Densities and Apparent Molar Volumes of Atmospherically Important

ARTICLE pubs.acs.org/JPCA

Densities and Apparent Molar Volumes of Atmospherically Important Electrolyte Solutions. 2. The Systems Hþ-HSO4--SO42--H2O from 0 to 3 mol kg-1 as a Function of Temperature and Hþ-NH4þ-HSO4--SO42--H2O from 0 to 6 mol kg-1 at 25 C Using a Pitzer Ion Interaction Model, and NH4HSO4-H2O and (NH4)3H(SO4)2-H2O over the Entire Concentration Range S. L. Clegg*,†,‡ and A. S. Wexler‡ † ‡

School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, U.K. Air Quality Research Center, University of California at Davis, Davis, California 95616, United States ABSTRACT: A Pitzer ion interaction model has been applied to the systems H2SO4-H2O (0-3 mol kg-1, 0-55 C) and H2SO4-(NH4)2SO4-H2O (0-6 mol kg-1, 25 C) for the calculation of apparent molar volume and density. The dissociation reaction HSO4-(aq) T Hþ(aq) þ SO42-(aq) is treated explicitly. Apparent molar volumes of the SO42- ion at infinite dilution were obtained from part 1 of this work,1 and the value for the bisulfate ion was determined in this study from 0 to 55 C. In dilute solutions of both systems, the change in the degree of dissociation of the HSO4- ion with concentration results in much larger variations of the apparent molar volumes of the solutes than for conventional strong (fully dissociated) electrolytes. Densities and apparent molar volumes are tabulated. Apparent molar volumes calculated using the model are combined with other data for the solutes NH4HSO4 and (NH4)3H(SO4)2 at 25 C to obtain apparent molar volumes and densities over the entire concentration range (including solutions supersaturated with respect to the salts).

1. INTRODUCTION In part 1 of this work1 we develop equations for apparent molar volumes and densities of aqueous solutions of several electrolytes, including sulfuric acid, from 0 to 60 C and with extrapolations to low temperatures and to very high concentrations including the hypothetical liquid melts. For aqueous H2SO4 (and many other solutes) data are sparse and quite often inaccurate for the very dilute solutions for which small errors in measured densities result in large errors in the apparent molar volume. Apparent molar volumes of H2SO4 also vary very steeply with molality in dilute solutions because of the effect of the varying dissociation of the HSO4- ion (e.g., see Figure 7 of Clegg et al.2). This is also true of the apparent molar volumes of acid sulfates, such as NaHSO4,3 and of the mixtures of aqueous (NH4)2SO4 with H2SO4 that are of interest in this work. In order to determine accurate values of apparent molar volumes of H2SO4 in dilute aqueous solutions, we have applied a Pitzer ion interaction model4 to apparent molar volumes of H2SO4-H2O from 0 to 60 C and 0 to 3 mol kg-1 and from 0 to 6 mol kg-1 at 25 C, representing explicitly the effect of HSO4dissociation. We also fitted the equations to (NH4)2SO4-H2O from 0 to 8 mol kg-1 at 25 C, and then developed a model of r 2011 American Chemical Society

mixture Hþ-NH4þ-HSO4--SO42--H2O for all compositions at 25 C and from 0 to 6 mol kg-1 total solute molality. The results obtained here are used in part 1 to develop a treatment of density from 0 to 100 wt % H2SO4, and in this work are combined with other data to obtain values of the apparent molar volumes of aqueous ammonium bisulfate (NH4HSO4) and letovicite ((NH4)3H(SO4)2) over the entire concentration range at 25 C.

2. THEORY Relationships between apparent molar volumes V φ (cm3 mol-1) and densities F (g cm-3) are given in section 2 of part 1. Densities of water used in this work were calculated using eq 16 of Kell,5 and all densities given in units of g mL-1 or kg L-1 were converted to g cm-3 where appropriate using the relationship 1.0 mL (old) = 1.000028 cm3.6 The apparent molar volume of the solute in an aqueous solution can be expressed in terms of the differential of the excess Received: September 20, 2010 Revised: December 16, 2010 Published: March 25, 2011 3461

dx.doi.org/10.1021/jp1089933 | J. Phys. Chem. A 2011, 115, 3461–3474

The Journal of Physical Chemistry A

ARTICLE

Table 1. Sources of Data for Aqueous (NH4)2SO4-H2SO4 Mixtures conc. range

unit

compositiona

t range (C)

quantityb

unit

used

note

ref

18.09-61.86

wt %

a

25

F

g cm-3

yes

40-97

wt %

a

25

F

g cm-3

yes

66.67

wt %

a

25

F(t, 25)

0.225-8.013

mol dm-3

a

25

F

g cm-3

no

d

19.0-61.5

wt %

a

25

F

g cm-3

yes

e

0.5707-8.463

mol kg-1

b

25

F

g cm-3

yes

40-78

wt%

b

25

F

g cm-3

yes

10-60 10-60

wt % wt %

c1 c2

-30 to 80 -30 to 80

F F

g cm-3 g cm-3

yes yes

19 19

20-40

wt %

a

-30 to 80

F

g cm-3

yes

19

20-40

wt %

b

-20 to 80

F

g cm-3

yes

19

100

wt %

a

25

F

g cm-3

yes

18

17 (pycnometer) c

yes

17 21 20 20 17 (pycnometer)

c

17

Compositions are as follows: a, NH4HSO4; b, (NH4)3H(SO4)2; c1, nNH4/(nNH4 þ nH) = 0.167; c2, nNH4/(nNH4 þ nH) = 0.333. b F(t,u) is the density of the solution at t C relative to that at u C. c Tang and Munkelwitz17 present only a fitted equation. The value of the density of pure water is given as 0.9971, suggesting units of g mL-1 (old), although this seems unikely given the date of publication. However, the equation is used here only to obtain densities of supersaturated solutions for which the molality is high enough that uncertainty of the exact unit used is not significant. d Quoted by Stelson and Seinfeld20 as a personal communication from Irish et al.; see text. e Data of Tang, interpolated from Figure 3 of Stelson and Seinfeld.20 a

Figure 1. Apparent molar volumes of the SO42- ion at infinite dilution (Vφ¥(2Hþ, SO42-)), as a function of temperature (t). Key: square, eq 15a (NaCl path); dot, eq 15b (NH4Cl path); plus, eq 15c (NH4NO3 path); line, eq 16. The vertical lines on the plot represent the upper and lower limits of predictions of eq 4 of Millero9 (stated uncertainty (0.3 cm3 mol-1). Inset: differences between values of Vφ¥(2Hþ, SO42-) calculated by eq 15 and eq 16. Key: dashed line, eq15a - eq16; solid line, eq15b - eq16; dash-dot line, eq15c - eq16.

Gibbs energy (Gex) of the solution with respect to pressure (P) V φ ¼ V φ¥ þ ð1=ns Þð∂Gex =∂PÞT , nW

ð1Þ

where V φ¥ (cm3 mol-1) is the value of the apparent molar volume at infinite dilution and ns is the number of moles of solute. In the partial differential the temperature (T) and number of moles of water solvent (nW) are held constant. The excess Gibbs energy per unit mass of water (wW/kg) can be written in

Figure 2. Apparent molar volumes of the HSO4- ion at infinite dilution (Vφ¥(Hþ, HSO42-)), as a function of temperature (t). Key: dot, this work (Pitzer model fits, see text); circle, Hovey and Hepler;10 line, eq 17.

terms of the stoichiometric mean activity coefficient of H2SO4 (γ() and the osmotic coefficient of the solution (φst)2,4 Gex =ðwW RTÞ ¼ 3mH2 SO4 ðln γ ( þ φst Þ

