Article pubs.acs.org/IECR
Determination of Volume Properties of Aqueous Vanadyl Sulfate at 283.15 to 323.15 K Wei-Guo Xu,†,‡ Ye Qin,† Fei Gao,† Jian-Guo Liu,*,† Chuan-Wei Yan,† and Jia-Zhen Yang‡ †
State Key Laboratory for Corrosion and Protection, Institute of Metal Research, Chinese Academy of Sciences, 62 Wencui Road, Shenyang 110016, China ‡ College of Chemistry, Liaoning University, Shenyang 110036, China S Supporting Information *
ABSTRACT: This paper reports the densities of aqueous VOSO4 that were measured by an Anton Paar DMA 4500 M from 0.0500 mol·kg−1 to 3.000 mol·kg−1 at 283.15 to 323.15 K. The apparent molar volume, ϕVB, the partial molar volume of VOSO4, V̅ B, the apparent molar expansibility of the solute, ϕE, and coefficient of thermal expansion of the solution, α, were calculated. First, these properties described above have been discussed with the temperature and concentration variation. Then the values of the apparent molar volume, ϕVB, were fitted to Pitzer’s model for volumetric properties by the method of the least squares, which (1)V V allowed the partial molar volume of the VOSO4 at infinite dilution, V̅ 0B, and Pitzer’s parameters, β(0)V MX , βMX , and CMX, to be obtained. The small standard deviations of the fitting show that Pitzer’s model is also appropriate for representing the volumetric properties of aqueous solutions of VOSO4. Using the data from literature, the partial molar volume of the ion VO2+ at infinite dilution, V̅ 0C, was discussed and the result showed that the negative value of V̅ 0C can be attributed to strong hydration of VO2+ as other divalent ions.
1. INTRODUCTION Over the past few decades, there have been considerable activities devoted to finding new cleaner energy sources to cope with environmental pollution. Wind, solar, and tidal energy are the ideal choices, however, they are noncontinuous energy, which needs to be combined with a large-scale energy storage device.1 The all vanadium redox flow battery (VRFB), an effective energy-storage system proposed by Skyllas−Kazacos et al.2−4 is one of the promising candidates. In the VRFB, the electrolyte is the most important components, which is not only the conductor of ions but also the energy-storage medium. The VRFB employs the V(II)/V(III) and V(IV)/V(V) redox couples in the negative and positive half-cell electrolytes, respectively, with sulfuric acid as the supporting electrolyte. The chemistry of vanadium has recently attracted considerable attention from both industry and the academic community because reliable thermodynamic data are needed both to help in providing an adequate description of interactions of various species in the VRFB solution and to provide clues to optimize the overall performance of the VRFB.5−8 Knowledge on the volumetric properties of the electrolyte solution for VRFB will give us a deep understanding of the volume changes in the battery charge and discharge processes.9 To this end and as a continuation of our previous investigation of vanadium electrolyte,10−15 here, we report the densities of aqueous solution of VOSO4 which were measured from (0.0500 to 3.0000) mol·kg−1 over the temperatures range from 283.15 to 323.15 K. Values of apparent molar volume, ϕ VB, partial molar volume, V̅ B, and the coefficient of thermal expansion of the solution, α, were calculated. The values of Pitzer’s parameter for volumetric properties were obtained by fitting the parameters of Pitzer’s model equations to experimental data. In comparison with our previous work,15 © 2014 American Chemical Society
the experiments are more accurate and the results are discussed more in-depth.
