Measurement and Prediction of Molar Volumes of Bivalent Metal Sulfates and Their Mixtures in Aqueous Solution

Abstract

Densities of aqueous solutions of MnSO$_4$, CoSO$_4$, NiSO$_4$ and CuSO$_4$ have been measured by vibrating tube densimetry up to near-saturation concentrations over the temperature range 293.15 $\le T/$K $\le$ 343.15 at 5 K intervals and at 0.1 MPa pressure. Apparent molar volumes, $V_\phi$, calculated from the densities revealed close similarities among all four electrolytes, with their $V_\phi$ values essentially differing by constant concentration-independent addends for each salt at each temperature. It was found that there is an almost linear relationship between $V_\phi$(MSO$_4$,aq) and the M–O bond length of the hydrated cations obtained from structural studies. This relationship can be used to predict values of $V_\phi$(MSO$_4$,aq) for other sulfate salts over wide ranges of concentration and temperature, providing that the requisite structural data are available. Measurements of selected ternary, quaternary and quinary mixtures of these electrolytes at constant total concentrations showed that their $V_\phi$ values almost always exhibit linear mixing (Young’s rule) behavior at all temperatures investigated. This finding can be exploited to predict the volumetric properties of solutions containing complex mixtures of bivalent metal sulfates.

1 Introduction

Aqueous solutions of the sulfate salts of the bivalent first transition row metal ions have vast applications in agriculture [1], marine science [2], medicine [3], and so on. Arguably, however, their most important roles are those that each plays (to varying degree) in the hydrometallurgical extraction, separation, recovery and purification of their respective metals [4].

The volumetric properties of metal sulfate solutions are required industrially for process engineering purposes [5]. However, while these solutions have been reasonably well characterized at near-ambient temperatures, they have been surprisingly little-studied at the higher temperatures more relevant to most practical processes (Table 1). To give just one example, the electrolytic deposition of nickel from acidified NiSO$_4$(aq) solutions, the standard method for nickel electrorefining [6], is invariably conducted at temperatures around 333 K. Similar conditions apply with respect to the hydrometallurgical processing of the other first transition row metals.

This paper presents a detailed investigation of the densities of the aqueous solutions of MnSO$_4$, CoSO$_4$, NiSO$_4$ and CuSO$_4$ over the temperature range 293.15 $\le T$/K $\le$ 343.15 and up to near-saturation concentrations. Iron(II) sulfate was omitted from this study because of the difficulty of maintaining constant its oxidation state and chemical speciation in near-neutral solutions [7, 8]. Measurements of ZnSO$_4$(aq) were unnecessary as they have been reported elsewhere under the present conditions [9].

In addition to measurements of the binary metal(II) sulfate solutions (containing just one solute + the solvent), data have also been obtained for representative ternary (two solutes + the solvent), quaternary (three solutes + the solvent) and one quinary (four solutes + the solvent) solutions. These data are of particular practical interest as industrial process solutions almost invariably contain mixtures of solutes. Such data can also be used as a stern test of a simple linear mixing model (Young’s rule) [10] which, if proven to be applicable, can be used to predict with good accuracy the volumetric properties of mixtures over wide ranges of conditions. Confirmation of the applicability of Young’s rule for any given system represents a major improvement over the purely empirical models that are nowadays routinely employed in the computer packages used for process design and optimization. Conformance to Young’s rule also minimizes the need for economically prohibitive experimental efforts that are required to define the multi-dimensional property spaces of real systems.

