A Temperature-Dependent Thermodynamic Model Derived from Heat

Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX-XXX

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A Temperature-Dependent Thermodynamic Model Derived from Heat Capacity of Metal Chloride Aqueous Solutions Xin Yi, Jiugang Hu,* Xueying Zhang, Min Sun, and Shijun Liu* School of Chemistry and Chemical Engineering, and Hunan Provincial Key Laboratory of Efficient and Clean Utilization of Manganese Resources, Central South University, Changsha, Hunan 410083, China S Supporting Information *

ABSTRACT: In this work, a temperature-dependent thermodynamic model based on heat capacity was proposed to predict the osmotic and mean ion activity coefficients of metal chloride aqueous solutions. The CaCl2−H2O system was used to verify the model and the results show the predicted values can well agree with the reported data, indicating that the proposed thermodynamic model is reliable. Meanwhile, the heat capacities at constant pressure of both CuCl2−H2O and NiCl2−H2O systems in the range from 298.15 to 363.15 K were calculated from the enthalpy changes determined by the calorimetric experiments. Therefore, the osmotic and mean ion activity coefficients of the two systems within 4 mol·kg−1 were predicted in a wide temperature range with the established model. The calculated results were well consistent with the data in the literature and further indicate the thermodynamic model was reliable and convenient.

1. INTRODUCTION Accurate thermodynamic data of metal chloride aqueous solutions are widely needed in various fields including solution chemistry,1 geology,2 and hydrometallurgy.3,4 For instance, Awakura determined the activities of HCl−MCln solutions to help understand the complex reactions in chlorine metallurgy.3 On the basis of thermodynamic analysis, Liu and coauthors disclosed the formation of diverse copper chloride complexes in hydrothermal fluids.2 Moreover, these practical fields generally require the thermodynamic data over a wide range of temperature, pressure, and composition. Although thermodynamic studies of metal chloride solutions such as osmotic coefficient, activity coefficient, enthalpy, and heat capacity have taken place for many years, most of the data are limited to the condition at 298.15 K and the data at other temperatures were not adequate. Therefore, thermodynamic studies that extend beyond a moderate temperature are still needed to understand the chemical reactions involving metal chloride in various industrial processes.5,6 For the thermodynamic properties at temperatures besides 298.15 K, some efforts have been made to obtain the data at higher molalities and temperatures from both experimental and theoretical aspects. For example, Mussini measured the activity coefficients of the CaCl2−H2O system at 298.15, 323.15, and 343.15 K by the electromotive force method.7 Moore obtained the mean ion activity coefficients of NiCl2−H2O and CoCl2− H2O system at 303.15 K with the vapor pressure method.8 Gilchrist measured the osmotic coefficients of CaCl2−H2O system at 313.15 K by isopiestic method.9 However, these © XXXX American Chemical Society

experimental methods are time-consuming and costly. Meanwhile, only limited thermodynamic results at the corresponding temperature and molality can be obtained through individual experiment, indicating that it is extremely strenuous to directly determine the thermodynamic data over a wide temperature range through experimental methods. Therefore, through limited experiments and appropriate existing thermodynamic parameters, establishing an appropriate theoretical model is necessary and meaningful to predict the thermodynamic properties of aqueous electrolytes at random temperature and molality. The Pitzer ion interaction model is one of the classic theories and it has been widely used for studying the thermodynamic properties of electrolyte solutions.10−12 The thermodynamic properties of solution systems, such as osmotic coefficient, mean ion activity coefficient, enthalpy, and heat capacity, can be well described through the use of a finite number of parameters. Chaudhari calculated the enthalpies of the LiCl−H2O system from 303.15 to 373.15 K by using the limited vapor pressure data.13 Dai predicted the osmotic and mean ion activity coefficients of NiCl2−H2O system by the dilution enthalpy data at 298.15, 303.15, and 308.15 K with the model established.14 Phutela calculated the parameters of the Pitzer equation of the CaCl2−H2O system at 298.15 K and further analyzed the behavior at high molality.15 The heat capacity at constant pressure is an important thermodynamic property, which is Received: May 29, 2017 Accepted: October 31, 2017

A

DOI: 10.1021/acs.jced.7b00483 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

f γ = −Aϕ[I1/2/(1 + bI1/2) + 2 ln(1 + bI1/2)/b]

more easily obtained than other thermodynamic data including vapor pressure and enthalpy via calorimetric experiments. Moreover, good regularity between temperature and heat capacity is well-known. Therefore, a temperature dependent thermodynamic model based on heat capacity is expected to calculate the osmotic coefficient and mean ion activity coefficient of electrolytes. In this work, a temperature-dependent thermodynamic model based on heat capacity was derived through the thermodynamic theories and Pitzer ion interaction model. The CaCl2− H2O system was used to verify the model due to its abundant thermodynamic data in the existing literature. Meanwhile, the proposed model was used to predict the osmotic coefficient and mean ion activity coefficient of CuCl2−H2O and NiCl2−H2O systems.

(0) (1) γ BMX = 2βMX + 2βMX [1 − (1 + αI1/2 − α 2I /2)

exp(−αI1/2)]/α 2I

Φ

L = ν|z Mz X|(AL /2b) ln(1 + bI1/2) − 2νMνXR L L T 2[mBMX + m2(νMz M)CMX ]

G /n w RT = f (I ) + 2 ∑ ∑ λij(I )mimj j

∑ ∑ ∑ μijk mimjmk i

j

k

⎛ ∂ΦL ⎞ C P = ΦCP0 + ⎜ ⎟ ⎝ ∂T ⎠ P

(5-b)

By combining the apparent molar enthalpy equation of Pitzer (4), the apparent molar heat capacity can be expressed as follows: Φ

C P = ΦCP0 + ν|z Mz X|(AJ /2b) ln(1 + bI1/2) J J − 2νMνXRT 2[mBMX + m2(νMz M)CMX ]

where ϕ represents the osmotic coefficient, z is the number of charges and v is the number of ion; M and X represent cation and anion; b is a universal parameter with the value 1.2 kg1/2·mol1/2; α has the value 2.0 kg1/2·mol1/2; Aϕ is the Debye−Hückel parameter for osmotic coeffcient which can be (1) got from the literature16; β(0) MX and βMX are two defined parameters of Pitzer equation.11 From the formula nB ∂Gex ln γ± = n w νmRT ∂nB

