kgp, the formation of calcium-phosphate ion pairs was

ignored. Subsequently, in the calculation of "true" solubility products, K D c

A

, allowance was made for ion-pair formation.

The equilibria, in addition to those for the dissociation

constants of phosphoric acid, are defined by the following

equations

.
-8 -

CaHP04 (s) = Ca 2+

+ HP04

2 “

, KD c

A

= (Ca 2

) (HP04

2 ~

) (1)

Ca 2+

+ HP04

2 " = CaHP04

°

, Kv =

(CaHp q4

0

\ x (Ca 2

) (HPO^~) (2)

Ca 2+

+ H2 P04 " = CaH2 P04

+

, K = (CaH2 PQ4 +

) Y (Ca 2

) (H2 P04 ) (3)

The relationships between the solubility product K DCa and the

association constants K and K follow from the substitution

x y

of eq 1 and the second dissociation constant of phosphoric acid

into eq 2 and 3.

K = (CaHPCU 0

) (4)

X Koc ,

K = K, (CaH^PO^*) (5)

y Kdca (H+

)

In order to solve the above equations for K0 c

A

, K , and K , x y

approximate values for these constants K° , K° and K°, were

chosen. These values were improved by a least squares adjustment procedure (7) based on the method of Deming (8)

,

which

(7) T. M. Gregory, E. C. Moreno, and W. E. Brown, to be published

(8) W. E. Deming, "Statistical Adjustment of Data", John Wiley

and Sons, Inc., New York, 1943, Chapter 4.

s imultaneous ly adjusted the weighted observables: pH, initial

phosphoric acid, and final calcium and phosphorus concentrations.

In this procedure, the weighted sum of squares was minimized

subject to three condition functions based on electroneutrality,

solubility product of CaHP04 , and congruent dissolution of
the solid, respectively:

2 [Ca 2 + ]+ [H+ ]+ [CaHgP04

+ ]-[0H~]-[HgP04 “]-2 [HP04

s "]-3 [P04

3 “] =0 ( 6 )

[M-a] [P-a] fCa a + 1 ftb

II O (7)

.[.Ml . . - 1

X

= 0

( 8 ) [p-p 0 i where ( 4 )

, - (H+

) 2 (H+

) + 1 , K3

/V (9)

Ki K2 Kg/, U (H+ )/ 3

K, ( 9) / K2 (io) and

K3 (11) are the first, second and third dissociation constants

( 9) L. F. Nims, J. Am. Chem. Soc. _5_6, 1110 (1934) . (10) R. G. Bates and S. F 0 Acree, J. Res 0 NBS _30, 129 (1943)

RP1524.

(11) C. E. Vanderzee and A. S. Quist, J. Phys. Chem. .65., 118

(1961) .

o f phosphoric acid,* /, , f 2 and f 3 are the activity coefficients

of H2 P04 -

, HP04 2- and P04

3 ", respectively; and a, the total

concentration of ion pairs in a given solution is given by

a = |[MfP]~H [M4-P] 2 -4[MP-XK0CA //Ca s+ ]}^ (10)

where M and P are the total calcium and phosphorus concentrations. In these equations, parentheses indicate molar activities and brackets indicate molar concentrations.
- 10-

Ionic strengths were calculated from the equation

I=| ZChZi (11)

where Cj is the concentration of the ith ion and z t is its

charge. Activity coefficients were calculated using the

equation (6)

Is ) -log f i = A t Z?

| i “'ii

" 0,31

j (12)

where At , a temperature dependent constant, was given the

values 0.492, 0.500, 0.509 and 0.521 at 5, 15, 25 and 37°c,

respectively (12). The activity coefficient for CaH2 P04

+ was

(12) G. G. Manov, R. G. Bates, W. J. Hamer, and S. F. Acree,

J. Am. Chem. Soc. _65, 1765 (1943) .

assumed to be equal to that for H2 P04 “

, and the activity coefficients of uncharged species were assumed to be unity. The

calculations were made by a computer using a program incorporating the above procedure (13)

(13) Provided by E. C. Moreno and T. M. Gregory.

RESULTS

Table I summarizes the experimental data and the derived

constants for the equilibria approached from under saturation

at four temperatures. The first column gives the compositions
- 11-

of initial phosphoric acid; the next three columns give the

pH and calcium and phosphorus concentrations of the equilibrated

solutions. The values in parentheses give the differences

between the last significant figures in the experimental and

the adjusted values. A negative value indicates that the experimental value is the larger of the two. The adjusted values

that differ from the experimental values by more than two times the

experimental error are marked with asterisks. The next column

gives the percent calcium calculated to be in the form of ion

pairs; the next gives the "apparent" solubility product constants calculated without assuming ion-pair formation; and

the final column gives the "true" solubility product constants

and the ion-pair association constants as defined by equations

1, 2 and 3, respectively.