ð2Þ

where mH2SO4 (mol kg-1) is the molality of the acid, and γ( is related to the conventional single ion activity coefficients (γ) and molalities (m) of Hþ and SO42- by γ ( ¼ ½ðmHþ γHþ Þ2 mSO4 2 - γSO4 2 - =ð4mH2 SO4 3 Þ1=3 ð3Þ The stoichiometric osmotic coefficient of the solution is related to the water activity (aw) and the osmotic coefficient (φ) on a 3462

dx.doi.org/10.1021/jp1089933 |J. Phys. Chem. A 2011, 115, 3461–3474


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Table 2. Fitted Pitzer Volume Parameters for the Hþ-HSO4--SO42--H2O and Hþ-NH4þ-HSO4--SO42--H2O Systemsa 0-3 mol kg-1 aqueous H2SO4, 0-55 C H-HSO4 parameter

b

H-SO4 value

parameter

25 C ∂/∂T ∂2/∂T2

β

0V

-4

-0.13986503 10 -0.38748731 10-5 0.52339211 10-7

25 C ∂/∂T ∂2/∂T2

β1V

25 C ∂/∂T ∂2/∂T2

Cφ0V

25 C ∂/∂T

Cφ1V

b

value

25 C ∂/∂T ∂2/∂T2

β

0V

0.82133713 10-4 -0.91582790 10-5 0.66837786 10-6

0.25043118 10-3 -0.15658368 10-4 0.44492276 10-6

25 C ∂/∂T ∂2/∂T2 ∂3/∂T3

β1V

-0.20136104 10-3 0.95233864 10-4 0.64832464 10-6 -0.71906477 10-7

0.16304210 10-4 0.36764430 10-5 -0.82156159 10-7

25 C ∂/∂T ∂2/∂T2

Cφ0V

-0.45307591 10-4 -0.51957489 10-5 -0.30955817 10-6

0.30416476 10-5 0-6 mol kg-1 aqueous H2SO4, 25 C

parameter

value

β β1V Cφ0V Cφ1V

NH4-SO4

H-SO4

H-HSO4 b

0-8 mol kg-1 (NH4)2SO4, 25 C

parameter

0V

b

value

β β1V Cφ0V Cφ1V

0.23439654 10-3 0.37890784 10-5 -0.31924735 10-3

parameter

0.10531962 10

0V

-3

β β1V C0V C1V 0V

-0.33848198 10-4

b

value 0.23301748 10-4 0.13810648 10-3 -0.57461412 10-6 0.10213613 10-4

0-6 mol kg-1 aqueous H2SO4-(NH4)2SO4 mixtures, 25 C NH4-HSO4 parameter

b

β0V β1V Cφ0V Cφ1V

ternary interactions b

value

parameter

-0.278555 10-5

ΨVHSO4,SO4,NH4 ΘVH,NH4

-0.270581 10-6 0.608890 10-3

value 0.97657 10-5 -0.11623 10-4

This table contains all Pitzer model volume parameters determined in this study. For the 0-55 C model of dilute aqueous H2SO4, the table lists the value of the parameter at 25 C, and up to three differentials of the parameter with respect to temperature (also at 25 C) from which the value of the parameter at any temperature can be calculated. Where there are blanks, the parameter has a value of zero. b Parameters Cφ0V and Cφ1V, and their differentials, for the interaction of cation M and anion X are related to the equivalent parameters C0V and C1V by: CφV = CV 3 2(zMzX)1/2, where zM and zX are the magnitudes of the charges on the ions. a

free ion basis by lnðaw Þ ¼ - ðMw =1000Þ3mH2 SO4 φst φ ¼ 3mH2 SO4 φst =ðmHþ þ mHSO4 - þ mSO4 2 - Þ

ð4aÞ

are given in Table 6 of Clegg et al.2 For any temperature and total molality of H2SO4 the molalities of the individual ionic species are obtained by iterating to satisfy the following equation Kd ¼ aHþ aSO4 2 - =aHSO4 ¼ mHþ mSO4 2 - =mHSO4 - γHþ γSO4 2 - =γHSO4 -

ð4bÞ

where Mw (18.0152 g mol-1) is the molar mass of water. The equation for the osmotic coefficients arises from the fact that the water activity of the solution is the same, however the concentration is expressed, and a similar argument applies to the activities of the solutes. See Robinson and Stokes7 for a formal definition of mean activity coefficients and their relationship to those of the ions. Both ln γ( and φst can be calculated using the Pitzer ion interaction model using the relationships above and eqs 3-7 from the study of Clegg et al.2 The Hþ, HSO4-, and SO42molalities determined in that work are based upon available activity, enthalpy, heat capacity, and degree of dissociation data for 0-6.0 mol kg-1 acid and for temperatures from 0 to 55 C. The fitted interaction parameters in the model equations

ð5Þ where Kd (mol kg-1), the dissociation constant of H2SO4, is given by eq 21 of Clegg et al.2 Finally, the Gex/(wWRT) in eq 1 and the differentials with respect to pressure of the ion interaction parameters are obtained by numerical differentiation of eq 2. Thus, for example, the value of parameter β0H,HSO4 in the activity coefficient equations at some fixed temperature and pressure, P, is given by β0 H, HSO4 ðPÞ ¼ β0 H, HSO4 ðPr Þ þ ðP - Pr Þβ0V H, HSO4

ð6Þ

where Pr is the reference pressure and β0VH,HSO4 is the partial differential of β0H,HSO4 with respect to pressure P. Analogous relationships apply to the other ion interaction parameters (see 3463



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Figure 3. √ (a) Measured and fitted stoichiometric apparent molar volumes (Vφ) of H2SO4 from 0 to 55 C, plotted against the square root of H2SO4 molality ( m). All apparent molar volumes are offset by an amount (t/t - 25) cm3 mol-1, where t (C) is temperature and t is equal to 1 C. The experimental temperatures (C) are noted on the figure. Key: dot, International Critical Tables; square, Hovey and Hepler;10 circle, Myhre et al.,25 Lindstrom and Wirth,3 Larson et al.,26 Rhodes and Barbour,27 Klotz and Eckert,28 Campbell et al.,29 Joshi and Kandpal,30 Tollert.31 (b) √ The difference between measured and fitted apparent molar volumes (ΔVφ, observed - calculated), as a function of the square root of H2SO4 molality ( m). Key: the symbols have the same meanings as in (a), with the exception of: diamond, Myhre et al.;25 circle, Lindstrom and Wirth;3 star, Larson et al.;26 triangle, Rhodes and Barbour;27 inverted solid triangle, Klotz and Eckert;28 cross, Campbell et al.;29 star, Joshi and Kandpal;30 plus, Tollert.31 The dotted lines show the effect of a (0.00005 g cm-3 error in a measured density. (c) Same as (b), except plotted against temperature (t).

Table 6 of Clegg et al.2 for a full list). Values of these differentials with respect to pressure are obtained, as functions of temperature, by fitting to apparent molar volume data. The equation for the apparent molar volume of H2SO4 in a pure aqueous H2SO4 solution is V φ ðH2 SO4 Þ ¼ ð1 - RÞV φ¥ ðHþ , HSO4 - Þ þRV φ¥ ð2H þ , SO4 2 - Þ þ ð1=mH2 SO4 ÞRTð∂ðGex =wW Þ=∂PÞT , nW

ð7Þ

where symbols V φ¥(2Hþ, SO42-) and V φ¥(Hþ, HSO4-) are the apparent molar volumes at infinite dilution of the pairs of ions (2Hþ, SO42-) and (Hþ, HSO4-). The equations and parameters for aqueous (NH4)2SO4 (yielding NH4þ-SO42- interactions in the model) are given in section 5 below, and those for aqueous H2SO4 (yielding Hþ-HSO4- and Hþ-SO42- interactions) in section 4. The equation used for the calculation and fitting of apparent molar volumes of the aqueous acid ammonium sulfate mixtures is once again based upon the excess Gibbs energy of the solution per kg 3464



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of liquid water4 Gex =wW ¼ RT