(5)

⎛ ∂B L ⎞ J BMX = ⎜ MX ⎟ ⎝ ∂T ⎠ P

(5-1)

⎛ ∂C L ⎞ J CMX = ⎜ MX ⎟ ⎝ ∂T ⎠ P

(5-2)

Φ 0 CP

where is the apparent molar heat capacity at infinite dilution and J represents heat capacity; AJ is Debye−Hückel parameter for heat capacity. Hence, it can be inferred that

the mean ion activity coefficient equation of Pitzer can be obtained:

⎛ ∂β (i) ⎞ (i , L) βMX = ⎜⎜ MX ⎟⎟ ⎝ ∂T ⎠ P

γ ln γ±MX = |z Mz X|f γ + 2mνMνXBMX γ + 2m2(νMνX)3/2 CMX /ν

(5-a)

Φ

(2)

(2-2)

(4-2)

where ΔHs is the enthalpy of solution per mole of solute at infinite dilution. As heat capacity at constant pressure is the first temperature derivative of enthalpy, the apparent molar heat capacity ΦCP can be shown as

ϕ ϕ − 1 = |z Mz X|f ϕ + 2mνMνXBMX /ν

ϕ (0) (1) BMX = βMX + βMX exp( −αI1/2)

⎛ ∂C ⎞ L CMX = ⎜ MX ⎟ ⎝ ∂T ⎠ P

Φ

the osmotic coefficient equation of Pitzer can be expressed as

(2-1)

(4-1)

ΔHs = ΦΔHs + ΦL

1 ∂Gex νmRT ∂n w

f ϕ = −AϕI1/2/(1 + bI1/2)

⎛ ∂B ϕ ⎞ L BMX = ⎜⎜ MX ⎟⎟ ⎝ ∂T ⎠ P

where L is the apparent molar enthalpy, L represents enthalpy, AL is Debye−Hückel parameter for enthalpy. At finite concentration, the enthalpy for solution ΔHs can be express as

(1)

where G is the excess Gibbs energy; nw is the number of kilograms of water; R is the gas constant with the value of 8.314 J·mol−1·K−1; I is the ionic strength; mi, mj, mk are the molalities of all solute species; λij represents the short-range interaction between solute i and j; μijk is for triple interaction among solute particles. According to the formula

ϕ + 2m2(νMνX)3/2 CMX /ν

(4)

Φ

ex

ϕ−1=−

(3-3)

where γ represents the mean ion activity coefficient; CMX is another defined parameters of Pitzer equation,11 like β(0) MX and β(1) MX. Similarly, for apparent molar enthalpy, the Pitzer equation can be expressed as

ex

+

(3-2)

γ ϕ CMX = 3CMX /2

2. THEORETICAL ASPECTS 2.1. Model Derivation. The Pitzer ion interaction model is based on statistical mechanics. For a single electrolyte MX, the function for excess Gibbs free energy is i

(3-1)

(3) B

(6) DOI: 10.1021/acs.jced.7b00483 J. Chem. Eng. Data XXXX, XXX, XXX−XXX


Article

Table 1. Thermodynamic Data of CaCl2−H2O System in the Literature data typea

T/K

molality range/mol·kg−1

data sources

ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ γ γ γ L L L L Cp Cp Cp Cp

298.15 308.15/318.15 313.15 298.15 382.0/413.8 298.15 323.15 303.15/323.15/343.15 298.15/308.15 298.15/328.15/343.15 298.15 298.15 298.15 303.15 293.15 298.15 298.15 348.15−373.15 323−600

0.0877−10.77 0.3043−7.031 0.36−3.005 0.0908−10.771 0.8028−3.2485 2.63−8.83 0.2438−2.7684 1.002−7.885 0.00299−0.075054 0.005828−0.0968 0.00325−0.2989 1.1102 × 10−4∼6.5 0.025−3 0.1896−0.9899 0.034693−1.1102 0.5177−5.6803 0.01256−0.328 0.07520−0.98452 0.05−6.4

Robinson17 Bechtold18 Gilchrist9 Stokes19 Holmes20 Rard21 Davis22 Patil23 Mcleod24 Mussini7 Briggs25 Wagman26 Perachon27 Leung28 Rechard29 Perron30 Perron31 Saluja32 Gates33

Table 3. Characteristics of Chemicals

Notation: ϕ, osmotic coefficient; γ, activity coefficient; L, enthalpy; Cp, heat capacity at constant pressure.

Table 2. Parameters of Osmotic Coefficient Equation Calculated from the Reported Data for the CaCl2−H2O System at Different Temperatures β(0)

β(1)

103Cϕ

298.15 303.15 313.15 323.15 333.15 343.15 353.15 363.15 373.15

0.3159 0.3155 0.3175 0.3220 0.3285 0.3372 0.3477 0.3598 0.3731

1.6140 1.6198 1.5573 1.4293 1.2508 1.0297 0.7808 0.5174 0.2524

−0.9603 −0.8642 −1.1645 −1.8456 −2.5904 −3.5476 −4.6171 −5.7517 −6.9026

source

initial mole fraction purity

Alfa Aesar Alfa Aesar Alfa Aesar

0.999 0.999 0.999

⎛ ∂C ⎞ L = ⎜ MX ⎟ CMX ⎝ ∂T ⎠ P

(7)

⎛ ∂ 2β (i) ⎞ ⎛ ∂β (i) ⎞ (i , J ) ⎟ + 2 ⎜ MX ⎟ βMX = ⎜⎜ MX 2 ⎟ T ⎜⎝ ∂T ⎟⎠ ⎝ ∂T ⎠ P P

(8)