2. EXPERIMENTAL SECTION AND RESULTS 2.1. Chemicals and Instruments. VOSO4·nH2O(s) (≥97 mass %, Shanghai Chemical Co., China) was recrystallized twice from water.17 The standard stock solution was prepared and its molality was determined by the method of gravimetric determination.18 Deionized water was prepared by an ultrapure water machine (WP-UPL-100C, Sichuan Woter Co., China); its conductivity was ≤2 × 10−5 S·m−1. 2.2. Experimental Process. All aqueous VOSO4 to be studied were freshly prepared by weight with correction for air buoyancy. The molality of all studied solutions were known within ± 0.02%. An Anton Paar DMA 4500 M oscillating U− tube densitometer was used to measure the density of the samples. The temperature in the cell was regulated to ± 0.01 K with solid state thermostat. Before the measurement, the apparatus was calibrated once a day with dry air and freshly double-distilled degassed water. Then the values of density of pure water were measured by the calibrated apparatus and were in good agreement with those of literiture 19 within experimental error. Finally, the density of the samples with molalities from 0.0500 mol·kg−1 to 3.000 mol·kg−1 was measured at 5 K interval from 283.15 to 323.15 K. 2.3. Densities of the Aqueous Solution. The density values of aqueous VOSO4 with various molalities measured at different temperatures are listed in Table 1. All the density Received: Revised: Accepted: Published: 7217
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Table 1. Value of the Densities (g·cm−3) of Aqueous Solutions of VOSO4 at 283.15 to 323.15 K densities, ρ(g·cm−3) m/mol·kg
283.15 K
288.15 K
293.15 K
298.15 K
303.15 K
308.15 K
313.15 K
318.15 K
323.15 K
0 0.05003 0.07006 0.1000 0.2001 0.2997 0.3990 0.5003 0.7996 0.9995 1.200 1.500 1.801 2.003 2.198 2.493 2.791 3.002
0.99971 1.00733 1.01031 1.01469 1.02927 1.04350 1.05756 1.07158 1.11253 1.13917 1.16525 1.20306 1.23947 1.26313 1.28540 1.31843 1.34941 1.37194
0.99911 1.00668 1.00963 1.01398 1.02844 1.04256 1.05651 1.07043 1.11111 1.13758 1.16351 1.20114 1.23739 1.26095 1.28315 1.31610 1.34701 1.36949
0.99822 1.00572 1.00866 1.01297 1.02733 1.04135 1.05521 1.06904 1.10947 1.13579 1.16159 1.19904 1.23515 1.25863 1.28077 1.31363 1.34447 1.36691
0.99706 1.00451 1.00743 1.01171 1.02598 1.03990 1.05368 1.06743 1.10764 1.13383 1.15951 1.19681 1.23278 1.25618 1.27826 1.31105 1.34183 1.36423
0.99566 1.00307 1.00596 1.01022 1.02440 1.03825 1.05195 1.06562 1.10563 1.13171 1.15727 1.19442 1.23028 1.25363 1.27564 1.30836 1.33908 1.36145
0.99405 1.00141 1.00429 1.00853 1.02263 1.03640 1.05003 1.06363 1.10346 1.12942 1.15489 1.19190 1.22765 1.25094 1.27289 1.30555 1.33622 1.35856
0.99223 0.99956 1.00242 1.00663 1.02066 1.03436 1.04793 1.06146 1.10112 1.12698 1.15235 1.18925 1.22489 1.24809 1.27004 1.30264 1.33325 1.35557
0.99022 0.99751 1.00036 1.00455 1.01851 1.03215 1.04566 1.05914 1.09864 1.12440 1.14969 1.18647 1.22202 1.24514 1.26706 1.29963 1.33018 1.35248
0.98804 0.99530 0.99813 1.00230 1.01620 1.02977 1.04323 1.05665 1.09600 1.12167 1.14688 1.18356 1.21903 1.24210 1.26398 1.29650 1.32701 1.34928
According to definition of the coefficient of thermal expansion of the solutions, α, is written as
values in Table 1 are the average of the measured values for three times.
3. DISCUSSION 3.1. Densities of the Aqueous Solution. Figure 1 is the density of aqueous VOSO4 vs the temperature and the molality.