X i

mi ð1 - φ þ ln γi Þ

ð8Þ

where mi is the molality of ion i, γi is its activity coefficient, φ is the osmotic coefficient of the solution, and wW is the mass of water in kilograms. The apparent molar volume of the solution V φ is defined in terms of the moles of solute (NH4)xH(2-x)SO4, and the Gibbs energy of mixing is on a stoichiometric or total basis (in terms of the ions Hþ, NH4þ, and SO42- only). Equation 8 can be written for this system þ Gex =wW ¼ RT½ðmHðTÞ þ mNH4 þ þ þ þ mSO4 2ðTÞ Þð1 - φst Þ þ mHðTÞ ln γHðTÞ þ þ mNH4 þ ln γNH4 ðTÞ þ mSO4 2ðTÞ ln γSO4 2ðTÞ

ð9Þ

where φst is the stoichiometric osmotic coefficient and the molalities of the ions Hþ and SO42- and their activity coefficients are both total values (neglecting formation of HSO4-). This equation can be rewritten in terms of the molality of the H2SO4 (mA) and the (NH4)2SO4 (mS) and their stoichiometric mean activity coefficients γA and γS Gex =wW ¼ 3RTððmA þ mSÞð1 - φst Þ þ mA 3 ln γA þ mS 3 ln γSÞ

ð10Þ

The Pitzer model of this system yields conventional single ion activity coefficients γHþ, γNH4þ, γHSO4-, and γSO42and an osmotic coefficient (φ) on a free ion basis. These quantities are related to those in the equation above as follows φst ¼ φðmHþ þ mHSO4 þ mSO4 2 - þ mNH4 þ Þ=ð3mA þ 3mSÞ γA ¼ ½ðmHþ γHþ Þ2 mSO4 2 - γSO4 2 - =ð4mA3 Þ1=3

ð11aÞ ð11bÞ

γS ¼ ½ðmNH4 þ γNH4 þ Þ2 mSO4 2 - γSO4 2 - =ð4mS3 Þ1=3 ð11cÞ where the molalities of Hþ and SO42- are both free values, taking into account HSO4- formation. Finally, the apparent molar volume of the solute (NH4)xH(2-x)SO4, V φ, is given by

Figure 4. The difference between densities of aqueous H2SO4, (NH4)2SO4, and Na2SO4 at 25 C and values predicted using the DebyeH€uckel limiting law expression in eq 10 of part 1 of this work (together with the appropriate values of the apparent molar volumes at infinite √ dilution) (ΔF) plotted as a function of the square root of molality ( m). Note that the apparent molar volumes obtained from eq 10 in part 1 can be converted to densities using eq 1a or eq 1b of the same work.

3. DATA Sources of data for densities of aqueous solutions of the electrolytes are listed in Table 6 (H2SO4) and Table 8 ((NH4)2SO4) of part 1 of this work. Sources of densities for Hþ-NH4þ-HSO4--SO42--H2O solutions are listed in Table 1 above, including those for ammonium bisulfate (NH4 HSO4 ) and letovicite ((NH 4 )3 H(SO 4 )2 or (NH 4 )1.5 H 0.5 SO 4 ) which are the most commonly studied mixtures. The general treatment and fitting of the data are as described in section 5 of part 1.

V φ ¼ ½mSO4 2 - V φ¥ ð2Hþ , SO4 2 - Þ

4. DILUTE AQUEOUS H2SO4 FROM 0 TO 55 C Experimental molar volumes V φ(H2SO4) can be expressed as the sums of the apparent molar volumes of the two pairs of ions (2Hþ, SO42-) and (Hþ, HSO4-) present in solution10

þ mHSO4 - V φ¥ ðHþ , HSO4 - Þ þ mNH4 þ V φ¥ ðNH4 þ Þ

V φ ðH2 SO4 Þ ¼ ð1 - RÞV φ ðHþ , HSO4 - Þ þ RV φ ð2Hþ , SO4 2 - Þ

þ RTð∂ðGex =wW Þ∂PÞT , nW =ðmA þ mSÞ

ð13Þ

ð12Þ

where the differential is with respect to pressure, P, at constant composition and temperature, and V φ¥(NH4þ) is the conventional apparent molar volume of the NH4þ ion at infinite dilution. This is obtained by recalling that V φ¥(Hþ) is zero by definition and, therefore, V φ¥(NH4þ) is equal to (V φ¥((NH4)2SO4) - V φ¥(2Hþ, SO42-))/2, which is equal to 18.02 cm3 mol-1 at 25 C. This value is very close to the 17.9 cm3 mol-1 recommended by Roux et al.8 and 17.87 cm3 mol-1 given by Millero.9

φ

where V (H2SO4) is the stoichiometric apparent molar volume calculated from solution density using eq 1 of part I of this work1 and R is the degree of dissociation which is equal to mSO42-/(mSO42- þ mHSO4-). Changes in the degree of dissociation of the HSO4- ion at low molalities are reflected in a more than factor of 2 difference between V φ(H2SO4) at infinite dilution and the value at 0.25 mol kg-1; see Figure 26 of part 1.1 Hovey and Hepler10 summarized previous determinations of the infinite dilution apparent molar volumes V φ¥(2Hþ, SO42-) 3465



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Table 3. Calculated Densities (F) and Apparent Molar Volumes of Aqueous H2SO4 from 0 to 50 Ca t