⎛ ∂ 2C ⎞ 2 ⎛ ∂C ⎞ J MX ⎟ + ⎜ MX ⎟ CMX =⎜ 2 T ⎝ ∂T ⎠ P ⎝ ∂T ⎠ P

(9)

where i = 0 or 1 in eqs 6 to 8. For the apparent molar heat capacity equation of Pitzer (5), the parameters can be fitted by the following equations:

a

T/K

chemical name Copper chloride dihydrate Nickel chloride hexahydrate Silver nitrate

(0, J ) βMX = P1/T + P2 + P3T

(10)

(1, J ) βMX = P4 /T + P5 + P6T

(11)

J CMX = P7/T + P8 + P9T

(12)

where P1 to P9 are defined as the fitting parameters. Processing eqs 6 and 8 with definite integration, the parameter β(0) MX can be shown as (0, L) T 2βMX =

T

(0, J ) (0, L) + 298.152 βMX (298.15) ∫298.15 (T 2)βMX

(13) (0) βMX =

T

(0, L) (0) + βMX (298.15) ∫298.15 βMX

(14)

where β(0) MX is the parameter of Pitzer eq 2 for pure salts (0,J) at arbitrary temperature; β(0,L) MX , βMX are the corresponding parameters for enthalpy and heat capacity; β(0) MX(298.15) is the

Figure 1. Comparison of osmotic coefficients (a) and mean ion activity coefficient (b) of CaCl2−H2O system between calculated values and literature values at 298.15 K. C



Article

Table 4. Enthalpy Changes (ΔTT0H) of CuCl2 Aqueous Solutions with Different Temperature Ranges and Molalities at 0.1 MPaa T/K 297.80 302.77 307.72 312.68 317.63 297.80 302.76 307.72 312.68 317.64 297.75 302.70 307.66 312.62 317.57 297.76 302.73 307.69 312.64 317.60 297.78 302.74 307.69 312.65 317.61 297.76 302.73 307.68 312.64 317.59

ΔTT0H/ J T0 = 292.82 114.985 240.096 366.077 492.666 619.327 T0 = 292.85 110.460 230.441 351.818 473.561 595.642 T0 = 292.79 106.137 220.779 337.072 453.746 570.529 T0 = 292.79 102.038 213.776 326.725 439.851 553.579 T0 = 292.73 123.249 227.200 332.404 438.253 544.499 T0 = 292.81 132.060 265.942 400.936 536.812 672.974

T/K

ΔTT0H/ J −1

K, m = 0.2971 mol·kg , 322.63 747.576 327.56 874.307 332.52 1002.064 337.48 1130.050 342.47 1259.018 K, m = 0.5433 mol·kg−1, 322.61 718.279 327.59 841.442 332.52 963.616 337.47 1086.504 342.43 1209.830 K, m = 0.7083 mol·kg−1, 322.53 687.858 327.49 805.461 332.45 923.304 337.41 1041.350 342.36 1159.326 K, m = 1.0184 mol·kg−1, 322.55 667.420 327.51 781.801 332.46 896.231 337.41 1010.907 342.37 1126.031 K, m = 1.4481 mol·kg−1, 322.56 650.895 327.52 757.836 332.47 864.859 337.43 972.361 342.39 1080.093 K, m = 1.8989 mol·kg−1, 322.55 809.922 327.50 947.059 332.46 1084.889 337.42 1223.088 342.38 1361.608

T/K

ΔTT0H/ J

T/K

ΔTT0H/ J

T/K

ΔTT0H/ J

T/K

ΔTT0H/ J

−1

w = 6.28961 g 347.37 1393.607 352.35 1514.882 357.30 1643.281 362.25 1771.786

297.76 302.72 307.68 312.63 317.59

w = 6.25219 g 347.39 1333.316 352.34 1456.685 357.30 1580.404 362.25 1703.946

297.76 302.72 307.67 312.62 317.57

w = 6.11561 g 347.31 1277.434 352.27 1395.877 357.22 1514.143 362.18 1632.673

297.81 302.78 307.73 312.69 317.65

w = 6.16862 g 347.32 1241.107 352.28 1356.568 357.23 1471.918 362.18 1587.357

297.76 302.73 307.68 312.64 317.59

w = 6.03361 g 347.34 1187.803 352.30 1295.894 357.25 1403.895 362.20 1511.991

297.81 302.78 307.74 312.69 317.65

w = 8.12419 g 347.33 1500.121 352.28 1638.858 357.24 1778.053 362.19 1917.096

297.81 302.78 307.73 312.69 317.65

T0 = 292.81 K, m = 2.0118 mol·kg , w = 6.21759 g 95.569 322.53 610.653 347.32 196.945 327.50 715.467 352.27 299.842 332.45 820.189 357.23 403.014 337.41 925.416 362.18 506.841 342.36 1030.686 T0 = 292.82 K, m = 2.4242 mol·kg−1, w = 6.43538 g 95.340 322.53 610.550 347.38 196.590 327.49 715.395 352.33 299.231 332.46 820.855 357.25 402.429 337.46 927.316 362.20 506.145 342.42 1033.248 T0 = 292.82 K, m = 2.9540 mol·kg−1, w = 9.26781 g 137.238 322.60 848.041 347.41 277.272 327.56 992.594 352.37 418.645 332.52 1137.686 357.32 561.138 337.48 1283.243 362.28 704.392 342.45 1429.487 T0 = 292.81 K, m = 3.4545 mol·kg−1, w = 10.29420 g 148.680 322.55 911.359 347.33 298.807 327.50 1066.456 352.28 450.383 332.46 1222.573 357.24 603.251 337.42 1379.327 362.19 756.727 342.38 1536.650 T0 = 292.94 K, m = 4.0357 mol·kg−1, w = 10.50971 g 147.821 322.61 897.545 347.39 295.185 327.57 1050.386 352.35 444.377 332.52 1203.618 357.30 594.305 337.48 1357.775 362.26 745.490 342.43 1512.151 T0 = 292.82 K, m = 4.4524 mol·kg−1, w = 10.28568 g 145.920 322.60 859.347 347.41 286.042 327.56 1004.954 352.37 427.747 332.52 1151.221 357.32 570.790 337.48 1298.045 362.28 714.787 342.45 1445.628