α≡
1 ⎛⎜ ∂V ⎞⎟ 1 ⎛ ∂ρ ⎞ =− ⎜ ⎟ ρ ⎝ ∂T ⎠ P , m V ⎝ ∂T ⎠ P , m
At constant molality and pressure, eq 1 is converted to the partial derivative with respect to temperature and (∂ρ/∂T)p is obtained ⎛ ∂ρ ⎞ ⎜ ⎟ = A + 2A (T − 273.15) 1 2 ⎝ ∂T ⎠ P
(3)
Substituting eq 3 into eq 2, the α values of aqueous VOSO4 with various molality at different temperatures were calculated and listed in Table S2 in Supporting Information. In addition, the coefficient of thermal expansion of pure water, α0, was included in the row of m = 0 mol/kg in Table S2. As an example, Figure 2 is the plot of the coefficient of thermal expansion, α, of aqueous VOSO4 vs the molalities at 318.15, 308.15, 298.15, and 288.15 K, and the plots at other temperatures are placed in Supporting Information (Figure S1). As seen from Figure 2 and Supporting Information Figure S1, the coefficients of thermal expansion of aqueous VOSO4 increase linearly with the temperature elevating and change with the molality augmenting like a parabola (at the beginning increase and then decrease). 3.2. Apparent Molar Volume of the Aqueous VOSO4. The apparent molar volume, ϕVB, was derived from the measured solution densities. Their values are calculated with the following equation:16
Figure 1. Plot of density of aqueous VOSO4 vs the temperature and the molality.
As seen from Figure 1, the density values of aqueous VOSO4 decrease with the temperature elevating and increase with the molality augmenting. At constant molality and pressure, an empirical equation can be obtained ρ = A 0 + A1(T − 273.15) + A 2 (T − 273.15)2
(2)
ϕ
(1)
where ρ is density of the solution, T is absolute temperature, and Ai is the empirical constant. The values of ρ obtained at different temperatures have been fitted against (T − 273.15) by the least squares to eq 1 and the obtained values of the parameters in eq 1 with the correlation coefficients r and standard deviations, s, are listed in Supporting Information Table S1 in Supporting Information.
VB =
1000(ρ0 − ρ) + mMBρ0 mρρ0
(4)
where ρ0 and ρ are the density of pure water and aqueous VOSO4 solutions, respectively, m is the molality, and MB the molar mass of VOSO4. The values of apparent molar volume calculated with eq 4 are listed in Table 2. As an example, Figure 3 is a plot of the apparent molar volume, ϕVB, against the molality m at 318.15, 308.15, 298.15, and 288.15 K; Figure S2 is 7218
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Figure 3. Plot of the apparent molar volume, φVB, against the molality m and the different temperatures. From top to bottom: ▶ 318.15 K, ▷ 308.15 K, ▲ 298.15.15 K, △ 288.15 K.
Figure 2. Plot of the coefficient of thermal expansion of aqueous VOSO4 vs the molality at the different temperatures. From top to bottom: ▶ 318.15 K, ▷ 308.15 K, ▲ 298.15.15 K, △ 288.15 K.
⎛ ∂ ϕV ⎞ B ⎟⎟ ϕE = ⎜⎜ ∂ T ⎝ ⎠P
a plot at other temperatures and placed in Supporting Information. As seen from Table 1, Figure 3, and Supporting Information Figure S2, the values of the apparent molar volume increase when m and T increase. When the molality of the solution is kept constant, the relationship between ϕVB and T can be expressed as the following empirical formula: ϕ
VB = B0 + B1(T − 273.15) + B2 (T − 273.15)2
(6)
The equation of calculating to ϕE can be derived from the derivative of eq 5 with respect to T ⎛ ∂ ϕV ⎞ B ⎟⎟ = B1 + 2B2 (T − 273.15) ϕE = ⎜⎜ T ∂ ⎝ ⎠P
(5)
where Bi is an empirical constant. The data of the apparent molar volume in Table 2 were fitted to eq 5 and the obtained values of parameters in eq 5 with the correlation coefficients r and the standard deviation, s, were listed in Table S3 in Supporting Information. As seen from Table S3, the values of the fitting correlation coefficients are greater than 0.999 and the values of the fitting standard deviation are less than the experimental error. The apparent molar expansibility of the solute, ϕE, is defined by
(7)
The ϕE values of ϕE were calculated with eq 7 and are listed in Supporting Information Table S4. As an example, Figure 4 is a plot of the apparent molar expansibility of the solute, ϕE, against the molality m at 318.15, 308.15, 298.15, and 288.15 K, Figure S3 is a plot at other temperatures and placed in Supporting Information. As seen from Table S4, Figure 4 and Supporting Information Figure S3, the values of ϕE, decrease as m and T increase.