Table 1 Selected literature publications on the volumetric behavior of MnSO$_4$(aq), CoSO$_4$(aq), NiSO$_4$(aq) and CuSO$_4$(aq)

2 Experimental Section

2.1 Reagents

Sources and purities of the reagents used are given in Table 2. All salts were commercial materials recrystallized once from slightly acidified aqueous solutions. Concentrated stock solutions were prepared by dissolving the purified solid metal(II) sulfate hydrate in ultra-pure water (Ibis Technology, Australia; resistivity >18 M$\Omega\cdot$cm) and acidifying with sulfuric acid to pH $\approx$ 4 to minimize cation hydrolysis. These solutions were allowed to stand for at least 12 h, filtered (Millipore 0.5 μm) and, standardized (±0.2%) by titration with a commercial concentrated volumetric standard containing ethylenediaminetetraacetate (EDTA), using either Eriochrome Black T or murexide indicator [23]. Occasional cross-checks were made by evaporative gravimetry, which involved sub-boiling evaporation at $\sim$340 K to superficial dryness followed by heating to constant mass at about 570 K. Such assays had a typical reproducibility of ±0.05% and agreed with the titrimetric results to within 0.2%. Working solutions were prepared by mass dilution of the stock solutions with ultra-pure water. Mixed electrolyte solutions, covering all the ternary, quaternary and one quinary solution, were prepared by mass from $\sim$1.0 mol$\cdot$kg$^{-1}$ sample solutions, so as to obtain approximately constant stoichiometric molalities of the various metal ions present. Buoyancy corrections were applied throughout.

Table 2 Sample sources and purities

2.2 Density Determinations

Densities were determined with an Anton Paar (Graz, Austria) DMA 5000 M glass vibrating-tube densimeter (vtd). The calibration of this instrument with water and air, and the measurement protocol have been described in detail elsewhere [24]. Measurements were performed isoplethically over the temperature range 293.15 $\le$ T/K $\le$ 343.15 at 5 K intervals. The temperature of the solutions in the densimeter tube was controlled to ± 0.002 K using the in-built thermostat of the vtd. The experimental pressure of (102 ± 2) kPa was obtained from the in-built sensor of the densimeter. Reproducibility of the measured densities was generally within ± 10 μg·cm$^{-3}$.

3 Results and Discussion

3.1 Densities and Apparent Molar Volumes

The experimental density differences between the sample solutions and pure water $\Delta \rho$ were converted to apparent molar volumes $V_\phi$ with the usual equation

$$\begin{aligned} V_\phi = \frac{M}{\Delta \rho +\rho _\textrm{w}} - \frac{\Delta \rho }{m \rho _\textrm{w}(\Delta \rho + \rho _\textrm{w})} \end{aligned}$$

(1)

where $\rho _\textrm{w}$ is the density of pure water at the measurement conditions, which was calculated from the IAPWS-95 EOS [25]. In Eq. 1, m is the molality (mol$\cdot$kg$^{-1}$) and M is the molar mass of the anhydrous solute. The latter were calculated using the current IUPAC atomic masses [26] and had values (in g$\cdot$mol$^{-1}$) of 150.99 (MnSO$_4$), 154.99 (CoSO$_4$), 154.75 (NiSO$_4$) and 159.60 (CuSO$_4$). Each of these molar masses has an uncertainty of approximately ±0.02 g$\cdot$mol$^{-1}$ mostly ascribable to sulfur.

The values of $\Delta \rho$ and $V_\phi$ at molalities m, temperatures T and pressure p = 0.1 MPa for the binary MSO$_4$(aq) solutions are given in Tables 3, 4, 5 and 6. These results are compared graphically in Figs. 1, 2, 3 and 4 with selected literature data (Table 1) and are discussed in the following sub-sections.

Table 3 Experimental density differences $\Delta \rho$ and apparent molar volumes $V_\phi$ for aqueous solutions of MnSO$_4$ at Molalities m, Temperatures T and Pressure = 0.1 MPa

Table 4 Experimental density differences $\Delta \rho$ and apparent molar volumes $V_\phi$ for aqueous solutions of CoSO$_4$ at Molalities m, Temperatures T and Pressure = 0.1 MPa

Table 5 Experimental density differences $\Delta \rho$ and apparent molar volumes $V_\phi$ for aqueous solutions of NiSO$_4$ at Molalities m, Temperatures T and Pressure = 0.1 MPa

Table 6 Experimental density differences $\Delta \rho$ and apparent molar volumes $V_\phi$ for aqueous solutions of CuSO$_4$ at Molalities m, Temperatures T and Pressure = 0.1 MPa