1136.385 1242.050 1348.070 1453.979

1139.462 1245.703 1351.500 1458.103

1575.754 1722.265 1868.651 2015.428

1694.151 1852.079 2010.683 2169.254

1667.285 1822.782 1978.242 2134.208

1593.276 1741.186 1888.961 2037.100

Standard uncertainties: u(T) = 0.05 K, u(P) = 0.001 MPa, u(w) = 0.00024 g, u(m) = 0.0048 mol·kg−1, u(ΔTT0H) = 0.0041 ΔTT0H. Notation: m, molality of CuCl2 aqueous solution; w, mass of CuCl2 aqueous solution. a

parameter of eq 2 at 298.15 K, β(0,L) MX (298.15) is the enthalpy parameter of eq 4 at 298.15 K. Therefore, the derivative result is

which can be used to verify the proposed thermodynamic model. The experimental conditions cover the wide molality range from dilute to near saturation and wide temperature range, as listed in Table 1. These collected data including osmotic coefficient (ϕ), activity coefficient (γ), enthalpy (L), and heat capacity at constant pressure (Cp) are reasonably used. Due to good regularity, the data of Perron30,31 at 298.15 K, Saluja,32 and Gates33 at other temperatures were referenced to calculate the parameters of apparent molar heat capacity equation of Pitzer eq 5. The data of Wagman26 and Perachon27 were utilized to fit the enthalpy parameters of eq 4 at 298.15 K, and the data of Robinson,17 Stokes,19 and Rard21 at 298.15 K were used to fit the osmotic parameters of eq 2. The fitted enthalpy parameters and the parameters of eq 2 are shown in the Supporting Information, Table S1. Combining the apparent molar heat capacity data from the literature and eq 5 gives the parameters of the apparent molar heat capacity equation of Pitzer. Moreover, by fitting with eqs 10, 11, and 12, the defined parameters P1 to P9 for CaCl2−H2O system can be obtained. Then the parameters in the osmotic coefficient equation of Pitzer at different temperatures can be calculated through eq 15.

(0) βMX = (T /2 + 298.152 /2T − 298.15)P1

+ (T 2/6 + 298.153 /3T − 298.152 /2)P2 + (T 3/12 + 298.154 /4T − 298.153 /3)P3 (0, L) + (298.15 − 298.152 /T )βMX (298.15) (0) + βMX (298.15)

(15)

For the parameters β(1) MX and CMX, the forms of equation are the same with β(0) MX when the parameters are P4 to P9. As a consequence, the osmotic coefficient and mean ion activity coefficient can be calculated at arbitrary temperature through the equation above. 2.2. Model Verification. The thermodynamic properties of the CaCl2−H2O system have been well investigated because it is widely used in biology and industrial fields.16 Thus, there is a large number of reliable thermodynamic data in the literature, D



Article

Table 5. Enthalpy Changes (ΔTT0H) of NiCl2 Aqueous Solutions with Different Temperature Ranges and Molalities at 0.1 MPaa ΔTT0H/ J

T/K 297.86 302.79 307.74 312.70 317.66 297.83 302.77 307.72 312.69 317.67 297.87 302.82 307.78 312.72 317.70 297.85 302.82 307.79 312.76 317.71 297.84 302.80 307.77 312.73 317.70

T0 = 292.91 141.728 290.082 439.312 589.170 739.151 T0 = 292.89 139.434 286.238 433.764 582.277 731.445 T0 = 292.90 126.007 257.784 390.646 523.708 658.517 T0 = 292.88 119.595 244.196 369.640 495.856 622.253 T0 = 292.87 119.297 245.991 373.818 502.188 631.537

T/K

ΔTT0H/ J −1

K, m = 0.1432 mol·kg , 322.63 889.731 327.62 1041.140 332.58 1191.843 337.50 1341.521 342.44 1491.981 K, m = 0.3571 mol·kg−1, 322.62 880.031 327.61 1030.100 332.59 1180.243 337.54 1329.560 342.50 1479.385 K, m = 0.9968 mol·kg−1, 322.64 792.832 327.62 928.749 332.58 1064.564 337.54 1200.743 342.50 1337.211 K, m = 1.2401 mol·kg−1, 322.66 749.257 327.62 877.053 332.58 1005.306 337.55 1134.198 342.51 1263.134 K, m = 1.5304 mol·kg−1, 322.66 761.263 327.62 891.546 332.57 1022.041 337.54 1153.461 342.51 1285.197

T/K

ΔTT0H/ J

ΔTT0H/ J

T/K

w = 7.21405 g 347.39 1642.907 352.36 1794.588 357.35 1947.019 362.30 2098.347

297.85 302.81 307.78 312.73 317.71

w = 7.37147 g 347.42 1628.062 352.39 1778.382 357.38 1929.368 362.33 2079.168

297.84 302.80 307.78 312.76 317.71

w = 7.28341 g 347.45 1473.617 352.39 1609.885 357.36 1747.035 362.31 1883.631

297.87 302.83 307.80 312.76 317.72

w = 7.04327 g 347.45 1391.775 352.40 1520.834 357.36 1650.225 362.32 1779.601

297.83 302.81 307.78 312.76 317.73

w = 7.42306 g 347.47 1416.905 352.42 1548.502 357.39 1680.707 362.35 1812.640

297.86 302.82 307.80 312.76 317.72

T0 = 292.89 108.425 224.008 340.669 457.625 575.986 T0 = 292.89 107.686 221.537 336.749 452.782 568.848 T0 = 292.90 102.694 212.156 322.791 434.071 546.139 T0 = 292.88 97.010 199.520 302.818 407.240 512.288 T0 = 292.89 96.761 198.654 302.008 405.902 510.658

T/K

ΔTT0H/ J

T/K

−1

K, m = 1.8741 mol·kg , 322.65 694.010 327.62 813.293 332.58 932.805 337.55 1052.951 342.51 1173.172 K, m = 2.2406 mol·kg−1, 322.65 685.329 327.61 802.852 332.58 921.102 337.55 1039.765 342.52 1158.754 K, m = 2.6084 mol·kg−1, 322.69 659.143 327.65 772.549 332.61 886.502 337.58 1001.154 342.55 1116.196 K, m = 3.6142 mol·kg−1, 322.69 617.878 327.67 724.571 332.64 831.645 337.58 938.585 342.56 1046.826 K, m = 4.1021 mol·kg−1, 322.69 616.400 327.66 722.826 332.64 830.065 337.60 937.377 342.56 1045.103