Table 2. Values of the Apparent Molar Volumes ϕVB /cm3·mol−1 of Aqueous VOSO4 from 283.15 to 323.15 K apparent molar volumes, ϕVB (cm3·mol−1) m/mol·kg
283.15 K
288.15 K
293.15 K
298.15 K
303.15 K
308.15 K
313.15 K
318.15 K
323.15 K
0.05003 0.07006 0.1000 0.2001 0.2997 0.3990 0.5003 0.7996 0.9995 1.200 1.500 1.801 2.003 2.198 2.493 2.791 3.002
10.51 11.48 12.90 14.72 16.16 16.94 17.97 19.59 20.51 21.46 22.73 24.02 24.87 25.60 26.59 27.87 28.36
11.42 12.53 13.98 15.78 17.19 17.95 18.93 20.47 21.33 22.23 23.44 24.66 25.48 26.17 27.11 28.34 28.81
12.69 13.59 15.01 16.75 18.11 18.82 19.78 21.23 22.05 22.90 24.05 25.21 26.00 26.66 27.56 28.75 29.20
13.46 14.38 15.85 17.53 18.90 19.57 20.50 21.88 22.66 23.47 24.57 25.68 26.44 27.07 27.93 29.10 29.52
14.14 15.19 16.57 18.24 19.54 20.20 21.12 22.44 23.18 23.96 25.01 26.09 26.82 27.43 28.25 29.39 29.79
14.93 15.84 17.19 18.83 20.13 20.78 21.66 22.92 23.63 24.38 25.39 26.43 27.14 27.73 28.53 29.64 30.03
15.29 16.36 17.76 19.35 20.63 21.24 22.12 23.33 24.02 24.74 25.71 26.73 27.42 27.98 28.75 29.85 30.22
15.76 16.77 18.17 19.80 21.05 21.64 22.50 23.67 24.34 25.04 25.99 26.97 27.66 28.19 28.94 30.02 30.37
16.21 17.21 18.60 20.18 21.45 22.00 22.86 23.98 24.62 25.30 26.22 27.18 27.85 28.37 29.09 30.16 30.50
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In order to obtain the value of the derivative in the specified temperature, the empirical eq 10 is used to fit the apparent molar volume data ϕ
VB = C0 + C1m1/2 + C2m + C3m3/2 + C4m2
(10)
where Ci are empirical parameters. These fitting parameters with the correlation coefficients, r, and the standard deviation, s, are listed in Table S6 in Supporting Information. As seen from Table S6, the values of the fitting correlation coefficients are greater than 0.999 and the values of the fitting standard deviation are less than the experimental error. At constant pressure and temperature, we can take partial derivative according to eq10 with respect to m: ⎛ ∂ ϕV ⎞ B ⎜⎜ ⎟⎟ ⎝ ∂m ⎠
Figure 4. Plot of the apparent molar expansibility of the solute, ϕE, against the molality m and the different temperature T. From top to bottom: △ 288.15 K, ▲ 298.15.15 K, ▷ 308.15 K, ▶ 318.15 K.