F

Vφ

t

F

Vφ

t

F

Vφ

t

F

Vφ

0.0

0

0.9998395

8.930

20

0.9982042

13.662

30

0.9956473

14.566

50

0.9880363

14.466

0.00050

0

0.9998836

9.907

20

0.9982455

15.207

30

0.9956878

16.652

50

0.9880758

18.319

0.0010

0

0.9999270

10.606

20

0.9982859

16.275

30

0.9957270

18.044

50

0.9881131

20.588

0.0020

0

1.0000122

11.695

20

0.9983644

17.851

30

0.9958029

20.015

50

0.9881843

23.460

0.0050

0

1.0002606

13.836

20

0.9985906

20.682

30

0.9960198

23.345

50

0.9883858

27.663

0.0100

0

1.0006608

15.925

20

0.9989522

23.170

30

0.9963650

26.089

50

0.9887058

30.664

0.0200

0

1.0014354

18.247

20

0.9996498

25.676

30

0.9970305

28.699

50

0.9893254

33.201

0.0500 0.100

0 0

1.0036694 1.0072572

21.389 23.717

20 20

1.0016654 1.0049217

28.683 30.632

30 30

0.9989574 1.0020836

31.620 33.365

50 50

0.9911345 0.9940971

35.682 36.956

0.200

0

1.0141799

25.997

20

1.0112568

32.335

30

1.0081987

34.763

50

0.9999434

37.825

0.300

0

1.0209021

27.289

20

1.0174484

33.236

30

1.0142046

35.447

50

1.0057221

38.194

0.400

0

1.0274924

28.162

20

1.0235373

33.836

30

1.0201301

35.882

50

1.0114428

38.415

0.500

0

1.0339835

28.801

20

1.0295425

34.277

30

1.0259871

36.193

50

1.0171078

38.572

0.600

0

1.0403931

29.295

20

1.0354746

34.622

30

1.0317818

36.434

50

1.0227170

38.699

0.700

0

1.0467298

29.694

20

1.0413399

34.903

30

1.0375173

36.629

50

1.0282704

38.809

0.800 0.900

0 0

1.0529977 1.0591976

30.029 30.320

20 20

1.0471423 1.0528844

35.140 35.345

30 30

1.0431952 1.0488163

36.795 36.941

50 50

1.0337677 1.0392088

38.909 39.003

1.00

0

1.0653291

30.581

20

1.0585678

35.526

30

1.0543810

37.072

50

1.0445936

39.094

1.10

0

1.0713908

30.822

20

1.0641936

35.690

30

1.0598893

37.194

50

1.0499226

39.181

1.20

0

1.0773810

31.048

20

1.0697625

35.840

30

1.0653410

37.309

50

1.0551962

39.267

1.30

0

1.0832979

31.266

20

1.0752750

35.979

30

1.0707362

37.419

50

1.0604152

39.351

1.40

0

1.0891397

31.477

20

1.0807313

36.110

30

1.0760748

37.525

50

1.0655805

39.434

1.50

0

1.0949048

31.685

20

1.0861317

36.234

30

1.0813570

37.628

50

1.0706933

39.514

1.60 1.70

0 0

1.1005921 1.1062004

31.891 32.095

20 20

1.0914764 1.0967654

36.353 36.469

30 30

1.0865829 1.0917529

37.729 37.828

50 50

1.0757548 1.0807664

39.592 39.668

1.80

0

1.1117292

32.299

20

1.1019986

36.581

30

1.0968678

37.926

50

1.0857294

39.741

1.90

0

1.1171787

32.503

20

1.1071761

36.690

30

1.1019281

38.022

50

1.0906450

39.812

2.00

0

1.1225497

32.706

20

1.1122978

36.797

30

1.1069349

38.116

50

1.0955144

39.880

2.10

0

1.1278204

32.918

20

1.1173452

36.912

30

1.1118609

38.222

50

1.1003321

39.949

2.20

0

1.1330596

33.109

20

1.1223278

37.028

30

1.1167362

38.324

50

1.1051020

40.017

2.30

0

1.1382600

33.285

20

1.1272768

37.133

30

1.1215913

38.411

50

1.1098323

40.081

2.40 2.50

0 0

1.1433981 1.1484743

33.456 33.623

20 20

1.1321745 1.1370210

37.236 37.337

30 30

1.1263983 1.1311572

38.495 38.578

50 50

1.1145168 1.1191553

40.144 40.206

2.60

0

1.1534885

33.788

20

1.1418161

37.435

30

1.1358680

38.660

50

1.1237478

40.267

2.70

0

1.1584412

33.949

20

1.1465602

37.533

30

1.1405309

38.740

50

1.1282943

40.329

2.80

0

1.1633327

34.108

20

1.1512534

37.629

30

1.1451458

38.819

50

1.1327949

40.390

2.90

0

1.1681636

34.265

20

1.1558959

37.724

30

1.1497131

38.897

50

1.1372499

40.451

3.00

0

1.1729344

34.419

20

1.1604878

37.818

30

1.1542329

38.975

50

1.1416596

40.512

m

m

t

F

Vφ

t

F

Vφ

t

F

Vφ

0.0 0.00050

10 10

0.9996997 0.9997422

11.862 13.050

25 25

0.9970449 0.9970858

14.213 16.003

40 40

0.9922158 0.9922558

14.777 17.624

0.0010

10

0.9997838

13.889

25

0.9971255

17.221

40

0.9922940

19.427

0.0020

10

0.9998655

15.165

25

0.9972027

18.984

40

0.9923674

21.851

0.0050

10

1.0001020

17.577

25

0.9974241

22.060

40

0.9925759

25.673

0.0100

10

1.0004819

19.821

25

0.9977773

24.679

40

0.9929071

28.608

0.0200

10

1.0012162

22.203

25

0.9984582

27.244

40

0.9935462

31.239

0.0500

10

1.0033357

25.251

25

1.0004273

30.218

40

0.9954032

33.986

0.100 0.200

10 10

1.0067490 1.0133581

27.380 29.374

25 25

1.0036148 1.0098328

32.069 33.619

40 40

0.9984306 1.0043828

35.499 36.606

0.300

10

1.0197905

30.486

25

1.0159249

34.411

40

1.0102530

37.103

0.400

10

1.0260998

31.243

25

1.0219258

34.927

40

1.0160594

37.402

0.500

10

1.0323121

31.807

25

1.0278508

35.301

40

1.0218079

37.610

0.600

10

1.0384422

32.249

25

1.0337083

35.592

40

1.0275008

37.771

3466



ARTICLE

Table 3. Continued t

F

Vφ

t

F

Vφ

t

F

Vφ

0.700

10

1.0444989

32.610

25

1.0395030

35.829

40

1.0331387

37.903

0.800 0.900

10 10

1.0504874 1.0564110

32.914 33.176

25 25

1.0452377 1.0509141

36.030 36.204

40 40

1.0387215 1.0442487

38.019 38.124

1.00

10

1.0622718

33.408

25

1.0565332

36.359

40

1.0497200

38.223

1.10

10

1.0680710

33.616

25

1.0620955

36.500

40

1.0551348

38.318

1.20

10

1.0738094

33.807

25

1.0676013

36.631

40

1.0604928

38.411

1.30

10

1.0794873

33.984

25

1.0730509

36.755

40

1.0657939

38.503

1.40

10

1.0851051

34.150

25

1.0784444

36.872

40

1.0710381

38.594

1.50

10

1.0906627

34.308

25

1.0837818

36.984

40

1.0762259

38.684

1.60 1.70

10 10

1.0961600 1.1015967

34.460 34.606

25 25

1.0890634 1.0942894

37.093 37.199

40 40

1.0813579 1.0864352

38.774 38.862

1.80

10

1.1069724

34.748

25

1.0994600

37.303

40

1.0914590

38.949

1.90

10

1.1122868

34.887

25

1.1045755

37.405

40

1.0964307

39.035

2.00

10

1.1175394

35.025

25

1.1096363

37.505

40

1.1013520

39.118

2.10

10

1.1226891

35.179

25

1.1146221

37.613

40

1.1061952

39.213

2.20

10

1.1277689

35.334

25

1.1195508

37.720

40

1.1109943

39.303

2.30

10

1.1328331

35.466

25

1.1244502

37.815

40

1.1157764

39.377

2.40 2.50

10 10

1.1378412 1.1427933

35.596 35.723

25 25

1.1293001 1.1341005

37.908 37.999

40 40

1.1205123 1.1252019

39.449 39.519

2.60

10

1.1476896

35.848

25

1.1388514

38.088

40

1.1298452

39.589

2.70

10

1.1525304

35.971

25

1.1435530

38.176

40

1.1344422

39.658

2.80

10

1.1573157

36.093

25

1.1482053

38.262

40

1.1389930

39.726

2.90

10

1.1620460

36.213

25

1.1528086

38.348

40

1.1434978

39.793

3.00

10

1.1667217

36.331

25

1.1573630

38.433

40

1.1479568

39.860

m

a

Units and symbols: m (mol kg-1), molality of H2SO4; t (C), temperature; F (g cm-3), density; V φ (cm3 mol-1), apparent molar volume of H2SO4.

and V φ¥(Hþ, HSO4-) and used their own measurements up to 1.0 mol kg-1 and an extended Debye-H€uckel expression to redetermine V φ(Hþ, HSO4-) from 10 to 55 C. The molalities of Hþ, HSO4-, and SO42- present at each total concentration and temperature were calculated using the model of Pitzer et al.11 Values of V φ¥(2Hþ, SO42-) were obtained by difference from the known apparent molar volumes at infinite dilution of other electrolytes V φ¥ ð2Hþ , SO4 2 - Þ ¼ V φ¥ ðNa2 SO4 Þ - 2V φ¥ ðNaClÞ þ 2V φ¥ ðHClÞ φ¥

þ

ð14Þ

A further estimate of V (2H , SO4 ) at 25 C was also made by Hovey and Hepler10 using eq 14 and values of V φ¥(K2SO4) and V φ¥(KCl) in place of those for the sodium salts. In our analysis we have determined V φ¥(2Hþ, SO42-) in a similar way, but as the average of three different paths and using apparent molar volumes at infinite dilution from our own critical evaluations elsewhere in this work 2-

V φ¥ ð2Hþ ; SO4 2- Þ ¼ V φ¥ ðNa2 SO4 Þ - 2V φ¥ ðNaClÞ þ 2V φ¥ ðHClÞ

ð15aÞ ¼ V φ¥ ððNH4 Þ2 SO4 Þ - 2V φ¥ ðNH4 ClÞ þ 2V φ¥ ðHClÞ ð15bÞ ¼ V φ¥ ðNa2 SO4 Þ - 2V φ¥ ðNaNO3 Þ þ 2V φ¥ ðHNO3 Þ φ¥