Fig. 1

Comparison at: a T = 298.15 K and b T = 318.15 K of present (black dots) and literature (other symbols [11,12,13]) values of apparent molar volumes $V_\phi$(MnSO$_4$,aq) at p = 0.1 MPa. Orange star: $V^\circ$ calculated from the ionic volumes tabulated by Marcus [27] (Color figure online)

Fig. 2

Comparison of present (black dots) and literature (other symbols [12, 14]) values of apparent molar volumes $V_\phi$ for CoSO$_4$(aq) at T = 298.15 K and p = 0.1 MPa. Orange star: $V^\circ$ calculated from the ionic volumes tabulated by Marcus [27] (Color figure online)

Fig. 3

Comparison at: a T = 298.15 K and b T = 328.15 K of present (black dots) and literature (other symbols [12, 14,15,16]) values of apparent molar volumes $V_\phi$(NiSO$_4$,aq) at p = 0.1 MPa. Orange star: $V^\circ$ calculated from the ionic volumes tabulated by Marcus [27] (Color figure online)

Fig. 4

Comparison at: a T = 298.15 K and b T = 328.15 K of present (black dots) and literature (other symbols [11, 12, 14, 17,18,19,20,21,22]) values of apparent molar volumes $V_\phi$(CuSO$_4$,aq) at p = 0.1 MPa. Orange star: $V^\circ$ calculated from the ionic volumes tabulated by Marcus [27] Red star: using unpublished $V^\circ$(Cu$^{2+}$,aq) value [28] (Color figure online)

MnSO₄(aq). Compared with the other bivalent metal sulfate solutions investigated here (see below), few studies have been reported for MnSO$_4$(aq), even at 298.15 K (Fig. 1, Table 1). The present results are slightly higher (by $\sim$1 cm$^3 \cdot$mol$^{-1}$) than the literature values [11,12,13] but show the same trend with respect to concentration (Fig. 1a) and are consistent with the infinite dilution volume $V^\circ$, calculated from the $V^\circ$(ion) values tabulated by Marcus [27]. At other temperatures, only the results of Przepiera and Zielenkiewicz [13] are available for comparison. Their values at T = 318.15 K (the highest temperature reported) are again lower than the present results by about 1 cm$^3 \cdot$mol$^{-1}$ (Fig. 1b).

CoSO₄(aq). Independent studies of the volumetric properties of CoSO$_4$(aq) at T= 298.15 K [12, 14] are in unusually good agreement with the present results (Fig. 2). However, in marked contrast, no reliable $V_\phi$ values appear to be available for CoSO$_4$(aq) solutions at other temperatures.

NiSO₄(aq). The volumetric properties of NiSO$_4$(aq) have been studied over wide of ranges of concentration and temperature. At T = 298.15 K (Fig. 3a), the present results and the literature data fit within a span of $\sim \pm$0.5 cm$^3 \cdot$mol$^{-1}$. At higher temperatures (Fig. 3b), the pycnometric results of Phillips [15] and Isono [16] show similar agreement with the present data.

CuSO₄(aq). The volumetric properties of CuSO$_4$(aq) at 298.15 K have been studied extensively over a wide concentration range (Fig. 4a). With the exception of the results of Puchalska et al. [19] (hexagons) at m $\ge$ 0.2 mol$\cdot$kg$^{-1}$, and the obvious outliers from the historic study by Herz [11] (triangles down), the various independent $V_\phi$ values are in reasonable agreement with each other and with the present results at this temperature. Thus, almost all the reported $V_\phi$ results lie within a span of $\sim$1 cm$^3 \cdot$mol$^{-1}$ over the whole concentration range, with the present values lying comfortably within the spread of the literature data (Fig. 4a).

In contrast to the other systems studied here, numerous $V_\phi$ values have been reported in the literature for CuSO$_4$(aq) at temperatures over the range 283.15 $\le$ T/K $\le$ 333.15 (Fig. 4). The spread in the literature data and their relation to the present values mimic those at 298.15 K (Fig. 4a). Similar results were obtained at other temperatures (data not shown).