= P ,T
Substitution of the above equation into eq 9 yields VB̅ = ϕVB +
As an alternative of eq 7, Harned and Owen20 gave an equation to calculate the apparent molar expansibility of solute from densities and their temperature coefficients ⎛ 1000 ⎞ ⎟⎟(α − α0) + α ϕVB ϕE = ⎜⎜ m ρ ⎝ 0 ⎠
1 3 C1m1/2 + C2m + C3m3/2 + 2C4m2 2 2
(11)
The values of the partial molar volume were calculated according to eq 11 and are listed in Table 3. As an example, Figure 5 is a plot of the partial molar volume of the solute, V̅ B, against the molality m at 318.15, 308.15, 298.15, and 288.15 K. Figure S4 (Supporting Information) is a plot at other temperatures and placed in Supporting Information. As seen from Table S7 (Supporting Information), Figure 5, and Supporting Information Figure S4, the values of V̅ B decrease when T increases, and increase when m increases. 3.4. Determination of Pitzer’s Parameters. Pitzer’s theory on electrolyte solution on the basis of the strictly statistical mechanics is a semiempirical theory which can be applied in a high concentration of an electrolyte solution.21 The apparent molar volume can be expressed in Pitzer’s equation for volumetric properties22
(8)
where ρ0 is the density of pure water. The values of the apparent molar expansibility of solute calculated according to eq 8 are in good agreement with that calculated from eq 7. 3.2. Partial Molar Volumes of Solute. The relationship between the apparent molar volume, ϕVB (m, T), and the partial molar volume, is ⎛ ∂ ϕV ⎞ B ⎟⎟ VB̅ = ϕVB + m⎜⎜ ∂ m ⎝ ⎠P , T
1 C1 3 + C2 + C3m1/2 + 2C4m 1/2 2m 2
(9)
Table 3. Values of the Partial Molar Volume, V̅ B/ cm3·mol−1, Calculated According to Eq 11 partial molar volume, V̅ B (cm3·mol−1) m/mol·kg
283.15 K
288.15 K
293.15 K
298.15 K
303.15 K
308.15 K
313.15 K
318.15 K
323.15 K
0.05003 0.07006 0.1000 0.2001 0.2997 0.3990 0.5003 0.7996 0.9995 1.200 1.500 1.801 2.003 2.198 2.493 2.791 3.002
13.21 14.38 15.98 17.99 19.46 20.26 21.39 23.60 25.11 26.76 29.16 31.51 32.97 34.17 35.60 36.90 37.12
13.31 14.47 16.06 17.98 19.38 20.14 21.21 23.34 24.83 26.47 28.87 31.20 32.66 33.83 35.20 36.38 36.48
13.15 14.30 15.89 17.83 19.24 20.01 21.08 23.17 24.62 26.22 28.54 30.84 32.28 33.46 34.87 36.17 36.40
13.18 14.34 15.90 17.80 19.17 19.90 20.94 22.98 24.41 25.99 28.31 30.60 32.03 33.19 34.57 35.82 36.00
13.17 14.32 15.88 17.74 19.09 19.80 20.82 22.82 24.24 25.80 28.11 30.38 31.80 32.95 34.31 35.53 35.68
13.08 14.22 15.78 17.66 19.01 19.73 20.75 22.72 24.10 25.63 27.89 30.13 31.54 32.70 34.10 35.40 35.65
13.17 14.31 15.85 17.66 18.96 19.64 20.63 22.56 23.94 25.49 27.76 30.01 31.41 32.54 33.87 35.04 35.15
13.17 14.30 15.84 17.64 18.92 19.59 20.56 22.46 23.83 25.36 27.62 29.85 31.25 32.37 33.69 34.84 34.95
13.15 14.28 15.81 17.59 18.87 19.53 20.49 22.36 23.71 25.23 27.47 29.69 31.08 32.21 33.53 34.71 34.83
