Oxygen solubility and diffusivity in hot concentrated H3PO4

Electroanalytical Chemistry and Interfacial Electrochemistry, 57 (1974) 281-289

281

© Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

OXYGEN SOLUBILITY AND DIFFUSIVITY IN HOT CONCENTRATED H3PO4 K. KLINEDINST, J. A. S. BETT, J. MACDONALD and P. STONEHART

Materials Engineerincj and Research Laboratory, Pratt & Whitney Aircraft, Middletown, Conn. 06457 (U.S.A.) (Received 19th April 1974; in revised form llth September 1974)

LIST OF SYMBOLS

a A Co v D

electrode radius (cm) electrode area (cm 2) uniform initial concentration of oxygen (mol cm-3) kinematic viscosity of electrolyte (cm 2 s-1) diffusion coefficient (cm 2 s-1) rotational velocity (radians s-1) E diffusion activation energy (cal* mol- x) F Faraday constant (A s mol-1) id diffusion current (A) L length of wire electrode (cm) n number of electrons involved in the overall electrochemical reduction or oxidation of a molecule of the diffusing substance R ideal gas constant (cal K-1 mol-1) t time of electrolysis (s) T absolute temperature (K) AH enthalpy of solution (cal mol- 1) AS entropy of solution (cal K - 1 mol- 1) Porous gas diffusion fuel cell electrodes rarely operate under the limiting conditions dictated by the heterogeneous kinetic reaction rate constants, although this condition is desired. For this reason, mathematical models of porous electrode structures are generated with a view to optimizing gas mass transport to the electrocatalytic reaction sites. Solubility-diffusivity data for the reacting gas molecules in the electrolyte are then required to develop the mathematical models. In this instance we are concerned with the solubility~liffusivity of oxygen in hot phosphoric acid over the concentration range of 85-96 wt. ~o. Due to the extremely difficult handling problems posed by the electrolyte and the parametric values required, it is worthwhile examining the available electroanalytical techniques. The best known technique entails semi-infinite linear diffusion to a fiat plate, where the diffusion controlled current density is given by id/ A = n F c o (D/ltt) ~ * 1 ca1=--4.184 J.

(1)

282

K. KLINEDINST, J. A. S. BETT, J. MACDONALD, P. STONEHART

but this provides no means for separating the solubility and diffusivity terms. In addition, as shown by Bard 1, the method is sensitive to convection and requires extensive shielding and precise orientation. A second approach for obtaining solubility-diffusivity products entails controlling the hydrodynamics (thickness of the diffusion layer) by some mechanical means such as a rotating disc where the diffusion current density is given by 2 id/A = 1.61 (O/v) ~ (v/o~) ~ [ 1 + 0.35 (D/v) °'3 6] -1 Oc ° nF

(2)

Again, the solubility and diffusivity terms cannot be separated and the mechanical problems associated with sealing the disc into a mount to be used in the hot phosphoric acid environment make hydrodynamic techniques unattractive, even with other configurations such as the translating electrode of Bopp and Mason 3. Trace impurities 4- 6 in the phosphoric acid cause poisoning of the electrocatalyst and the poisoning is accelerated by increased mass transport, as has been described previously by Stonehart 7 for rotating disc electrodes, so that the use of enhanced mass transport systems is not acceptable. A third approach is to examine the diffusion controlled current density to the internal surface of an annulus of length L as shown by Klinedinst and Stevenson 8, where the diffusion current density is given by id = 4 nF Dc o ~L exp ( - 5.784 Dt/a 2)

(3)

Here, the solubility and diffusivity terms may be separated and although the technique is admirable for diffusion in metal-oxygen systems, it was found that phosphoric acid convection precluded use of the technique in this system. Finally, a complementary technique to the annular electrode was chosen 9, also entailing cylindrical diffusion but to a shielded wire electrode. Both the solubility and diffusivity terms can be determined from the experimental current-time parameters and convection of the electrolyte is minimized by an external shield and the presence of the wire electrode itself. Separation of the solubility and diffusivity parameters is described more fully in the following section. THEORETICAL CONSIDERATIONS

The general solution to the analogous heat conduction problem was first derived by Carslaw and Jaeger 1° and the solution to the mass transport problem was given by Crank 11. The amount of diffusing substance per unit area reaching the cylindrical wire in unit time is given by ~c - D