ΔTT0H/ J

w = 6.99880 g 347.49 1294.122 352.45 1414.752 357.41 1535.475 362.37 1656.215 w = 7.13470 g 347.49 1277.990 352.47 1397.632 357.42 1516.637 362.39 1636.123 w = 7.03981 g 347.49 1230.847 352.46 1346.421 357.41 1461.677 362.40 1577.930 w = 7.10253 g 347.50 1154.551 352.46 1262.988 357.43 1371.841 362.39 1480.592 w = 7.30102 g 347.52 1153.151 352.47 1261.212 357.45 1370.074 362.41 1478.552

Standard uncertainties: u(T) = 0.05 K, u(P) = 0.001 MPa, u(w) = 0.00037 g, u(m) = 0.0055 mol·kg−1, u(ΔTT0H) = 0.0041 ΔTT0H. Notation: m, molality of NiCl2 aqueous solution; w, mass of NiCl2 aqueous solution. a

Table 6. Heat Capacities at Constant Pressure (CP) of CuCl2−H2O System with Different Temperatures and Molalities at 0.1 MPaa CP/J·g−1·K−1

a

m mol·kg−1

298.15K

303.15K

313.15K

323.15K

333.15K

343.15K

353.15K

363.15K

0.2971 0.5433 0.7083 1.0184 1.4481 1.8989 2.0118 2.4242 2.954 3.4545 4.0357 4.4524

3.974 3.856 3.773 3.630 3.458 3.295 3.272 3.156 3.028 2.918 2.806 2.727

3.989 3.873 3.790 3.648 3.477 3.314 3.292 3.178 3.051 2.941 2.830 2.752

4.017 3.901 3.820 3.679 3.510 3.347 3.328 3.217 3.090 2.982 2.872 2.796

4.040 3.923 3.842 3.703 3.537 3.375 3.356 3.250 3.121 3.017 2.907 2.831

4.057 3.939 3.858 3.722 3.557 3.395 3.378 3.275 3.145 3.045 2.934 2.857

4.070 3.949 3.867 3.734 3.571 3.411 3.393 3.293 3.160 3.066 2.953 2.874

4.077 3.954 3.867 3.741 3.578 3.418 3.401 3.305 3.168 3.081 2.964 2.881

4.081 3.953 3.865 3.740 3.578 3.420 3.401 3.309 3.168 3.087 2.968 2.880

Standard uncertainties: u(T) = 0.05 K, u(P) = 0.001 MPa, u(m) = 0.0048 mol·kg−1, u(CP) = 0.0024 CPο.

temperatures. To verify the applicability of the derived thermodynamic model, the osmotic coefficient data from Robinson,17 Stokes,19 and Rard21 as well as the mean ion activity coefficient data from Mcleod,24 Mussini7 and Briggs25 were selected to compare with the calculated results at 298.15 K. As shown in Figure 1, the calculated results can agree well with the osmotic and mean ion activity coefficients of CaCl2−H2O system in the literature. Although the limited experimental data of mean ion

The detailed values on the parameters of osmotic coefficient equation of Pitzer for CaCl2−H2O system at different temperatures are listed in Table 2. By substituting the parameters in Table 2 into eq 2 and 3, the osmotic coefficients and mean ion activity coefficients at different temperatures can be predicted. Figure S1 in Supporting Information presents the calculated osmotic coefficients and mean ion activity coefficients of CaCl2−H2O system at different E



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Table 7. Heat Capacities at Constant Pressure (CP) of NiCl2−H2O System with Different Temperatures and Molalities at 0.1 MPaa CP/J·g−1·K−1 m mol·kg

−1

0.1432 0.3571 0.7035 0.9968 1.2401 1.5304 1.8741 2.2406 2.6084 3.6142 4.1021 a

298.15K

303.15K

313.15K

323.15K

333.15K

343.15K

353.15K

363.15K

4.072 3.932 3.713 3.549 3.452 3.336 3.220 3.110 3.018 2.789 2.702

4.074 3.938 3.730 3.567 3.471 3.355 3.239 3.131 3.040 2.813 2.726

4.078 3.950 3.756 3.597 3.504 3.388 3.273 3.166 3.079 2.854 2.768

4.083 3.960 3.782 3.622 3.531 3.414 3.301 3.195 3.111 2.888 2.803

4.089 3.967 3.798 3.640 3.551 3.434 3.321 3.217 3.136 2.916 2.829

4.096 3.973 3.808 3.652 3.565 3.448 3.335 3.231 3.155 2.937 2.848

4.103 3.977 3.810 3.659 3.572 3.455 3.343 3.239 3.166 2.951 2.860

4.111 3.978 3.806 3.659 3.573 3.455 3.345 3.240 3.170 2.959 2.863

Standard uncertainties: u(T) = 0.05 K, u(P) = 0.001 MPa, u(m) = 0.0055 mol·kg−1, u(CP) = 0.0024 CP.