3.2 Comparison of Bivalent Metal Sulfate Solutions

Comparison of the data obtained for the present electrolytes with those of other bivalent metal sulfates (MgSO$_4$, FeSO$_4$, ZnSO$_4$ and CdSO$_4$) provides a clear-cut example of the well-known chemical similarities of the aqueous solutions of these salts (Fig. 5). Like the more-limited range of bivalent metal sulfates studied previously (see Fig. 5 in Ref. [14]) the $V_\phi$ values of the present salts differ (in essence) only by fixed, almost concentration-independent addends. Inclusion of these addends confirms that, as found previously [14], all $V_\phi$(MSO$_4$,aq) can be made to lie on a single common line, from infinite dilution to near-saturation concentrations, at temperatures up to at least 343.15 K (the maximum temperature investigated here).

Fig. 5

Comparison at: a T = 298.15 K and b T = 338.15 K of apparent molar volumes $V_\phi$(MSO$_4$,aq) at p = 0.1 MPa. Filled symbols, this work: pink circles, MnSO$_4$(aq); red squares, CoSO$_4$; green triangles, NiSO$_4$(aq); blue diamonds, CuSO$_4$(aq). Unfilled symbols, literature: red squares, MgSO$_4$(aq) [29]; orange triangles, FeSO$_4$(aq) [30]; black circles, ZnSO$_4$(aq) [9]; purple diamonds, CdSO$_4$(aq) [12]. (Note: data at T = 338.15 K were used in order to include MgSO$_4$(aq)) (Color figure online)

3.3 Prediction of Apparent Molar Volumes of MSO$_4$(aq) Solutions

A plot of $V_\phi$(MSO$_4$,aq) against $r_\mathrm {M-O}$, the metal-oxygen bond length in the hydrated bivalent first-row transition metal ions, obtained from structural studies, produces an almost-straight line (Fig. 6a). This strongly suggests that the observed differences in $V_\phi$(MSO$_4$,aq) in Fig. 5 are largely determined by the size (volume) of the hydrated cations. Furthermore, because $r_\mathrm {M-O}$ does not vary much with temperature over the present range of interest, the relationship in Fig. 6a persists up to at least T = 343.15 K with an essentially unchanged slope (Fig. 6b). An interesting exception to this systematic behavior is found for CuSO$_4$(aq). Thus, while the value of $V_\phi$(CuSO$_4$,aq) conforms well with the other MSO$_4$(aq) at T = 298.15 K (Fig. 6a), it deviates significantly as the temperature is increased (Fig. 6b).

Fig. 6

Apparent molar volumes $V_\phi$(MSO$_4$,aq) at: a T = 298.15 K and b T = 343.15 K at concentration m = 1.0 mol$\cdot$kg$^{-1}$ at 0.1 MPa pressure as a function of $r_\mathrm {M-O}$ in the hydrated cations. Values of $V_\phi$(MSO$_4$,aq): present results (M = Mn, Co, Ni and Cu); literature values (M = Zn [9] and Fe [30]). The unfilled square for Fe shows the effect of increasing $r_\mathrm {Fe-O}$ from 0.212 nm to 0.213 nm (see text). The lines are best fits to the data (excluding Cu). Values of $r_\mathrm {M-O}$ are taken to be independent of T

By inspecting the isopleths of the different MSO$_4$(aq) solutions at, say, m = 1.0 mol$\cdot$kg$^{-1}$, it becomes apparent that temperature dependency of $V_\phi$(CuSO$_4$,aq) differs significantly from the other $V_\phi$(MSO$_4$,aq). Isopleths of MnSO$_4$(aq), CoSO$_4$(aq) and NiSO$_4$(aq), show the expected behavior, i.e., a weak maximum at $T \approx$ 323 K, similar to ZnSO$_4$(aq) and MgSO$_4$(aq). For CuSO$_4$(aq), the maximum, however, lies at or just outside the upper temperature limit of the present study, T = 343 K. This may be partly due to the Jahn–Teller effect [8, 31], causing a distortion of the otherwise octahedral coordination sphere possessed by the Cu$^{2+}$ ions. In this context, it is noteworthy that CuSO$_4$(aq) solutions show a marked increase in the concentration of contact ion-pairs with increasing temperature [32]. Such a change is likely to lead to an increase in $V_\phi$.