With few exceptions, the differences between the experimental and adjusted values for the initial acid concentration

and final pH and calcium and phosphorus concentrations are

within the experimental errors. The errors in the solubility

products and association constants are standard errors based

on (c[N - b) degrees of freedom; where c[ is the number of

condition functions (two for the super saturation experiments

and three for all others) , N is the number of equilibrated

solutions in the set and b is the number of adjustable parameters.
- 12 -

Table II gives the same information as Table I for the

equilibrations from super saturation , except that pH and

calcium concentration before equilibrating with CaHP04 are

also given. It can be seen that in each instance the pH

decreased on equilibration, indicating that precipitation had

occurred. The differences between the calcium and phosphorus

concentrations before and after equilibration are so small

that errors in analyses preclude calculation of the Ca/P ratio

of the material that precipitated during equilibration. Since

the data in Table II form sets that are indistinguishable from

corresponding sets in Table I, we assumed that the precipitate

in the equilibrations from super saturation was CaHP04 . This

was confirmed by a plot of the chemical potential of

Ca(0H) 2 against that of H3 P04 (14, 15) which revealed that

(1 4) E. C. Moreno, W. L. Lindsay, and G. Osborn, Soil Sci.,

90 , 59 (1960).

(15) K. MacGregor and W. E. Brown, Nature 205 , 359 (1965).

the Ca/P ratio of the equilibrating

phase is 1.000 at both temperatures, 25 and 37°, at which

super saturation experiments were made. The data were

treated so that each of the different sets of experiments

yielded independent values for the three constants, KS p , Kx

and Ky. The weighted means of these constants, as given in

Tables I and II, are listed in Table III 0 In each instance.
-13-

the constants given in Tables I and II are within two

standard deviations of the over-all mean value. Also given

in Table III are the weighted mean values of the constants

derived from a treatment in which activity coefficients were

calculated through use of the extended Debye-Huckel equation.

Although there are objections to the use of the Debye-Huckel

equation in mixed electrolytes (16), it is apparent that the

derived equilibrium constantsare not greatly influenced by

the method used to calculate activity coefficients. The

values using the Debye-Huckel equation are included to permit

direct comparison with other solubility data for calcium phosphates

(17, 18, 19)

(16) E. A. Guggenheim, "Thermodynamics", 3rd ed. , NorthHolland Publishing Co., Amsterdam, 1957, p. 355.

(17) R. W. Mooney and G. J. Meisenhelter , J. Cnem. Eng. Data

J5, 373 (1960) . (18) E. C. Moreno, W. E. Brown and G. Osborn, Soil Sci., Soc.

Proc. , 24, 94 (1960) . (.19) E. C. Moreno, T. M. Gregory and W. E. Brown, J. Res. NBS

7 2A (Phys. and Chem.) No. 6, 773 (1968) . Standard heats and entropies for the dissolution reaction

for CaHP04 and the association reactions for CaHPO* ° and

CaH2 P04

+ relate to their respective equilibrium constants

by the linear function

( 13 )
-14-

The least squares results for each of the three reactions

are given in Table IV. The data points in the plots of Kx

and Ky vs 1/T were quite scattered, as indicated by the large

errors in the derived quantities for the association reactions.

These errors are much larger than the indicated errors in the

same quantities for the dissolution reaction. This is attributed to the fact that the concentrations of the ion pairs

were always small compared to the concentrations of unpaired ions.

DISCUSSION

Under the conditions used in these experiments, equilibrium was attained in a relatively short time, apparently

because the solid phase was well crystallized and free of

extraneous phases. Under these conditions, more basic calcium

phosphates, Cae H2 (P04 ) 6

• 5H2 0 , R-Ca3 (P04 ) 2 and Ca^ 0 (P04 ) s (OH) 2 were not formed in significant amounts during the equilibration

process even though under some of the conditions CaHP04 was

metastable with respect to some of these salts. This is shown

by the fact that essentially the same results were obtained

whether equilibrium was attained (1) by leaching or by much

longer batch equilibrations, or (2) by approaching equilibrium

from supersaturated conditions.
-15-

The two values listed in the literature for the "apparent"

solubility product of CaHP04 at 25° were calculated from the

composition of solutions that had pH values much lower than

those in this work. Elmore and Farr (5) calculated a K^p

value for a single point (0.826% CaO and 2.387% P s 0 5 ) for

which the pH was taken to be 2.80. Their value, 3.3 x 10”7

,

would not differ greatly from the value one would get by

extrapolating our data to this pH. The extrapolation cannot

be done accurately because of the large curvature in the low

pH region. The other value, 2.18 x 10“7 (2), was calculated

from the invariant point composition for the solution in

equilibrium with CaHP04 and Ca (H 2 P04 ) 2 «H2 0. This solution

has a pH of about 1.01. Considering the uncertainties in the

calculation and the difficulty of extrapolating our data to

such a low pH, this value of K' is reasonably close to ours. 2=5 ir The dependence of apparent solubility product constants

on the pH of the equilibrated solution is consistent with

the formation of the ion-pairs, CaHP04

° and CaH2 P04

+

. The

fraction of calcium complexed, pj-, shows the same trend with

respect to pH as the apparent solubility products. Both have

minima at the same pH. When corrections are made for ion-pair

formation, "true" solubility product constants that are
-16-

independent of pH are obtained. This fact is shown by the

small errors in the K values listed in Table III.