~r

4 (c 1 - c 0) D r=a -

~2~

e- o,2t o

du u [ j 2 (au) + y 2

(au)]

(4)

Jo and Y0 are, of course, the zero th order Bessel functions. For large times, the integral can be approximated as a power series expansion: e- Du2t o

du

~z2

u [ j 2 ( a u ) + y2(au)] = 2-

1 iog(aX)-2~

7 [log(4X)-2~] 2 - "'"

where X = D t / a z and y =0.57722 is Euler's constant. For intermediate times, an approximation of the integral equation is

(5)

283

OXYGEN SOLUBILITY AND DIFFUSIVITY I N H 3 P 0 ~

e-°"2t u[J2(au)+ Y~(au)] = 4

du

E(nX)-++½-¼

+ ~- - ...

(6)

At the shortest times, the abbreviated power series is

f~e_DU2t

du 7t2 u [j2 (au) + Yg (au)] = 4-- [½+ (nX)- ~]

(7)

In terms of the current arising from the electrochemical oxidation or reduction of the diffusing species, eqn. (4) becomest2 :

id 4nF(cl-co)D f f e_O,,2t du A n 2a u [jg (au) + Yg (au)]

(8)

This equation, with cl =0 and n---4, was used to abstract the oxygen diffusion coefficient, D, and solubility, c0, from the experimentally obtained current-time relationships. In particular, for all experimental times, the following form of eqn. (8) was found to represent accurately the data (substituting eqn. (6)) as given in ref. 13:

id = 8~LFDc o [ (reX)- ~ + ½- ¼ (X/n) ½+ X/8 - 0.1469X g + 0.203X 2 - 0.315X ~ + 0.536X 3 - . . . ]

(9)

This equation is solved for D and for the productDco (from which cocan be obtained) by fitting the experimental data (plotted as log id VS. log 0 to the general solution of eqn. (9) (plotted as log 0 vs. log X, where 0 = rcid/32LFDco) 9. Considering only the smallest times, the following solution of eqn. (8) (substituting eqn. (7)) was found to epresent accurately the experimental data.

id = 4z~LFOc o [1 + 2a(rcOt)-~]

(10)

EXPERIMENTAL

Concentrated phosphoric acid electrolyte was prepurified and contained within a Pyrex chamber (250 ml) with a hydrogen reference electrode contained within a capillary-tipped probe extending into the acid. A platinum foil served as the counter electrode. The working electrode, used to meastlre diffusion-limited oxygen reduction currents, was a platinum wire (0.38 mm diam.) sealed at the lower end into a glass bead and at the upper end into a 3 mm O.D. Pyrex tube, leaving 1.5-1.7 cm of the wire exposed to the phosphoric acid solution. The 3 mm Pyrex tube fitted snugly into a 3.4 mm I.D. Pyrex tube with a flared end to seat the glass bead at the electrode tip. By sliding the smaller tube within the larger tube, a sample of the phosphoric acid solution could be withdrawn into the volume between the wire and the outer tube, shielding it from the bulk of the solution and ensuring that no gross convective movement of the acid sample could occur. A heating tape powered by a proportional band temperature-controller was used to heat the cell. The electrode potential was controlled by a Wenking potentiostat, Model 70HC3 and cell currents were recorded on a Varian F-100 time-base recorder. Fisher 85 % phosphoric acid was purified by a three step process. It was first heated with hydrogen peroxide to oxidize some of the impurities. Secondly, it was circulated over a large platinum black surface, maintained at a low potential by bubbling hydrogen to remove adsorbable impurites. Thirdly, it was circulated over