The simple relationship evident from Fig. 6 creates the possibility of predicting values of $V_\phi$(MSO$_4$,aq) over wide ranges of m and T, providing the appropriate structural information (i.e., $r_\mathrm {M-O}$) exists. Such data are available, in both solid and solution states, for many compounds from X-ray and neutron scattering experiments and related techniques such as EXAFS. This pathway for estimating $V_\phi$ values is potentially valuable for electrolytes that are difficult (or even impossible) to study experimentally. Some of these are now considered in the following paragraphs.

${V_\phi }$(FeSO₄,aq). As noted in the Introduction above, FeSO$_4$(aq) was not included in the present study due to its tendency to oxidize and hydrolyze in near-neutral aqueous solutions [7, 8]. This is unfortunate because the volumetric data for such solutions are of considerable interest industrially.

Taking $r_\mathrm {Fe-O}$ = 0.212 nm from the solution-XRD data of Ohtaki et al. [33] and using the relationship between $V_\phi$ and $r_\mathrm {M-O}$ (Fig. 6) enabled calculation of $V_\phi$(FeSO$_4$,aq) at T = 298.15 K as a function of concentration (Fig 7). The values so obtained are in good agreement with the experimental (vtd) data of Königsberger et al. [30], being lower on average by $\sim$0.6 cm$^3 \cdot$mol$^{-1}$. The experimental uncertainties in the results of Königsberger et al. are unknown but are likely to be significant given that FeSO$_4$ is a reactive 2:2 electrolyte. Accordingly, some confidence can be placed in the values predicted for $V_\phi$(FeSO$_4$,aq) using this approach at other m and T (Table S1, Supporting Information) where no experimental volumetric data are available. Such estimates represent a significant improvement on the existing database and are sufficiently accurate for most industrial applications.

Fig. 7

Comparison of literature values (triangles) [30] and present predictions (filled and unfilled circles) of $V_\phi$(FeSO$_4$,aq) at T= 298.15 K and at 0.1 MPa pressure. The unfilled circles indicate the effect of an increase of 1 pm in $r_\mathrm {Fe-O}$ (see text). Orange star: $V^\circ$ calculated from the ionic values tabulated by Marcus [27] The line is a visual guide only (Color figure online)

On the other hand, it needs to be noted that the predicted values of $V_\phi$(FeSO$_4$,aq) are very sensitive to the value chosen for $r_\mathrm {Fe-O}$. For example, adopting $r_\mathrm {Fe-O}$ = 0.213 nm (the unfilled square in Fig. 6), determined from a single-crystal XRD study [34] increases $V_\phi$(FeSO$_4$,aq) by $\sim$0.6 cm$^3 \cdot$mol$^{-1}$. While this change brings the calculated values into near-quantitative agreement with the experimental volumes [30], it is almost certainly fortuitous. Nevertheless, this tiny change (of just 1 pm!) also makes $r_\mathrm {Fe-O}$ more consistent with the $r_\mathrm {M-O}$ values of the other first-row transition metal ions (Fig. 6a), although this is again probably fortuitous.

$V_{\phi}$(VSO₄,aq). While the volumetric database for FeSO$_4$(aq) is limited, that for VSO$_4$(aq) is non-existent [35]. This is not surprising because V$^{2+}$(aq) is highly reactive [8]. It follows that measurements of the volumetric properties of VSO$_4$(aq) would require a disproportionate (and possibly fruitless) experimental effort. However, combining the value of $r_\mathrm {V-O}$ = 0.215 nm, from the single crystal XRD data of Montgomery et al. [34] with the correlation of Fig. 6 enables calculation of $V_\phi$(VSO$_4$,aq) over a wide range of m and T. A selection of these values is given in Table S2 in the Supporting Information.