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where Y is an extrapolation function and its value can be calculated from experimental data. The values of Y calculated at different temperature were fitted by least squares to eq 16. The (1)V V obtained values of the Pitzer’s parameters, β(0)V MX , βMX , and CMX with the standard deviation, s, correlation coefficients, r2, and values of partial molar volume of VOSO4 at infinite dilution (or the limiting partial molar volume), V̅ 0B, are also listed in Supporting Information Table S7. The small standards deviations and high correlation coefficients for the fitting to eq 16 imply that the volumetric properties of aqueous VOSO4 can be predicted with Pitzer’s model. The partial molar volume of VOSO4 at infinite dilution, V̅ 0B, is an important thermodynamic quantity which can show the hydration effect of the electrolyte in a solution. The value of V̅ 0B may be separable into the contributions of the cations and the anions, which should be constant from one electrolyte to the other Figure 5. Plot of the partial molar volume of the solute, V̅ B, against the molality m and the different temperature T. From top to bottom: △ 288.15 K, ▲ 298.15.15 K, ▷ 308.15 K, ▶ 318.15 K. ϕ
V̅B0 = Vc̅ 0 + Va̅ 0
where V̅ 0c and V̅ 0a are the volume values of the cations and the anions at infinite dilution, respectively. The value of V̅ 0a for SO42‑ is 26.8 cm3·mol−1 so that the value of V̅ 0c for VO2+ is −16.5 cm3·mol−1. In comparison with V̅ 0c = −34.0 cm3·mol−1 for Mg2+ and V̅ 0c = −25.3 cm3·mol−1 for Ba2+, the value of V̅ 0c for VO2+ seems reasonable because its ionic radius is larger.23 The negative value of V̅ 0c means that VO2+ has a strong hydration as other divalent ions. 3.5. Prediction of the Apparent Molar Volumes ϕVB of Aqueous VOSO4. The dependence of parametersV̅ 0B, β(0)VMX, β(1)VMX and CVMX on temperature is24,25
⎛A ⎞ VB = V B0 + v|z Mz |⎜ V ⎟ln(1 + bI1/2) + 2vMvXR ⎝ 2b ⎠ V 2 V T (mBM X + vMz Mm C MX )
(12)
where zM and zX are number of ionic charges of VOSO4 of the positive and negative ion in electronic units, vM and vX are the numbers of ions of each type in the formula of the VOSO4 and v = vM + vX, subscript M and X mean cation and anion of VOSO4, I is total ionic strength given by I = (1/2)Σimizi2, m is the molality of the electrolyte, R the gas constant, T thermodynamic temperatures, AV is is Debye−Hückel parameter for volume,15 the parameter b is given the value 1.2 and taken as being temperature independent. Pitzer’s parameters BVMX account for short-range interactions between M and X, the third Virial coefficient CVMX means for triple ion interactions and cannot be neglected at high concentrations (m > 2.0 mol·kg−1) V BMX
⎛ ∂B ⎞ (0)V (1)V = ⎜ MX ⎟ = βMX + g (x)βMX ⎝ ∂P ⎠ I , T
g (x ) =
2[1 − (1 + x)e−x] x2
V̅B0 = q0 + q1T + q2T 2
(18)
(0)V βMX = q3 + q4T
(19)
(1)V βMX = q5 + q6T
(20)
V CMX = q7
(21)
Thus, substitution of eq 18, 19, 20, 21 into the working eq 16 yields
(13)
Y = ϕVB − 4(AV /1.2)ln(1 + 1.2I1/2) (14)
⎛ ∂C ⎞ V CMX = ⎜ MX ⎟ ⎝ ∂P ⎠ I , T
= q0 + q1T + q2T 2 + 2RTmq3 + 2RT 2mq4 + 2RTmg (αI1/2)q5 + 2RT 2mg (αI1/2)q6 + 4RTm2q7 (22)
(15)
The values of extrapolation function Y calculated at different molalities and at different temperature were regressed using the least squares program so that 7 parameters, qi, were obtained and are listed in Table 4. The standard deviation s = 0.258 and correlation coefficients, r = 0.994. In terms of the values of parameters qi in Table 4, the values of the apparent molar volume, ϕVB, of aqueous VOSO4 with various molalities at different temperatures were calculated and are listed in Table 5.