284

K. KLINEDINST, J. A. S. BETT, J. MACDONALD, P. STONEHART

another large platinum black surface whilst bubbling oxygen to remove any impurities capable of adsorbing on the platinum surface at high potentials. The concentration of the acid was increased by evaporation of water and was decreased by the addition of triply-distilled water. The gases used in the experiments (argon and oxygen) were carefully humidified before entering the cell so that the acid concentrations would not change during the experimental determinations. This was accomplished by bubbling the gases through a large volume of water whose temperature was maintained to give an appropriate water vapor partial pressure 14. The gases were passed over the "surface of the acid (rather than bubbling through the acid) during the measurement periods so that the acid in the sample chamber was not disturbed by convective motion resulting from the bubbling but at the same time maintaining the partial pressures of the gases and water vapor over the electrolyte. The acid was heated to the desired temperature and saturated with oxygen by having the pre-humidified oxygen bubbling through it for about 1 h. The test electrode was cleaned by anodizing for a few seconds at 1.4 V, the bubbling stopped, and a sample of the oxygen-saturated acid withdrawn into the sample chamber. After the electrolyte sample was taken, the system was allowed to equilibrate for a few minutes in order to allow any residual convective movement of the acid sample to dampen out. The test electrode potential was then instantaneously set at a potential low enough to allow the measurement of the diffusion-limited oxygen reduction current (0.2 to 0.4 V vs. NHE) and the current recorded for a period of 2 min. In a number of cases, duplicate determinations were made at both 0.2 and 0.4 V, as well as at 0.3 V, in order to demonstrate that the measured currents were diffusion rather than activation limited. Identical current-time curves were required to prove the acceptability of the determination. Oxygen was removed from the acid by bubbling pre-humidified argon and a blank determination was made to see whether electroactive impurities present in the acid contribute~l to themeasured oxygen reduction currents. In every case, the background currents were negligible fractions of the currents measured with the oxygen-saturated acid. Finally, the acid concentrations were determined by specific gravity measurements 15. RESULTS

To test the sensitivity of the experimental method to changes in temperature, the i - t curves were measured as a function of temperature at constant acid concentration (95.7 wt. ~o HaPO4) (Fig. 1). As the temperature was varied from 100 to 150°C, sizable variations in the i-t curves were found. Similarly, the sensitivity of the method to changes in.acid concentration was examined by comparing the i-t curves measured at constant temperature (100°C) and the concentration was varied from 86 to 95.6 wt.~o H3PO4. Again, sizable differences between the i - t curves were seen indicating that the method is easily capable of distinguishing the effects of changing acid concentration as well as the effects of temperature variation for oxygen concentrations and electrolyte temperatures of interest in this work. For each acid concentration and temperature datum, triplicate determinations were made without any variation in the i - t parameters. The Dco2 and D values were obtained from the i-O and t - X relationships,

OXYGEN SOLUBILITY AND DIFFUSIVITY IN H3PO4

285

3O

-.......

=_,

10

1

2

3

4

TIME/S

Fig. 1. Diffusion-controlled oxygen reduction current vs. time. (l) 100°C, (2) I20°C, (3) 135°C, (4) 150°C.

0

0

0

0

0

% ..u.

{

[DO0

0

0

CI

O

I

I5 .

°,o

Fig. 2. Oxygen diffusion coefficient-solubility product (corrected to 1 atm. oxygen partial pressure) vs. HsPO ~ coneentration. (~) 100°C, (O) 120°C, (C3)135°C, (O) 150°C. Fig. 3. Oxygen diffusion coefficient vs. H3PO 4 concentration. (A) 100°C, (IS])120°C, (O) 150°C. respectively, resulting from the best fit of the data to the general solution of eqn. (9). The Dco2 v a l u e s w e r e corrected to one a t m o s p h e r e pressure assumin~ H e n r y ' s law and using the equilibrium water v a p o r ,pressure data of M c D o n a l d and Boyack 1°. The Dco~ values are plotted vs. wt. ~ H 3 P O 4 at constant t e m p e r a t u r e in Fig. 2. Best fit lines have been d r a w n through the points at each t e m p e r a t u r e and it can be

286

K. KLINEDINST, J. A. S. BETT, J. MACDONALD, P. STONEHART

seen that, to within experimental error, the Dco2 product varies linearly with acid concentration at constant temperature. The average deviation of the experimental points from the best fit lines is 3.5 ~ . The oxygen diffusivity values are shown in Fig. 3, also plotted vs. wt. ~ H 3 P O 4 at constant temperature. Again, best fit lines have been drawn through the points at each temperature and, as with the Dco2 values, the diffusion coefficient is seen to vary linearly with acid concentration. Due, however, to certain characteristics of the data reduction process, the average deviation of the latter data from the best fit lines was

5"11tI " 4.0

x~ \\\\

\

~,,,,

E uo 3.1

2.0 85

I 90

I

L

~ 3P04.1/ 95W.,r..o/°

IQO

Fig. 4. Oxygen solubility (correctedto 1 atm. oxygen partial pressure)vs. H 3PO4 concentration. (A) 100° C, ([3) 120°C, (O) 150°C, (11) 120°C (Ref. 16), (O) 150°C (Ref. 16). 8

3~

7.