Table 8. Thermodynamic Data of the CuCl2−H2O and NiCl2−H2O Systems in the Literature data type ϕ ϕ ϕ ϕ ϕ L L L ϕ ϕ ϕ ϕ L L L

T/K

molality range/mol·kg−1

CuCl2−H2O System 0.1085−2.769 2.236−5.750 0.1875−2.2482 3.5302−5.7861 0.1−5.0 0.0111−2.7753 4.8371−5.3886 0.1271−1.1066 NiCl2−H2O System 298.15 0.1188−2.123 298.15 1.1634−5.714 298.15 0.29976−6.1364 298.15 1.0521−4.9023 298.15−318.15 0.05115−4.0635 298.15 0.1031−0.9733 298.15 0.0005−2.2202 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15

from Gilchrist,9 Davis,22 and Patil23 were compared with the calculated values. It can be found from Figure S2 that the calculated results at 303.15, 313.15, 328.15, and 343.15 K also agree with the reported values, respectively. Although the available experimental data of mean activity coefficient at high temperature and high molality are limited, by comparing the results from Mcleod,24 Mussini,7 and Patil,23 it can be found from Figure S3 that the calculated values at 303.15, 313.15, 328.15, and 343.15 K can agree with the reported mean ion activity coefficient values of CaCl2−H2O solutions, respectively. Therefore, these results indicate that the proposed model is reasonable for predicting the osmotic and mean ion activity coefficients of metal chloride solutions at varied temperatures.

data sources Robinson36 Brown37 Downes38 Rard39 Stokes40 Wagman26 Partington41 Schreiber42 Robinson43 Stokes40 Rard39 Shults44 Dai14 Schreiber42 Wagman26

3. EXPERIMENTAL SECTION 3.1. Apparatus and Accuracy Test. The Tian−Calvet heat flow calorimeter (C80, SETARAM) was used in the experiment. Because the steel cell is easily corroded by chloride ions, the Teflon casings were put into the sample cell and reference cell. To confirm the reliability of the apparatus, the heat capacities at constant pressure of redistilled water from 298.15 to 363.15 K were measured. As shown in Table S2 in Supporting Information, the relative deviation between experiment and literature data35 was less than 0.45%, indicating the apparatus and experimental method were reliable.

activity coefficient is available, the calculated values in this work are consistent with the results calculated from electromotive force measurements of galvanic cells by Staples.34 Moreover, the osmotic coefficient data at temperatures other than 298.15 K

Figure 2. Literature data of osmotic coefficients for the (a) CuCl2−H2O system and (b) NiCl2−H2O system at 298.15 K. F



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Figure 3. Literature data of dilution enthalpies for the (a) CuCl2−H2O system and (b) NiCl2−H2O system at 298.15 K.

Table 9. Thermodynamic Parameters for Osmotic Coefficient Equation and Apparent Molar Enthalpy Equation at 298.15 Ka osmotic coefficient equation solution system CuCl2−H2O NiCl2−H2O a

β

(0)

0.2531 0.3805

β

(1)

1.9215 1.1301

apparent molar enthalpy equation

3

2

10 C

R

−7.5696 −4.6920

0.9891 0.9945

(0,L)

10 β

103 β(1,L)

105 CL

R2

−2.6404 −1.9220

3.6331 −2.6404

6.9327 3.4663

0.9921 0.9914

3

R is the correlation coefficient in equation fitting.

Table 10. Fitted Parameters in Equations 10 to 12 for Both CuCl2−H2O and NiCl2−H2O Systemsa

a

parameters

CuCl2−H2O system

R2

NiCl2−H2O system

R2

P1 P2 P3 P4 P5 P6 P7 P8 P9

0.3992 −2.4660 × 10−03 3.6801 × 10−06 32.8563 −0.1987 0.2941 × 10−03 −0.02181 0.1341 × 10−03 −1.9857 × 10−07

0.9889 0.9889 0.9889 0.9915 0.9915 0.9915 0.9931 0.9931 0.9931

0.2329 −1.5710 × 10−03 2.5920 × 10−06 19.3789 −0.1216 0.1896 × 10−03 −0.01591 0.1044 × 10−03 −1.6953 × 10−07

0.9923 0.9923 0.9923 0.9949 0.9949 0.9949 0.9968 0.9968 0.9968

R is the correlation coefficient in equation fitting.

3.2. Chemicals. All chemicals in the experiment were of guaranteed grade and used as received. The relevant information on chemicals was listed in Table 3. The near saturated stock solution was prepared by dissolving a certain amount of CuCl2·2H2O or NiCl2·6H2O into redistilled water, respectively. The solutions with different molalities were prepared by weight dilution of the stock solution. The molality of stock solutions can be confirmed by the AgCl precipitation method. Excess AgNO3 solution was added to the certain amount of stock solution. The mass of chloride ions can be determined through the mass of AgCl precipitation, and then the molalities of stock solutions can be calculated. All results are presented as the average value of triplicate analysis. The related data and the uncertainties are shown in Table S3.

from 293.15 to 363.15 K were measured by C80 calorimeter. The determined molality range is from 0.2971 to 4.4524 mol·kg−1 for CuCl2 solutions and from 0.1432 to 4.1021 mol·kg−1 for NiCl2 solutions. The detailed results were listed in Table 4 and Table 5, respectively. The cubic polynomial eq 16 was used to fit the experimental data in Table 4 and Table 5 and the parameters a0, a1, a2, a3 can be obtained for the CuCl2−H2O system and NiCl2−H2O system, which are shown in Table S4 and Table S5, respectively. Similarly, for the blank experiment (both sample cell and reference cell were empty), the enthalpy changes ΔTT0Hblank are listed in Table S6, and the related parameters can be obtained by fitting with eq 17 and listed in Table S7. ΔTT0H = a0 + a1T + a 2T 2 + a3T 3

(16)

where a0, a1, a2, and a3 are four parameters for fitting.

4. RESULTS AND DISCUSSION 4.1. Heat Capacity at Constant Pressure of CuCl2−H2O System and NiCl2−H2O System. The enthalpy changes ΔTT0H of CuCl2 and NiCl2 aqueous solutions in the temperature range

ΔTT0Hblank = b0 + b1T + b2T 2 + b3T 3

(17)

where b0, b1, b2, and b3 are four parameters for fitting of blank experiment. G



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contribution of the blank experiment, the heat capacities at constant pressure of sample solutions can be calculated as follows. Cp =