Other M²⁺ sulfates. The present method for predicting $V_\phi$(MSO$_4$,aq) can be easily extended to other bivalent metal sulfates as long as suitable structural data are available to fix the value of $r_\mathrm {M-O}$. Fortunately, the popularity and quality of structural investigations has created an extensive and reliable database of inorganic compounds suited to this purpose (see Persson [36]), including some species that are too reactive to persist in solution but which can be stabilized in the solid state.

It must be noted in passing that the correlation in Fig. 6 does not include bivalent main group sulfates, even though their plots of $V_\phi$(m) follow the pattern of the transition metals (Fig. 5). Since most of these are only sparingly soluble this is not of major consequence.

3.4 Mixtures of Bivalent Metal Sulfate Solutions

The volumetric properties of binary metal sulfate solutions are, as noted in the Introduction, important physicochemical quantities in their own right. However, in most industrial situations working solutions typically contain (often many) more than one solute. It is therefore of interest to investigate the behavior of mixtures of metal sulfate solutions. Because of the ‘dimensional explosion’ that is inevitable when trying to characterize the properties of complex mixtures, the present density measurements were restricted to one quinary, four quaternary and six ternary mixtures. Each system was investigated at a single composition corresponding to approximately equal concentrations of each constituent metal ion. Apart from limiting the experimental effort, this choice was made because departures from ideal mixing (see below) are typically maximal at, or close to, such composition ratios [10].

The experimental density differences, $\Delta \rho$, and the mean apparent molar volumes $V_{\phi ,\textrm{mix}}$ calculated from them for various mixtures are listed in Table 7. The $V_{\phi ,\textrm{mix}}$ values were obtained by substituting molality m in Eq. 1 by the total molality $m_\textrm{T} = \sum _i m_i$ (where $m_i$ is the molality of the salt i in the solution), and by replacing the molar mass M with the mean molar mass ${\bar{M}} = \sum _i m_i M_i /m_\textrm{T}$ where $M_i$ is the molar mass of the anhydrous salt i. A useful way of examining such data is to compare them with those calculated using Young’s rule [10]. This rule can be expressed as:

$$\begin{aligned} V_{\phi , \textrm{YR}} = \sum _i m_i V_{\phi ,i}/m_\textrm{T}. \end{aligned}$$

(2)

which corresponds to ideal (linear) mixing behavior. Note that Young’s rule requires no adjustable parameters and no experimental information beyond $V_{\phi ,i}$, the apparent molar volumes of the neat binary solutions for each solute at $m_\textrm{T}$.

Table 7 Experimental density differences $\Delta \rho$ and apparent molar volumes $V_{\phi ,\textrm{mix}}$ for aqueous mixtures of MnSO$_4$, CoSO$_4$, NiSO$_4$ and CuSO$_4$ at Molalities $m_i$, Temperatures T and Pressure = 0.1 MPa

The departures of the present data from Young’s rule, $\Delta V_{\phi ,\textrm{mix}}^*$ = $V_{\phi ,\textrm{mix}}$ – $V_{\phi ,\textrm{YR}}$ are listed in Table 7. The overall situation for the various mixtures is plotted in Fig. 8, with the position of the bars on the x-axis indicating the magnitude of the departure from ideal mixing while the length of the bar corresponds to the effect of temperature on $\Delta V_{\phi ,\textrm{mix}}^*$. In general, $\Delta V_{\phi ,\textrm{mix}}^*$ increases with increasing T (Fig. 9a) but more importantly, with values ranging from ca. $-$0.1 to +0.2 cm$^3 \cdot$mol$^{-1}$, all lie within the estimated experimental uncertainties (Table 7, Fig. 9b). It is interesting to note that the systems showing the largest departures (Figs. 8, 9) always involve Cu$^{2+}$. The small magnitude of $\Delta V_{\phi ,\textrm{mix}}^*$ (Fig. 10) is particularly pleasing for the more complex (i.e., quaternary and quinary) mixtures.