þ

2-

ð15cÞ

Values of V (2H , SO4 ) determined from each path are shown in Figure 1, together with values calculated using the

following fitted equation V φ¥ ð2Hþ , SO4 2 - Þ ¼ 14:21264 - 1936:9962 lnðT=Tr Þ - ð4:572729 105 Þð1=T - 1=Tr Þ þ 0:00241865ðT 2 - Tr 2 Þ ð16Þ

where T (K) is temperature and Tr is equal to 298.15 K. Next, we used eq 13 to fit densities from sources in Table 12 of part 1 from 0 to 3 mol kg-1 at individual temperatures from 0 to 55 C and with values of R at each concentration calculated using the model of Clegg et al.2 This enabled us to obtain values of V φ¥(Hþ, HSO4-) at each temperature and first estimates of the Pitzer model volume parameters for the interactions Hþ-SO42and Hþ-HSO4-. The values of V φ¥(Hþ, HSO4-) are shown in Figure 2, and are fitted by the equation V φ¥ ðHþ , HSO4 - Þ ¼ 36:495 - 7:140366ðT - Tr Þ þ 0:0257307ðT 2 - Tr 2 Þ - ð3:01625 10-5 ÞðT 3 - Tr 3 Þ ð17Þ where T (K) is temperature and Tr is equal to 298.15 K. The results of Hovey and Hepler10 (their Table 2) are also shown in Figure 2 and agree with eq 17 to within (-0.59, þ0.28) cm3 mol-1. Our interest is first of all in the 0-3 mol kg-1 molality range. All the data used in the individual determinations of V φ¥(Hþ, HSO4-) were next fitted simultaneously using eq 7 (with R allowed to vary in the partial differential), yielding the parameters listed in Table 2. Measured and fitted apparent molar volumes agree satisfactorily and are shown in Figure 3. Data for very low concentrations at 25 C are plotted in Figure 4 as differences between actual densities and those predicted for a 2:1 electrolyte 3467



ARTICLE

√ Figure 5. (a) Measured and fitted apparent molar volumes (V φ) of (NH4)2SO4 at 25 C, as a function of the square root of molality ( m). The data (see Table 17 of part 1 of this work) were fitted with the Pitzer model equation4 given by eq 8 in part 1. Key: plus, Tang and Munkelwitz17 (electrodynamic balance measurements); cross, Hervello and Sanchez;32 solid diamond, Fang et al.;33 circle, Albright et al.;34 solid triangle, Goldsack and Franchetto;35 line, the fitted model. The solid vertical arrow indicates a solution saturated with respect to (NH4)2SO4(s). (b) The difference between √ measured and fitted apparent molar volumes (ΔV φ, observed - calculated), plotted against the square root of molality of (NH4)2SO4 in solution ( m). The symbols have the same meanings as in (a); dotted lines, the effect of a (0.00005 g cm-3 √ error in measured density. (c) Measured and calculated densities (F) of aqueous (NH4)2SO4, plotted against the square root of (NH4)2SO4 molality ( m). Key: the symbols, line, and vertical arrow have the same meanings as in (a).

by the equation for the Debye-H€uckel limiting slope for volume. At 0.25 mol kg-1 the ΔF for H2SO4 is about a factor of 10 greater than for Na2SO4 and (NH4)2SO4. Apparent molar volumes and densities for aqueous H2SO4 calculated by eq 7, using parameters in Table 2, are listed in Table 3. Note that the listed values are given to two digits more than the likely uncertainty. A Pitzer model of apparent molar volume is also needed for the treatment of densities of aqueous Hþ - NH4þ - HSO4- - SO42solutions at 25 C described below. We therefore repeated the fit

described above for the larger molality range 0 - 6 mol kg-1 at this temperature. The model parameters are listed in Table 2.

5. Hþ-NH4þ-HSO4--SO42--H2O Stelson and Seinfeld12 have reviewed the data for densities of aqueous NH4HSO4 up to 1980 and identified a number of inconsistencies and errors in earlier work. First, the molar concentrations of NH4HSO4 and water listed in Table 1 of Irish and Chen,13 which were calculated from their own measured 3468



ARTICLE

Figure 6. (a) Measured and fitted apparent molar volumes of√(NH4)2-xHxSO4 in aqueous H2SO4-(NH4)2SO4 mixtures at 25 C, as a function of the square root of the total molality of H2SO4 þ (NH4)2SO4 ( mT). The molar ratio H2SO4:(NH4)2SO4 is indicated on the plot for each set of data. Apparent molar volumes of both solutes in pure aqueous solution are also shown. Key: square, Tang and Munkelwitz17 (pycnometric measurements); circle, Tang17 (electrodynamic balance measurements); cross, Perkin;21 solid square, Tang (interpolated from Figure 2 of Stelson and Seinfeld);12 solid triangle, Semmler et al.;19 star, Tang and Munkelwitz17 (pycnometric measurements); circle, Tang and Munkelwitz17 (electrodynamic balance measurements); lines, the fitted model. (b) The difference between measured and fitted apparent molar volumes (ΔV φ, observed - calculated), plotted √ against the square root of total solute molalty in solution ( mT). Key: the symbols have the same meanings as in (a); dotted lines, the effect of a (0.00005 g cm-3 error √ in measured density. (c) Calculated densities (F) of aqueous H2SO4, NH4HSO4, and (NH4)2SO4 plotted against the square root of solute molality ( m). Key: solid line, H2SO4; dashed line, (NH4)2SO4; dash-dot line, NH4HSO4.

densities, are in error. Second, densities listed by Tang and Munkelwitz14 were not determined experimentally but were calculated using an interpolation formula and a single measurement listed in the International Critical Tables15 which was wrongly assumed to be for the concentration of a saturated solution. Third, the density listed in the International Critical Tables is not a value of F(t, 4) as stated but is a density relative to water at the same temperature. Finally, Stelson and

Seinfeld12 note that the source of densities for ammonium bisulfate cited by Mellor16 is in fact for ammonium bisulfite (NH4HSO3). In a personal communication to Stelson and Seinfeld, Tang provided a set of seven measurements of densities at 25 C, from about 9 to 61.5 wt % NH4HSO4, which are displayed in their Figure 2.12 In a later publication Tang and Munkelwitz17 presented tabulated pycnometric density measurements 3469



ARTICLE

Table 4. Calculated Densities and Apparent Molar Volumes of Aqueous H2SO4-(NH4)2SO4 Mixtures at 25 Ca xA