2.G

E

,.J1.~

6.5

6 .ID 85

-1. 9/

/5

100

85

Fig. 5. Oxygen diffusion activation energy vs. H3PO4 concentration. Fig. 6. Oxygen enthalpy of solution vs. H3PO4 concentration.

OXYGEN SOLUBILITY AND DIFFUSIVITY IN H3PO 4

287

about 20 ~o, several times larger than the corresponding deviation found in the case of the Dco2 values. Oxygen solubility values, as a function of acid concentration and temperature, were calculated from the best fit lines drawn through the Dco~ and D values. These values refer to one atmosphere oxygen partial pressure and are plotted in Fig. 4. Smooth curves have been drawn through the constant temperature points. Also shown in the figure are the results reported by Gubbins and Walker 16. It is found that over sufficiently small temperature ranges, both gas diffusivities and gas solubilities exhibit exponential reciprocal temperature dependencies. It is expected, therefore, that the D and co2 values might vary with temperature according tO:

D = A exp(-E/RT)

(11)

Co2 = B exp ( - AH/R T)

02)

where E is the diffusion activation energy and AH is the enthalpy of solution. Linear variations of log D vs. 1/T were obtained from the data at different acid concentrations. Constant oxygen concentration lines were drawn and the diffusion activation energies calculated from the slopes of these lines are plotted as a function of acid concentration in Fig. 5 showing that the activation energy decreases by about 20 ~o as the acid concentration increases from 86 to 96 wt. ~ H3PO 4. Similarly, approximately linear variations of log Co2 vs. 1/T (at constant acid concentration) were obtained from the data of Fig. 4. Enthalpy of solution values calculated from the average slopes of the constant acid concentration lines are plotted in Fig. 6 as a function of acid concentration. It is evident that the enthalpy of solution decreases from around 2.5 kcal mol-1 at 86 wt.~ H3PO 4 to a value near zero at 96 w t . ~ H3PO 4. In addition to the dependence of the gas solubility upon the reciprocal of the

"7 x,"

IE

I

85

I 90

95

Fig. 7. Oxygen entropy of solution

100

vs.

H s P o , concentration.

288

K. KLINEDINST, J. A. S. BETT, J. MACDONALD,P. STONEHART

temperature, its dependence upon the logarithm of the temperature is also of interest 17, since:

As _ (0 In Co2 R

I13)

\O In T i p

where AS is the entropy of solution. Entropies of solution calculated from the average slopes of the log Co~vs. log T lines (at constant acid concentration) are plotted in Fig. 7 as a function of acid concentration. It is seen that the entropies are negative and, just as with the enthalpy of solution, the entropy decreases to a small value around 96 wt.~ H3P04. In the calculation of the thermodynamic quantities from the experimental data the activity of the gas was referred to the same standard state as the activity of the solute in solution and Henry's law was assumed. The enthalpy and entropy of solution values are, of course, assumed to be temperature independent to within experimental accuracy over the rang~ of temperatures examined. DISCUSSION The self-consistency of the data and the agreement between the diffusivity values calculated using eqn. (10) and those calculated using only eqn. (9) supported the accuracy of the results. In addition, the agreement between these results and the earlier, more limited values of Gubbins and Walker 16 suggests a high degree of ~onfidence in the derived oxygen solubility and diffusivity values. For instance, Gubbins and Walker reported an oxygen diffusivity of 2.2 x 10-6cm2 s- 1 in 85.0 wt. ~ IH3P O4 at a temperature of 60°C 16. For these conditions, a value of 2:0 × !0 -6 crn2 s ~ 1 was obtained from our data by extrapolation. Gubbins and Walker reported a value of 4.2 × 10-6 cm 2 s-1 for the diffusivity of oxygen in 85.0 wt. Y/ooH3PO~ at 83°C and our data yielded an extrapolated value of 4.3 x 10-6 c m 2 s - 1. While the solubility values reported earlier 16 do not have the same acid concentration and temperature dependencies as do the values reported here, they do nevertheless fall within the same ~ fairly narrow concentration range. It was previously pointed out that the diffusivity values derived from the experimental i-t curves show a significantly larger average deviation from the best-fit D vs. w t . ~ H~PO~ lines than do the corresponding diffusivity-solubility product values from the best-fit Dco2 vs. wt.~ H3PO4 lines (cf. Figs. 2 and 3). This situation results from the dependence of 0 upon X, as expressed by eqn. (9). While X changes by three orders of magnitude, 0 changes by only one order of magnitude. Due to this, the l o g / - l o g 0 relationship (from which the Dco2 value is calculated) can be more precisely specified than can the log X - t o g t relationship (used to calculate the D value), leading to a greater reliability for the Dco2 values than for the D values. The acid concentration dependence of the physical and chemical properties of phosphoric acid solutions have frequently been examined and it is well known that the percentage of orthophosphoric acid increases with nominal wt. ~ H3PO4 until a maximum concentration is reached at about 96 w t . ~ H3PO418. In general, the concentrations of condensed phosphate species increase above the'nominal 96 wt. ~ H3PO 4 concentration as the water and orthophosphoric acid concentrations decrease. Also, it has been reported recently that the rate of change of heat capacity