On the basis of the present equimolar measurements (Table 7, Figs. 8, 9 and 10), it is reasonable to conclude that Young’s rule holds for the present systems at all concentration ratios, at least up to $m_\textrm{T}$ = 1.0 mol$\cdot$kg$^{-1}$ and T = 343.15 K, and probably well beyond. Similar observations have been made by Chen [37] for ternary CoSO$_4$(aq) + NiSO$_4$(aq) mixtures at T = 298.15 K.

The applicability of Young’s rule to these mixtures makes possible the estimation, with some confidence, of the volumetric properties of solutions of even more complex composition, without any experimental data beyond that of the component binary solutions. This approach represents a significant advance over the computational packages currently used for process engineering that either rely on empirical corrections or ignore such effects altogether.

Fig. 8

Departures of $V_{\phi ,\textrm{mix}}$ from Young’s rule (Eq. 2) for equimolal mixtures of bivalent metal sulfate solutions at $m_\textrm{T}$ = 1.0 mol$\cdot$kg$^{-1}$ over the temperature range 293.15 $\le$ T/K $\le$ 343.15 K and at 0.1 MPa pressure. The width of the horizontal bar represents the spread of ($V_{\phi ,\textrm{mix}} - V_{\phi ,\textrm{YR}}$) with T for each mixture

Fig. 9

Departures of $V_{\phi ,\textrm{mix}}$ from Young’s rule (Eq. 2) for: a CoSO$_4$(aq)+CuSO$_4$(aq) and b CoSO$_4$(aq)+MnSO$_4$(aq) equimolal mixtures at $m_\textrm{T}$ = 1.0 mol$\cdot$kg$^{-1}$ at temperatures 293.15 $\le$ T/K $\le$ 343.15 and 0.1 MPa pressure

Fig. 10

Departures of $V_{\phi ,\textrm{mix}}$ from Young’s rule (Eq. 2) for an equimolal quinary mixture of MnSO$_4$(aq) + CoSO$_4$(aq) + NiSO$_4$(aq) + CuSO$_4$(aq) at $m_\textrm{T}$ = 1.0 mol$\cdot$kg$^{-1}$ at temperatures 293.15 $\le$ T/K $\le$ 343.15 and 0.1 MPa pressure

These findings supplement earlier work, which has shown strong similarities among the osmotic and activity coefficients, heats of dilution and heat capacities of the bivalent first-row transition metal sulfate solutions. It is clear that the differences in the interactions of the metal cations with each other and with sulfate anions are almost negligible. It is reasonable to assume that at least some of the other thermodynamic properties of these mixtures will also exhibit ideal mixing behavior. This is indeed the case for the heat capacity results reported by Chen [37] for CoSO$_4$(aq) + NiSO$_4$(aq) mixtures. Unfortunately, no other data appear to be available to further test this hypothesis.

4 Conclusions

The present density measurements greatly expand the database of the volumetric properties of aqueous solutions of bivalent metal sulfates. Comparison of the volumetric behavior of the various MSO$_4$(aq) solutions, as measured here and as reported in the literature, shows undeniable similarities. Moreover, a simple linear relationship is observed between the metal–oxygen bond length in the hydrated cations and the apparent molar volumes of the first-row transition metal sulfates in aqueous solution. The volumetric behavior of the experimentally difficult FeSO$_4$(aq) has been predicted by exploiting this relationship, with good agreement with literature data being obtained. Furthermore, it is shown that this approach can be used to calculate the volumetric properties of unmeasured bivalent metal sulfate solutions over wide ranges of temperature and concentration, providing the relevant structural information is available. Measurements of mixtures of various MSO$_4$(aq) show that their apparent molar volumes mostly follow linear (Young’s rule) mixing behavior. This observation can be used to predict the volumetric behavior of complex mixtures of these electrolytes.