F

Vφ

xA

F

Vφ

xA

F

Vφ

xA

F

Vφ

xA

F

Vφ

0.0

0.0

0.997045

50.25

0.25

0.997045

41.24

0.50

0.997045

32.23

0.75

0.997045

23.22

1.00

0.997045

14.21

0.0010

0.0

0.997126

50.56

0.25

0.997126

42.29

0.50

0.997126

33.98

0.75

0.997126

25.63

1.00

0.997126

17.24

0.0020

0.0

0.997208

50.69

0.25

0.997206

42.92

0.50

0.997205

35.05

0.75

0.997204

27.08

1.00

0.997203

19.01

0.0050

0.0

0.997450

50.95

0.25

0.997442

44.16

0.50

0.997434

37.06

0.75

0.997429

29.70

1.00

0.997424

22.11

0.010

0.0

0.997852

51.25

0.25

0.997826

45.36

0.50

0.997805

38.93

0.75

0.997789

32.04

1.00

0.997776

24.76

0.020

0.0

0.998651

51.68

0.25

0.998580

46.70

0.50

0.998525

40.93

0.75

0.998485

34.43

1.00

0.998456

27.36

0.050

0.0

1.001010

52.55

0.25

1.000785

48.56

0.50

1.000614

43.50

0.75

1.000495

37.37

1.00

1.000419

30.39

0.10 0.20

0.0 0.0

1.004855 1.012304

53.54 54.94

0.25 0.25

1.004364 1.011301

50.00 51.55

0.50 0.50

1.003988 1.010527

45.30 47.00

0.75 0.75

1.003740 1.010046

39.29 40.98

1.00 1.00

1.003593 1.009782

32.28 33.88

0.30

0.0

1.019493

56.00

0.25

1.018013

52.58

0.50

1.016872

48.01

0.75

1.016194

41.90

1.00

1.015844

34.68

0.40

0.0

1.026463

56.87

0.25

1.024537

53.38

0.50

1.023059

48.76

0.75

1.022221

42.52

1.00

1.021819

35.19

0.50

0.0

1.033238

57.62

0.25

1.030892

54.06

0.50

1.029103

49.37

0.75

1.028140

43.01

1.00

1.027722

35.56

0.60

0.0

1.039835

58.29

0.25

1.037094

54.65

0.50

1.035017

49.89

0.75

1.033963

43.41

1.00

1.033560

35.84

0.70

0.0

1.046266

58.89

0.25

1.043152

55.18

0.50

1.040809

50.35

0.75

1.039694

43.75

1.00

1.039339

36.06

0.80

0.0

1.052541

59.44

0.25

1.049076

55.66

0.50

1.046485

50.77

0.75

1.045338

44.05

1.00

1.045060

36.25

0.90 1.00

0.0 0.0

1.058668 1.064653

59.95 60.42

0.25 0.25

1.054872 1.060548

56.10 56.51

0.50 0.50

1.052051 1.057512

51.15 51.50

0.75 0.75

1.050899 1.056380

44.32 44.57

1.00 1.00

1.050724 1.056332

36.41 36.56

1.10

0.0

1.070504

60.87

0.25

1.066109

56.89

0.50

1.062873

51.83

0.75

1.061783

44.80

1.00

1.061883

36.69

1.20

0.0

1.076224

61.29

0.25

1.071558

57.25

0.50

1.068137

52.14

0.75

1.067110

45.01

1.00

1.067378

36.81

1.30

0.0

1.081818

61.69

0.25

1.076902

57.59

0.50

1.073309

52.43

0.75

1.072365

45.21

1.00

1.072815

36.93

1.40

0.0

1.087290

62.07

0.25

1.082143

57.90

0.50

1.078391

52.70

0.75

1.077548

45.40

1.00

1.078195

37.05

1.50

0.0

1.092645

62.43

0.25

1.087285

58.21

0.50

1.083386

52.96

0.75

1.082662

45.58

1.00

1.083516

37.16

1.60

0.0

1.097884

62.78

0.25

1.092330

58.49

0.50

1.088297

53.21

0.75

1.087708

45.75

1.00

1.088780

37.27

1.70 1.80

0.0 0.0

1.103013 1.108033

63.12 63.44

0.25 0.25

1.097283 1.102145

58.77 59.03

0.50 0.50

1.093127 1.097877

53.45 53.67

0.75 0.75

1.092689 1.097604

45.91 46.07

1.00 1.00

1.093987 1.099135

37.37 37.48

1.90

0.0

1.112948

63.76

0.25

1.106920

59.29

0.50

1.102551

53.89

0.75

1.102457

46.21

1.00

1.104226

37.58

2.00

0.0

1.117760

64.06

0.25

1.111609

59.53

0.50

1.107150

54.10

0.75

1.107249

46.36

1.00

1.109259

37.69

2.20

0.0

1.127086

64.64

0.25

1.120739

59.99

0.50

1.116133

54.49

0.75

1.116653

46.63

1.00

1.119158

37.89

2.40

0.0

1.136032

65.19

0.25

1.129552

60.42

0.50

1.124838

54.86

0.75

1.125827

46.88

1.00

1.128838

38.09

2.60

0.0

1.144616

65.71

0.25

1.138063

60.83

0.50

1.133280

55.20

0.75

1.134781

47.11

1.00

1.138309

38.29

2.80

0.0

1.152855

66.21

0.25

1.146285

61.21

0.50

1.141470

55.52

0.75

1.143524

47.33

1.00

1.147582

38.48

2.90 3.00

0.0 0.0

1.156851 1.160767

66.45 66.68

0.25 0.25

1.150293 1.154233

61.40 61.58

0.50 0.50

1.145474 1.149419

55.68 55.83

0.75 0.75

1.147818 1.152063

47.44 47.54

1.00 1.00

1.152149 1.156671

38.57 38.66

3.20

0.0

1.168368

67.14

0.25

1.161917

61.92

0.50

1.157137

56.11

0.75

1.160407

47.74

1.00

1.165588

38.82

3.40

0.0

1.175673

67.58

0.25

1.169350

62.25

0.50

1.164633

56.39

0.75

1.168562

47.92

1.00

1.174347

38.98

3.60

0.0

1.182696

67.99

0.25

1.176542

62.57

0.50

1.171917

56.65

0.75

1.176534

48.10

1.00

1.182956

39.13

3.80

0.0

1.189453

68.40

0.25

1.183506

62.87

0.50

1.178997

56.90

0.75

1.184330

48.27

1.00

1.191421

39.26

4.00

0.0

1.195956

68.79

0.25

1.190252

63.16

0.50

1.185881

57.14

0.75

1.191953

48.43

1.00

1.199744

39.39

4.20

0.0

1.202219

69.16

0.25

1.196790

63.44

0.50

1.192577

57.37

0.75

1.199409

48.58

1.00

1.207920

39.51

4.40 4.60

0.0 0.0

1.208255 1.214076

69.52 69.86

0.25 0.25

1.203131 1.209285

63.71 63.96

0.50 0.50

1.199093 1.205436

57.59 57.80

0.75 0.75

1.206700 1.213830

48.73 48.88

1.00 1.00

1.215941 1.223798

39.63 39.74

4.80

0.0

1.219695

70.20

0.25

1.215263

64.21

0.50

1.211614

58.00

0.75

1.220800

49.02

1.00

1.231478

39.87

5.00

0.0

1.225121

70.52

0.25

1.221075

64.44

0.50

1.217636

58.20

0.75

1.227615

49.16

1.00

1.238968

39.99

5.20

0.0

1.230368

70.82

0.25

1.226732

64.66

0.50

1.223507

58.38

0.75

1.234275

49.29

1.00

1.246255

40.13

5.40

0.0

1.235444

71.12

0.25

1.232243

64.87

0.50

1.229236

58.56

0.75

1.240782

49.42

1.00

1.253328

40.28

5.60

0.0

1.240361

71.40

0.25

1.237621

65.07

0.50

1.234831

58.73

0.75

1.247138

49.56

1.00

1.260176

40.43

5.80

0.0

1.245129

71.67

0.25

1.242874

65.26

0.50

1.240298

58.89

0.75

1.253344

49.69

1.00

1.266790

40.60

6.00

0.0

1.249757

71.93

0.25

1.248015

65.44

0.50

1.245646

59.05

0.75

1.259404

49.82

1.00

1.273165

40.78

m

Symbols and units: m (mol kg-1), molality of the solute (NH4)2-xHxSO4; xA, the acid mole fraction in the solute (equal to nH2SO4/(nH2SO4 þ n(NH4)2SO4) where prefix n indicates the number of moles); F (g cm-3), density; V φ (cm3 mol-1), apparent molar volume of the solute. a

(18.08-61.86 wt %, and 25 C) of which it is possible that some or all may be the same as communicated to Stelson and Seinfeld. Tang and Munkelwitz17 also give, as a fitted equation, the results of density measurements from 40 to 97 wt % at 25 C made using

an electrodynamic balance. Zelenyuk et al.18 have measured the density of pure liquid NH4HSO4 at 25 C. Data for other Hþ:NH4þ ratios are restricted to pycnometric (0.57-8.46 mol kg-1) and electrodynamic balance 3470