(a1 − b1) + 2(a 2 − b2)T + 3(a3 − b3)T 2 w

(18)

where w is the mass of the sample solution, the unit of CP is J·g−1·K−1. The enthalpy changes of CuCl2 and NiCl2 aqueous solutions in the temperature range from 293.15 to 363.15 K were measured by C80 calorimeter. The determined molality range is from 0.2971 to 4.4524 mol·kg−1 for CuCl2 solutions and from 0.1432 to 4.1021 mol·kg−1 for NiCl2 solutions. The detailed results were listed in Table 4 and Table 5, respectively. By fitting the original data of enthalpy changes with eq 16, the parameters a0, a1, a2, a3 can be obtained for CuCl2−H2O system and NiCl2− H2O system, which are shown in Table S4 and Table S5, respectively. Similarly, the parameters for eq 17 can be obtained by fitting the enthalpy changes listed in Table S6. The results are shown in Table S7. Therefore, on the basis of the fitted parameters, the heat capacity at constant pressure of solution can be calculated with eq 18. The detailed data on heat capacity at constant pressure for CuCl2−H2O system and NiCl2−H2O system are displayed in Table 6 and Table 7, respectively. During this experiment, the temperature precision of the calorimeter was 0.01 K and the precision for weighing was 0.00001 g. The following aspects could induce the deviation of heat capacity. On one hand, the molalities of the solutions were confirmed by AgCl precipitation method and the uncertainty is within 0.50%. On the other hand, the mass of the solution could be lost due to the volatilization at high temperature. However, by comparing the mass of the solution before and after calorimetric experiments, the mass deviation is lower than 0.16% for pure water and 0.20% for metal chloride aqueous solution, respectively. Accordingly, the hydrolysis of metal ions can be ignored. As a consequence, it is expected that the obtained data on heat capacity at constant pressure of both CuCl2−H2O system and NiCl2−H2O system are reliable. 4.2. Osmotic Coefficients and Mean Ion Activity Coefficients of CuCl2−H2O System and NiCl2−H2O System. The apparent molar heat capacity can be calculated as the following equation:

Figure 4. Temperature-dependent thermodynamic parameters of eq 2 for both the CuCl2−H2O system and NiCl2−H2O system.

As the heat capacity at constant pressure is the first temperature derivative of enthalpy change, after subtracting the

Φ

C P = MCP + 1000(CP − CPw )/m

(19)

Figure 5. Osmotic coefficients (a) and mean ion activity coefficients (b) calculated with the proposed thermodynamic model of CuCl2−H2O system at different temperatures and molalities. H



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Figure 6. Osmotic coefficients (a) and mean ion activity coefficients (b) calculated with the proposed thermodynamic model of NiCl2−H2O system at different temperatures and molalities.

Figure 7. Relative deviation of osmotic coefficients between the calculated and the reported values at 298.15 K. (a) CuCl2−H2O system; (b) NiCl2− H2O system.

where ΦCP is the apparent molar heat capacity; M is the molar mass of CuCl2 and NiCl2, g·mol−1; m is the molality of the solution, mol·kg−1; CP is the heat capacity at constant pressure of the solution; CwP is the heat capacity at constant pressure of pure water.35 After obtaining the apparent molar heat capacity with eq 19, the parameters of apparent molar heat capacity equation of Pitzer at different temperatures can be further obtained by fitting eq 5. Similar to that of the CaCl2−H2O system, the published thermodynamic data for the CuCl2−H2O system and NiCl2−H2O system including osmotic coefficient and enthalpy at 298.15 K in the molality range from 0.1 to 6 mol·kg−1 are collected and listed in Table 8. To obtain the important parameters β(0,L) MX (298.15) and (0) βMX(298.15) in eq 15, the reported data of osmotic coefficients and dilution enthalpies of both CuCl2−H2O and NiCl2−H2O systems at 298.15 K were plotted in Figure 2 and Figure 3, respectively. It can be seen that these data reported by different authors have good consistency, which can be favorable for parameter fitting with eqs 2 and 4. The fitted β(0,L) MX (298.15) and β(0) MX(298.15) for the CuCl2−H2O and NiCl2−H2O systems are presented in Table 9. Then by fitting the parameters in eq 5 with eqs 10, 11, and 12, the defined parameters P1 to P9 of

CuCl2−H2O and NiCl2−H2O systems can be obtained, respectively. The results are listed in Table 10. On the basis of eq 15, the thermodynamic parameters of eq 2 at different temperatures can be calculated for both CuCl2−H2O and NiCl2−H2O systems. As shown in Figure 4, it is clear that three parameters in eq 2 including β(0), β(1), and Cϕ are dependent on temperature. Moreover, there is a similar trend as temperature rises for both CuCl2−H2O and NiCl2−H2O systems. The value of β(0) linearly decreases when the temperature increases from 298.15 to 363.15 K; however, the value of β(1) for the CuCl2−H2O system is more sensitive to temperature than that of the NiCl2−H2O system, especially after 310 K, which could be attributed to the stronger ion association effects. Unlike β(0) and β(1), the value of Cϕ increases with the increase of temperature for both systems, while the Cϕ value of CuCl2−H2O system increases more obviously than that of NiCl2−H2O system. On the basis of these temperature-dependent thermodynamic parameters, the osmotic and mean ion activity coefficients of CuCl2 and NiCl2 solutions can be calculated with eqs 2 and 3. As shown in Figure 5, the osmotic coefficients show a growth trend after a sharp reduction with the increase of molality at I



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Notes

each temperature. At the same CuCl2 molality, the coefficient obviously decreases with rising temperature. Although the mean ion activity coefficient has a similar trend, it barely increases while the temperature is greater than 333.15 K. Similarly with that of the CuCl2−H2O system, the osmotic and mean ion activity coefficients of NiCl2−H2O system also present the same trend except that the growth degree is more obvious than that for the CuCl2−H2O system (Figure 6). Figure 7 shows the relative deviation between the calculated and the cited osmotic coefficients at 298.15 K. The maximum deviation is less than 0.018 for the CuCl2−H2O system and less than 0.014 for the NiCl2−H2O system, indicating the proposed thermodynamic model can reliably extend the data range to higher molalities and temperatures of metal chloride solutions.

The authors declare no competing financial interest.