Therefore, rearranging eq 12 yields the working equation for aqueous solutions of VOSO4 Y = ϕVB −
(17)
4AV ln(1 + bI1/2) b
(0)V (1)V g (α1I1/2) = V̅B0 + 2RTmβMX + 2RTmβMX V + 4RTm2CMX
(16)
Table 4. Values of Parameters, qi, of Eq 22, the Standard Deviation, s, of the Fits q0
q1
q2
q3
q4
q5
q6
q7
r
s
−175.46
1.10048
−0.0016
0.00163
−0.00000661
0.03146
−0.0000848
0.0000798
0.994
0.258
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Table 5. In Terms of Values of Parameters, qi, the values of ϕVB(Cal) (in cm3·mol−1) Calculated from Eq 22 apparent molar volumes, ϕVB (Cal) (cm3·mol−1) m/mol·kg
−1
0.05003 0.07006 0.1000 0.2001 0.2997 0.3990 0.5003 0.7996 0.9995 1.200 1.500 1.801 2.003 2.198 2.493 2.791 3.002 ϕ
283.15 K
288.15 K
293.15 K
298.15 K
303.15 K
308.15 K
313.15 K
318.15 K
323.15 K
11.33 12.01 12.84 14.85 16.25 17.31 18.19 20.10 21.07 21.91 23.03 24.11 24.83 25.54 26.66 27.86 28.76
12.32 12.99 13.82 15.80 17.17 18.21 19.07 20.93 21.86 22.67 23.76 24.80 25.50 26.19 27.28 28.46 29.35
13.22 13.88 14.71 16.67 18.01 19.03 19.86 21.66 22.56 23.34 24.39 25.39 26.07 26.73 27.80 28.95 29.82
14.04 14.71 15.53 17.46 18.77 19.77 20.58 22.32 23.19 23.94 24.94 25.90 26.56 27.21 28.24 29.36 30.21
14.80 15.46 16.27 18.18 19.47 20.43 21.22 22.90 23.73 24.45 25.41 26.34 26.97 27.59 28.59 29.69 30.52
15.48 16.14 16.95 18.83 20.09 21.03 21.80 23.42 24.22 24.90 25.82 26.71 27.31 27.92 28.89 29.95 30.76
16.08 16.74 17.54 19.39 20.62 21.54 22.28 23.84 24.60 25.25 26.13 26.97 27.55 28.13 29.06 30.09 30.88
16.62 17.28 18.07 19.90 21.11 22.00 22.72 24.22 24.94 25.56 26.39 27.19 27.75 28.30 29.21 30.21 30.97
17.07 17.73 18.52 20.32 21.50 22.37 23.06 24.50 25.18 25.77 26.55 27.31 27.84 28.37 29.24 30.20 30.95
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ACKNOWLEDGMENTS This project was supported by the National Natural Science Foundation of China (No.21003143 and No.21373009).
VB(Cal) = 0.22936 + 1.00195 × ϕVB s = 0.25464, r = 0.99875
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Figure 6 is a plot which is obtained by the calculated values of ϕVB against those of experimental values listed in Table 3. A
Figure 6. Plot of ϕVB(Cal) in Table 5 vs the corresponding value of ϕ VB in Table 2.
good linear, in which r equals 0.9988, shows that Pitzer’s model is appropriate for representing the volumetric properties of aqueous solutions of VOSO4.
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ASSOCIATED CONTENT
S Supporting Information *
Parameter values for the equations. This material is available free of charge via the Internet at http://pubs.acs.org.
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ABBREVIATIONS Ai = the empirical constant in eq 1 AV = Debye−Hückel parameter for volume Bi = an emperical constant in eq 5 BVMX = Pitzer’s volume parameter accounting for short-range interactions between M and X b = empirical constant in eq 12 Ci = empirical parameters in eq 10 CVMX = Pitzer’s volume parameter for triple ion interactions g(x) = a function in Pitzer’s theory; see eq 14 I = ionic strength M = molar mass MB = the molar mass of VOSO4 m = molality N = Avogadro’s constant p = pressure R = gas constant r = correlation coefficient s = standard deviation T = absolute temperature ϕ VB = the apparent molar volume V̅ 0B = the standard partial molar volume or the limiting partial molar volume of VOSO4 V̅ B = the partial molar volume of VOSO4 Vm = molecular volume x = 2.0 I1/2 Y = extrapolation function zM = number of ionic charges for VO2+ zX = number of ionic charges for SO42−
Greek Letters
α = the coefficient of thermal expansion of the solutions β(1)V MX = Pitzer’s volume parameter for short-range interactions between M and X ρ = density of the solution ρ0 = the density of water v = vM + vX
AUTHOR INFORMATION
Corresponding Author
*J.-G. Liu. Tel.: 0086-24-23998320. E-mail:
[email protected]. Notes
The authors declare no competing financial interest. 7222
dx.doi.org/10.1021/ie402040h | Ind. Eng. Chem. Res. 2014, 53, 7217−7223
Industrial & Engineering Chemistry Research
Article
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vM = the numbers of cation in the formula of the VOSO4 vX = the numbers of anion in the formula of the VOSO4 ϕE = the apparent molar expansibility of the solute
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REFERENCES
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