OXYGEN SOLUBILITY AND DIFFUSIVITY IN H3PO 4

289

with acid concentration shows a fairly abrupt discontinuity at about 96 wt. ~ H3 P 0419. The solubility of oxygen in phosphoric acid solutions becomes temperature independent at about 96 wt.~o HaPO4. Consequently, the enthalpy and entropy of solution approximate to zero at this acid concentration. While the meaning of these observations is still not clear, it does seem likely that the thermodynamic properties of theoxygen-phosphoricacid system are themselves functions of the structural and thermodynamic properties of the water-P20 5 system and that the concentration dependencies of the oxygen enthalpy and entropy of solution mirror the fundamental structural changes that occur in the acid as the nominal concentration approaches 96 wt. ~ H3PO 4. SUMMARY

The solubilities and diffusivities of oxygen in hot concentrated phosphoric acid solutions (100-150°C, 85-96 wt.~o HaPO4) were obtained from an analysis of the cylindrical diffusion current-time parameters to a wire electrode. Oxygen solubilities become temperature independent at about 96 w t . ~ H3PO 4 with the consequence that enthalpy and entropy terms become very small. In spite of the considerable enperimental difficulties of working with this system, the experimental results are very reproducible and self consistent. REFERENCES 1 A. J. Bard, Anal. Chem., 33 (1961) 11. 2 D. P. Gregory and A. C. Riddiford, J. Chem. Soc., (t956) 3756~ 3 G. R. Bopp and D. M. Mason, Electrochem. Tech., 2 (1964) 129; 4 S. B. Brummer, J. P. Ford and M. Sturmer, J. Phys. Chem., 69 (1965) 3434. 5 J. Bravaeos, M. Bonnemay, E. Levart and A. Pilla, C.R. Acad. Sci. Paris, 265 (1967) 337. 6 P. Stonehart and P. A. Zucks, Electrochim. Acta, 17 (1972) 2333. 7 P; Stonehart, Electrochim. Acta, 12 (1967) 1185. 8 K. Klinedinst and D. A. Stevenson, J. Electrochem. Soc., 120 (1973) 304: 9 0 . R . Brown, J. Electroanal. Chem., 34 (1972) 419. 10 H. S: Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, London, t959o pp. 335-336. 11 J. Crank, The Mathematics of Diffusion, Oxford University Press, London, 1964, p. 82. 12 P. Delahay, New Instrumental Methods in Electrochemistry, Interscience, New York, 1954, pp. 67-70. 13 J. C. Jaeger and M. Clarke, Proc. Roy. Soc, Ed~burgh, A61 (1942) 229. t4 D. I. MacDonald and J. R. Boyack, J. Chem. Eng. Data, 14 (1969) 380. 15 Monsanto Chemical Co., Technical Bulletin 1-239, St. Louis, Missouri, p. 2t. -.. 16 K: E. Gubbins and R. D. Walker, Jr., J. Electrochem. Soc., 112 (1965) 469; also Solubility and:Dif~sivity of Hydrocarbons and Oxygen in FuelCell Electrolytes, Final Report on Contract No. DA-49-186-AMC45(X), June 30, 1965, University of Florida, pp. 14 and 48. t 7 J. H. HiR!ebrand and R. L. Scott, Regular Solutions, Prentice-Hall, Englewood Cliffs, NA.I 1962, pp. 23-24. 18 R. F. Jameson~ J. Chem, Soc., (1959) 752. 19 Z. T. Wakefield, ]3. B: Luff and R.B. Reed; d: Ghem.:Eng. Data,: 17 (1972) ([20.