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Figure 7. (a) Measured and fitted apparent molar volumes (V φ) of NH4HSO4 at 25 C over the entire concentration range, as a function of the square √ root of the weight percent of NH4HSO4 in solution ( wt %). Key: plus, Tang and Munkelwitz17 (electrodynamic balance measurements); solid square, melt (extrapolated equation of Janz36); diamond, Perkin;21 triangle, Tang (interpolated from Figure 2 of Stelson and Seinfeld20); star, Semmler et al.;19 circle, Pitzer equation (this study); inverted solid triangle, Tang and Munkelwitz17 (pycnometric measurements); line, the fitted equation. The vertical φ arrow indicates the concentration of a saturated solution. (b) The difference between measured √ and fitted apparent molar volumes (ΔV , observed calculated), plotted against the square root of the weight percent of NH4HSO4 in solution ( wt %). The symbols have the same meanings as in (a); dotted lines, the effect of a (0.00005 g cm-3 error in measured √ density. (c) Measured and calculated densities (F) of aqueous NH4HSO4, plotted against the square root of NH4HSO4 weight percent in solution ( wt %). Key: the symbols and vertical arrow have the same meanings as in (a); line, the fitted equation. The inset shows the difference between densities calculated using the fitted model and the equation of Tang and Munkelwitz17 (model - Tang values). For clarity, in all three plots only a portion of the values calculated using the Pitzer model and equation of Tang and Munkelwitz17 are shown.

measurements (40-78 wt %) of aqueous letovicite densities,17 and measurements by Semmler et al.19 for several different Hþ: NH4þ ratios over a range of temperatures. The dissociation of the HSO4- ion varies as a function of temperature, concentration, and Hþ:NH4þ ratio, and this affects the density of the solution as it does for aqueous H2SO4. As well as investigating the available measurements for aqueous

NH4HSO4,12 Stelson and Seinfeld20 predicted densities using a combination of Young’s rule for mixing, data for degrees of dissociation of aqueous H2SO4 and NH4HSO4, and densities of solutions of several electrolytes including H2SO4 and (NH4)2SO4. The results agreed with the pycnometric data of Tang and Munkelwitz and the measurement of Perkin.21 It was confirmed that the molarities of water listed by Irish and Chen22 3471



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Table 5. Fitted Parameters for Apparent Molar Volumes of Aqueous NH4HSO4 and (NH4)1.5H0.5SO4 at 25 Ca V φ(NH4HSO4) knot

V φ((NH4)1.5H0.5SO4)

coefficient

knot

coefficient

0.0

32.22996636

0.0

41.24060101

0.0

33.14228746

0.0

41.29728203

0.0 0.0

39.08480586 43.26934631

0.0 0.0

42.21108250 47.15415820

0.2398372224

48.21833327

0.04972323401

51.60460966

0.6770004925

50.91380752

0.2940459148

56.34536507

1.066771052

56.23471107

0.9896009297

64.87802792

3.212903427

61.72457233

2.999983333

71.16464713

4.962862077

64.66251531

5.978795865

74.49077197

7.483314774

66.15037377

10.0

10.0 10.0

10.0 10.0

10.0

10.0

10.0 a

The equation is a normalized b-spline of degree 3 (order 4), and its evaluation is briefly described in the Appendix to part 1 of this work. The independent variable is the square root of weight percent of solute in both cases. The equations and fitted parameters are valid over the entire concentration range.

had been wrongly calculated (leading to an incorrect density), and an equation provided by Irish and Chen in a personal communication which had been fitted to their measurements was found to agree quite well with the predicted densities and other data. It is evident from the work summarized above that the uncertainties regarding densities of aqueous NH4HSO4 at 25 C are largely resolved. For letovicite (Hþ:NH4þ molar ratio equal to 1/3) densities have been measured by Tang and Munkelwitz,17 and by Semmler et al.19 who have also determined densities for other ratios both at 25 C and other temperatures. Clegg et al.23 have developed a Pitzer activity coefficient model for the Hþ-NH4þ-HSO4--SO42--H2O system at 25 C, and Clegg et al.24 later developed a mole fraction based model for a wide range of temperatures and applicable to very highly concentrated solutions. Both models include an explicit treatment of HSO4- dissociation, and the molality based model23 incorporates the treatment of aqueous H2SO4 which is used to model the apparent molar volumes and densities of the aqueous acid in section 4 above. It is clear from Table 1 that there are relatively few measurements for aqueous Hþ-NH4þ-HSO4--SO42solutions at temperatures other than 25 C. We have therefore used the data, at this temperature only, to develop a Pitzer model of densities for molalities up to 6 mol kg-1 and for all Hþ:NH4þ ratios. First, values of the NH4þ-SO42- volume interaction parameters were determined by fitting eq 8 in part 1 to measurements from sources listed Table 17 of that work. The results are shown in Figure 5, and the fitted coefficients are listed in Table 2. For the acid sulfate mixtures, the value of the partial differential in eq 12 was obtained numerically in the same way as for H2SO4. The model parameters that were fitted for this system are: β(0)VNH4,HSO4, C(0)VNH4,HSO4, C(1)VNH4,HSO4, θVH,NH4 and ψVNH4,HSO4,SO4. Their values are listed in Table 2. Calculated apparent molar volumes in the solutions are compared with measured values in Figure 6 and agree satisfactorily. A notable feature of the data is that V φ

of aqueous NH4HSO4 does not lie half way between values for H2SO4 and (NH4)2SO4 but is shifted toward the salt because of the dissociation equilibrium. Densities and apparent molar volumes calculated using our fitted model (eq 12) for various solution compositions are listed in Table 4.

6. NH4HSO4-H2O Using the model described above, we first generated apparent molar volumes of aqueous NH4HSO4 in dilute solutions. The value of the apparent molar volume at infinite dilution is equal to V φ¥(2Hþ, SO42-) þ V φ¥(NH4þ), or 32.23 cm3 mol-1. Note the use of V φ¥(2Hþ, SO42-) rather than V φ¥(Hþ, HSO4-), because HSO4- is entirely dissociated in an infinitely dilute solution. Zelenyuk et al.18 have measured the molar volume of the pure supercooled liquid NH4HSO4 at 25 C, obtaining 65.03 ( 0.85 cm3 mol-1. The equation of Tang and Munkelwitz,17 which is based upon electrodynamic balance measurements to >95 wt % NH4HSO4, yields 66.16 cm3 mol-1 when extrapolated. The equation for melt density as a function of temperature in Table 1 of part 1 (refitted to yield at 0 K the value for solid at 25 C) gives 66.15 cm3 mol-1. The three values are in quite close agreement and we have adopted 66.15 cm3 mol-1 in our calculations. Next, the apparent molar volumes noted above, and those from other sources listed in Table 1, were fitted as a function of molality. The data are difficult to represent with a simple polynomial because of the influence of HSO4- dissociation at low concentrations. Accordingly, a normalized cubic b-spline was used. (See the Appendix in part 1 of this work for details of how the spline was evaluated.) The results are shown in Figure 7 as both V φ(NH4HSO4) and F, together with the difference between our result and the equation of Tang and Munkelwitz17 for low molalities. The knots and coefficients of the spline equation are listed in Table 5. 7. (NH4)3H(SO4)2-H2O Data for this composition (letovicite) are limited to the pycnometer and electrodynamic balance measurements of Tang and Munkelwitz,17 and measurements at a few concentrations by Semmler et al.19 The latter were found to be in broad agreement with the other data, but somewhat more scattered, and were not fitted. The data were supplemented by additional values of the apparent molar volume at low concentrations calculated using the Pitzer model described above and then fitted by a spline equation in the same way as for NH4HSO4. In this case the value of the apparent molar volume at infinite dilution is (1.5V φ¥(NH4þ) þ V φ¥(2Hþ, SO4 2-)) or 41.24 cm-3 mol -1 for the solute (NH4)1.5H0.5SO4, or alternatively (3V φ¥(NH4þ) þ 2V φ¥(2Hþ, SO42-)) or 82.48 cm-3 mol-1 for solute (NH4)3H(SO4)2. The results are shown in Figure 8, and fitted parameters are listed in Table 5. 8. SUMMARY The application of the ion interaction model to H2SO4 solutions and their mixtures with (NH4)2SO4 allows the effect of HSO4- dissociation on volume properties to be described explicitly and yields more accurate apparent molar volumes and densities than have previously been available. The values of V φ¥(2Hþ, SO42-) used here were obtained by difference from our results 3472