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(1) Millero, F. J. The physical chemistry of natural waters. Pure Appl. Chem. 1985, 57, 1015−1024. (2) Liu, W.; Mcphail, D. C. Thermodynamic properties of copper chloride complexes and copper transport in magmatic-hydrothermal solutions. Chem. Geol. 2005, 221, 21−39. (3) Awakura, Y.; Kawasaki, Y.; Uno, A.; Sato, K.; Majima, H. Activities of water and HCl in Aqueous solution systems of HCl-MCln including CuCl2, NiCl2 and FeCl3. Hydrometallurgy 1987, 19, 137− 157. (4) Lundström, M.; Aromaa, J.; Forsén, O.; Hyvärinen, O.; Barker, M. H. Leaching of chalcopyrite in cupric chloride solution. Hydrometallurgy 2005, 77, 89−95. (5) Hu, G.; Chen, D.; Wang, L.; Liu, J.; Zhao, H.; Liu, Y.; Qi, T.; Zhang, C.; Yu, P. Extraction of vanadium from chloride solution with high concentration of iron by solvent extraction using D2EHPA. Sep. Purif. Technol. 2014, 125, 59−65. (6) Königsberger, E.; May, P.; Harris, B. Properties of electrolyte solutions relevant to high concentration chloride leaching I. Mixed aqueous solutions of hydrochloric acid and magnesium chloride. Hydrometallurgy 2008, 90, 177−191. (7) Mussini, T.; Pagella, A. Standard potentials of the sodium amalgam electrode at various temperatures, with related thermodynamic functions. J. Chem. Thermodyn. 1971, 16, 281−288. (8) Moore, T. E.; Gootman, E. A.; Yates, P. C. Activities of Transition Metal Chlorides in Aqueous Hydrochloric Acid Mixtures. I. Nickel(II) Chloride and Cobalt(II) Chloride. J. Am. Chem. Soc. 1955, 77, 298−304. (9) Gilchrist, M. A.; Baabor, J. S.; Delgado, E. J. Isopiestic study of (calcium chloride + water) and (calcium chloride + magnesium chloride + water) at T = 313.15 K. J. Chem. Thermodyn. 2001, 33, 405−411. (10) Gupta, A. R. Thermodynamics of electrolytes in mixed solvents. Application of Pitzer’s thermodynamic equations to activity coefficients of 1:1 electrolytes in methanol-water mixtures. J. Phys. Chem. 1979, 83, 2986−2990. (11) Pitzer, K. S. Thermodynamics of electrolytes. I. Theoretical basis and general equations. J. Phys. Chem. 1973, 77, 268−277. (12) Pitzer, K. S.; Mayorga, G. Thermodynamics of electrolytes. II. Activity and osmotic coefficients for strong electrolytes with one or both ions univalent. J. Phys. Chem. 1973, 77, 2300−2308. (13) Chaudhari, S. K.; Patil, K. R. Thermodynamic Properties of Aqueous Solutions of Lithium Chloride. Phys. Chem. Liq. 2002, 40, 317−325. (14) Dai, P.; Huang, H.; Ding, Z.; He, Y.; Liu, S. Osmotic coefficient and mean ion activity coefficient of NiCl2 aqueous solution at several temperatures. J. Chem. Thermodyn. 2016, 100, 72−78. (15) Phutela, R. C.; Pitzer, K. S. Thermodynamics of aqueous calcium chloride. J. Solution Chem. 1983, 12, 201−207. (16) Pitzer, K. S. Activity Coefficients in Electrolyte Solutions, 2nd ed.; CRC Press, 1991. (17) Robinson, R. A. Part II. The activity and osmotic coefficients of aqueous calcium chloride at 298.15 K. Trans. Faraday Soc. 1940, 36, 735−738. (18) Bechtold, M. F.; Newton, R. F. The Vapor Pressures of Salt Solutions. J. Am. Chem. Soc. 1940, 62, 1390−1393. (19) Stokes, R. H. A thermodynamic study of bivalent metal halides in aqueous solution. Part XIII. Properties of calcium chloride solutions up to high concentrations at 25°C. Trans. Faraday Soc. 1945, 41, 637− 641. (20) Holmes, H. F.; Baes, C. F.; Mesmer, R. E. lsopiestic studies of aqueous solutions at elevated temperatures I. KCl, CaCl2, and MgCl2. J. Chem. Thermodyn. 1978, 10, 983−996. (21) Rard, J. A.; Habenschuss, A.; Spedding, F. H. A Review of osmotic coefficients of aqueous CaCl2 at 25°C. J. Chem. Eng. Data 1977, 22, 180−186.

5. CONCLUSION A temperature-dependent thermodynamic model based on heat capacity was reasonably derived from Pitzer equation and the basic thermodynamic relationship of osmotic coefficient, mean ion activity coefficient, and enthalpy. The abundant thermodynamic data of CaCl2−H2O system were used to verify the proposed model. The results indicated that the predicted osmotic coefficients and mean ion activity coefficients of CaCl2−H2O system agree well with the reported values, suggesting that the proposed thermodynamic model is reliable. Meanwhile, the heat capacities at constant pressure of CuCl2−H2O and NiCl2−H2O systems in the range of 298.15 K−363.15 K were calculated from the enthalpy changes determined by the calorimetric experiments, in which the molality range is 0.2971 mol·kg−1 to 4.4524 mol·kg−1 for the former and 0.1432 mol·kg−1 to 4.1021 mol·kg−1 for the latter, respectively. Through the limited experiments and the derived thermodynamic model, the osmotic and mean ion activity coefficients of both CuCl2−H2O and NiCl2−H2O systems were predicted in a wide temperature range within the molality of less than 4 mol·kg−1. The calculated results are well consistent with the data in the literature and further indicate that the thermodynamic model based on heat capacity is reliable and convenient.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.7b00483. Osmotic coefficients; mean ion activity coefficients; comparison with literature values; fitted parameter of the CaCl2−H2O system; heat capacity of redistilled water; molalities of CuCl2 and NiCl2 stock solutions; paranmeters for equation 16 (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Jiugang Hu: 0000-0002-5702-9547 Funding

This work was financially supported by National Basic Research Program of China (2014CB643401), the National Natural Science Foundation of China (No.51134007) and the Hunan Provincial Science and Technology Plan Project of China (No. 2016TP1007). J



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A Temperature-Dependent Thermodynamic Model Derived from Heat

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