sp

The association constants, K and K , listed in Table

x y

III compare favorably with those obtained by other investigators using similar and different methods. Gergory, Moreno

and Brown obtained the values K = 479, 283, 264, and 431 and X

Ky = 9.7, 5.2, 3.7, and 3.3 from solubility measurements on

CaHP04 *2H2 0 at 5, 15, 25, and 37°C, respectively (7), when

the data was treated by the procedure outlined in this paper.

Their values compare favorably with ours.

Chughtai, Marshall and Nancollas (18) determined association

(20) A. Chughtai, R. Marshall and G 0 H. Nancollas, J. Phys.

Chem. 72./ 208 (1968) .

constants for CaHP04

° and CaH2 P04

+ from potentiometr ic measurements on systems containing sodium, potassium and chloride ions

in addition to those of the system Ca (OH) 2 -H3 P04 -H2 0. Their

values are Kx (25°c)= 5o48 x 10 2

, Kx (37 ct)= 6.81 x 10 2

, Ky ( 2 5 °C ) = 25.6 and Ky(37 <C)= 31.9; the values for exceed

ours by more than would be expected from our computed errors.
-17-

Davies and Hoyle ( 21) reported values of Kx = 5 x 10 2

(21) C. W. Davies and B » E. Hoyle, J. Chem. Soc., 4134 (1953).

and Ky = 11.8 from a study of the effect of calcium ions on

the pH of phosphate buffers at 25 °g. These values are in

reasonable agreement with ours.

The value for the standard heat of solution of CaHP04 reported in this work is about half the average value, AH = -9515

calories per mole, calculated by Mooney and Meisenhelter

(1 ) from the value of K' given for 25 <tby reference 2 and

their solubility measurement at 90^. The solubility constant at

90°C was based on a pH measured at 25°G. In view of the

uncertainties in both solubility constant values and the

large temperature range over which the calculation was made,

great reliance cannot be given to this value of AH. A value of

9515 calories per mole would yield a much larger variation in

solubility with temperature than would be consistent with our

experimental errors.

Table

I:

Solubility

and

Ion-Pair

Association

Constants

for

Equilibria

Approached

-18-

o-!O c\> rH O

o-IO Oi .H O

CoH4-> rd5-1 P

4-> rd

COu

CD bcseou

4-1

.a

a

-CO

t'obdoo

COco

•H4JP

rHowb

aj +j

rdup

-p

rd co

M-loco

•H

4-> •H

COO&ou oXaoXa

rd Ua

uo

in

+J

rd

CJ> G

•rH bu

rd

Cl) J

X

X

X

X

00 rH O CN o p • 00 0 CN o • o • CO o • O • +1 o +1 o

+1 +1

r-' +i

p +! Oo oo co r" • p CN • 00 CD f—1 • 9 rH « • CN in rH Oo

II II & II II it II a a

CO

X cn

X >1 * « «

00 00 OO CN

H r~ oo CN oop CO r-> 00

CN rH o O rH CN

p t-" r- r- 00 OO • • 0 • • 6 • • CN CN CN CN CN CN CN rH rH rH rH ! 1 rH rH

CO P ao P P OO rH 00 00 o O rH r- • • • • • • • 9 • • • • • •

r-' in P 00 in

O UO

U0 P 00

p CO OO 00

rH rH rH

ajp rH ,—v i CN CN 00 O + rH r—1

00 ys s—• + + + rH p sH o + i 1 rH 1—1

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-23-

Table IV j Thermodynamic Functions for CaHPO^

and CaHa P04

+

AH°

Reaction kcal mol 1

CaHP04 (s) = Ca 3+

+ HP04

2 ”

Ca e+

+ HP04 3 "= CaHPO ° 4

Ca 3+

+ H2 P04 " = CaH2 P04 +

-4.17 ± 0.04

(-17.4 ± 0 . 2) a 3.7 ± 1.1

(15.5 ± 4.6)

1.2 ± 0 o 9

(5.0 ± 3.8)

a values in kJ mol-1

b values in JK-1 mol-1

"1 cal = 4.184 J"

CaHP04°

AS 0 cal K

-1 mol -1

-45 0 63 ± 0.41

(-190.9 ± 1.7) b

24 ± 4

(100 ± 17)

8 o 2 ± 4.6

(34.3 ± 19.2)

USCOMM-NBS-DC