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Figure 8. (a) Measured and fitted apparent molar volumes (V φ) of (NH4)1.5H0.5 √SO4 (letovicite) at 25 C over the entire concentration range, as a function of the square root of the weight percent of (NH4)1.5H0.5SO4 in solution ( wt %). Key: plus, Tang and Munkelwitz17 (electrodynamic balance measurements); dot, Tang and Munkelwitz17 (pycnometric measurements); circle, calculated using the Pitzer model developed in this study; line, the fitted equation. The data of Semmler et al.,19 although broadly consistent with the other measurements, were not fitted. The vertical arrow indicates the concentration φ of a saturated solution. (b) The difference between measured √ and fitted apparent molar volumes (ΔV , observed - calculated), plotted against the square root of the weight percent of (NH4)1.5H0.5SO4 in solution ( wt %). The symbols have the same meanings as in (a); dotted lines, the effect of a (0.00005 g cm-3 error in measured density. √ (c) Measured and calculated densities (F) of aqueous (NH4)1.5H0.5SO4, plotted against the square root of (NH4)1.5H0.5SO4 weight percent in solution ( wt %). Key: the symbols and vertical arrow have the same meanings as in (a); line, the fitted equation. The inset shows the difference between densities calculated using the fitted model and the equation of Tang and Munkelwitz17 (model - Tang values). For clarity, in all three plots only a portion of the values calculated using the Pitzer model and equation of Tang and Munkelwitz17 are shown.

for other solutes (in part 1 of this work), and values of V φ¥(Hþ, HSO4-) were obtained by fitting density data at different temperatures. The results are in broad agreement with those of Hovey and Hepler10 and reduce the lower limit of temperature for which V φ¥(Hþ, HSO4-) are available from 10 to 0 C. The existing application of the Pitzer model to describe osmotic and activity coefficients of the H2SO4-(NH4)2SO4-H2O23

has been extended to describe volume properties at 25 C. Apparent molar volumes and densities can now be calculated for all relative compositions from H2SO4-H2O to (NH4)2SO4-H2O for total molalities of 0-6 mol kg-1. The results of this model have been combined with data for more concentrated solutions—notably the electrodynamic balance measurements of Tang—to obtain volume properties of aqueous NH4HSO4 and 3473


The Journal of Physical Chemistry A letovicite solutions over the entire concentration range and with improved accuracy at low to moderate concentrations compared to previous treatments.17,20 The densities and apparent molar volumes determined in this work for 0-3 mol kg-1 H2SO4 solutions have been combined with those obtained in part 1 for the rest of the concentration range, and are tabulated in the Supporting Information for that study. The combined model for H2SO4-H2O, that for H2SO4(NH4)2SO4-H2O, and the results for the two acid ammonium sulphates, have been included in the Extended Aerosol Inorganics Model (E-AIM, http://www.aim.env.uea.ac.uk/aim/aim.php). Apparent and partial molar volumes of the solutes, and densities of the solutions, can be calculated using the model.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT The authors gratefully acknowledge the support of the U.S. Department of Energy (grant number DE-FG02-08ER64530) and National Oceanic and Atmospheric Administration (grant number NA07OAR4310192), and the Natural Environment Research Council of the U.K. (grant number NE/E002641/1).

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(23) Clegg, S. L.; Milioto, S.; Palmer, D. A. J. Chem. Eng. Data 1996, 41, 455–467. (24) Clegg, S. L.; Brimblecombe, P.; Wexler, A. S. J. Phys. Chem. 1998, A102, 2127–2154. (25) Myhre, C. E. L.; Christensen, D. H.; Nicolaisen, F. M.; Nielsen, C. J. J. Phys. Chem. A 2003, 107, 1979–1991. (26) Larson, J. W.; Zeeb, K. G.; Hepler, L. G. Can. J. Chem. 1982, 60, 2141–2150. (27) Rhodes, F. H.; Barbour, C. B. Ind. Eng. Chem. 1923, 15, 850– 852. (28) Klotz, I. M.; Eckert, C. F. J. Am. Chem. Soc. 1942, 64, 1878– 1880. (29) Campbell, A. N.; Kartzmark, E. M.; Bisset, D.; Bednas, M. E. Can. J. Chem. 1953, 31, 303–305. (30) Joshi, B. K.; D., K. N. Phys. Chem. Liq. 2007, 45, 463–470. (31) Tollert, H. Z. Phys. Chem. A 1939, 184, 165–178. (32) Hervello, M. F.; Sanchez, A. J. Chem. Eng. Data 2007, 52, 752– 756. (33) Fang, S.; Zhang, J.-G.; Liu, Z.-Q.; Liu, M.; Luo, S.; He, C.-H. J. Chem. Eng. Data 2009, 54, 2028–2032. (34) Albright, J. G.; Mitchell, J. P.; Miller, D. G. J. Chem. Eng. Data 1994, 39, 195–2000. (35) Goldsack, D. E.; Franchetto, A. A. Electrochim. Acta 1977, 22, 1287–1294. (36) Janz, G. J. J. Phys. Chem. Ref. Data 1988, 17 (Suppl. 2), 312.

’ REFERENCES (1) Clegg, S. L.; Wexler, A. S. J. Phys. Chem. A 2011, DOI: 10.1021/ jp108992a. (2) Clegg, S. L.; Rard, J. A.; Pitzer, K. S. J. Chem. Soc., Faraday Trans. 1994, 90, 1875–1894. (3) Lindstrom, R. E.; Wirth, H. E. J. Phys. Chem. 1969, 73, 218–223. (4) Pitzer, K. S. Ion interaction approach: theory and data correlation. In Activity Coefficients in Electrolyte Solutions, 2nd ed.; Pitzer, K. S., Ed.; CRC Press: Boca Raton, FL, 1991; pp 75-153. (5) Kell, G. S. J. Chem. Eng. Data 1975, 20, 97. (6) Kell, G. S. J. Chem. Eng. Data 1967, 12, 66–91. (7) Robinson, R. A.; Stokes, R. H. In Electrolyte Solutions, 2nd ed. (Revised); Butterworths: London, 1965. (8) Roux, A.; Musbally, G. M.; Perron, G.; Desnoyers, J. E.; Singh, P. P.; Woolley, E. M.; Hepler, L. G. Can. J. Chem. 1978, 56, 24–28. (9) Millero, F. J. Geochim. Cosmochim. Acta 1982, 46, 11–22. (10) Hovey, J. K.; Hepler, L. G. J. Chem. Soc., Faraday Trans. 1990, 86, 2831–2839. (11) Pitzer, K. S.; Roy, R. N.; Silvester, L. F. J. Am. Chem. Soc. 1977, 99, 4930–4936. (12) Stelson, A. W.; Seinfeld, J. H. Atmos. Environ. 1982, 16, 355–357. (13) Irish, D. E.; Chen, H. J. Phys. Chem. 1970, 74, 3796–3801. (14) Tang, I. N.; Munkelwitz, H. R. J. Aerosol Sci. 1977, 8, 321–330. (15) Washburn, E. W., Ed. International Critical Tables of Numerical Data, Physics, Chemistry and Technology, Vol. III; McGraw-Hill: New York, 1928. (16) Mellor, J. W. A Comprehensive Treatise on Inorganic and Theoretical Chemistry, Vol. VIII, Suppl. I, N (Part I), p 503; John Wiley & Sons, New York. (17) Tang, I. N.; Munkelwitz, H. R. J. Geophys. Res. 1994, 99, 18801–18808. (18) Zelenyuk, A.; Cai, Y.; Chieffo, L.; Imre, D. Aerosol Sci. Technol. 2005, 39, 972–986. (19) Semmler, M.; Luo, B. P.; Koop, T. Atmos. Environ. 2006, 40, 467–483. (20) Stelson, A. W.; Seinfeld, J. H. J. Phys. Chem. 1981, 85, 3730– 3733. (21) Perkin, W. H. J. Chem. Soc. 1889, 680–749. (22) Irish, D. E.; Chen, H. J. Phys. Chem. 1970, 74, 3796–3801. 3474


Densities and Apparent Molar Volumes of Atmospherically Important

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