Thermodynamic Study of the Solubility Product Constant of K2Cr2O7

Article pubs.acs.org/jced

Thermodynamic Study of the Solubility Product Constant of K2Cr2O7 in Aqueous Solution and Related Ion-Pair Formation at 25 °C Mehran Aghaie*,† and Zahra Shahamat‡ †

Faculty of Chemistry, North Tehran Branch, Islamic Azad University, Tehran, Iran Department of Chemistry, Science and Research Branch, Islamic Azad University, Tehran, Iran

‡

ABSTRACT: The solubility of K2Cr2O7 in water at 25 °C was determined by using the evaporating method in the presence of various concentrations (0, 0.0500, 0.100, ..., 1.800 mol·L−1) of NaNO3. The values of solubility and solubility product constant was evaluated at zero ionic strength upon the extrapolation method. Using the value of solubility product constant obtained by the extrapolation method enables us to calculate the extension of ion-pairing in the saturated solution of K2Cr2O7 at 25 °C. On the other hand, the ion-association phenomenon could be explained by the Bjerrum theory or contact ion-pair model of Fuoss theory. Now, assuming that all of B+ ions and A2− ions are in the free state in the saturated solution of B2A (neglecting the ionic− association phenomenon) and estimating the mean activity coefficient, f±, in the consider solution upon a suitable model, such as extended Debye−Hückel theory, we can calculate another solubility product constant, Ksp(m), naming the modeled solubility product constant:1−4

1. INTRODUCTION In a saturated solution of an ionic compound, such as B2A (2B+,A2−), the following equilibrium at a constant temperature can be achieved: + 2− B2A (s) ⇔ 2B(aq) + A (aq)

(1)

K sp(a) = a+2a− = [B+]2 f+2 [A2 −]f− = 4s 3f±3

(2)

K sp(m) = [B+]2 [A2 −]f±3 = K sp(c)f±3

where Ksp(a) is the solubility product constant of B2A in terms of activities a+ and a−, f+ and f− are the respective activity coefficients of B+ and A2− ions in solution, f± is the mean activity coefficient, and s represents the solubility of B2A in terms of molarity. Equation 2 may also be represented as: K sp(a) = K sp(c)f±3

In general, the values of Ksp(th), Ksp(m), and Ksp(c) are not the same at the same conditions for the same ionic compound, and often the following trend may be observed: K sp(c) > K sp(m) > K sp(th)

(3)

(4)

where ΔG°f(i) represents the standard Gibbs free energy of formation of species (i) which may be found in the standard thermodynamic tables. Then, the thermodynamic solubility product constant, Ksp(th) = Ksp(a), of the considered ionic compound is:

+ 2− B(aq) + A (aq) ⇔ [BA]−ion‐pair

⎛ −ΔGdiss ° ⎞ K sp(th) = exp⎜ ⎟ ⎝ RT ⎠

(5)

2−

K sp(c) = [B ] [A ] = 4s

3

© 2013 American Chemical Society

(9)

x = the molarity of ion-pair in the saturated solution at T (10); s = solubility of B2A in the saturated solution in terms of

In turn, the value of Ksp(c) of the B2A compound can be determined experimentally by measuring the solubility, s/ mol·L−1, of B2A: + 2

(8)

This is partly due to the nonideal behavior of ions in the ionic solution and partly due to the ionic association phenomenon that almost always takes place in the ionic solutions.1−10 In fact, in an ionic solution, part of ions with opposite charges associate with each other to produce ion-pairs, ion-triplets, and so on.1−15 So, the trend in eq 8 is usually expected in the course of studing the ionic solution behaviors. Now, knowing, fairly, the rigorous value of Ksp(th) for the dissolution of an ionic compound such as B2A at temperature T and measuring its solubility, s, at the same temperature and estimating f± upon one suitable model, then we can evaluate the approximate extension ion-pair formation as follows:1−4

where Ksp(c) = [B+]2[A−] = 4s3 is the solubility product constant of B2A in terms of molarities. The value of Ksp(a) = Ksp(th) can be estimated from the standard Gibbs free energy, ΔG°diss, of the standard dissolution of B2A compound: ° = 2ΔG°f (B+) + ΔG°f (A2−) − ΔGf (B2A) ΔGdiss

(7)

Received: October 1, 2012 Accepted: December 12, 2012 Published: January 18, 2013

(6) 383

dx.doi.org/10.1021/je301134x | J. Chem. Eng. Data 2013, 58, 383−387

Journal of Chemical & Engineering Data

Article

molarity at T (11); (2s − x) = molarity of free B+ ions in the same conditions (12); (s − x) = molarity of free A2− ions in the same conditions (13). Hence K sp(th) = (2s − x)2 (s − x)f±3

The equilibrium constant, KIP, for the reaction of ion-pair formation (eq 20) is: + 2− B(aq) + A (aq) ⇔ [BA]−ion‐pair

(14)

or

KIP = ⎤ ⎡⎛ K ⎞ sp(th) 3 2 2 ⎢ ⎜ ⎟ [x − 5sx + 8s x] + ⎜ 3 ⎟ − K sp(c)⎥ = 0 ⎥ ⎢⎝ f ⎠ ⎦ ⎣ ±

(15)

KIP =

(22)

∫a

q

r 2e 2q / r dr

(23)

(24)

To calculate KIP upon eq 23, the values of r and q should be expressed in terms of meters, m. In the Bjerrum method, the distance q which defines the ionpair formation has a nearly arbitrary character. So, this is the main defect of this method. In addition, the fact that the ions, in the Bjerrum method, do not need to be in physical contact for ion-pair formation is another shortcoming of the theory. A more complete theory of ion pairing in the ionic solution is the Fuoss approach that consider the contact ion-pair formation and yields the following expression for KIP with respect to the very dilute solution of ionic compound BA(B+A−).12,14

(16)

+ − B(aq) + A (aq) ⇔ [BA]ion‐pair ;

KIP

(25)

(17)

KIP =

r = q = ( −z+z −e 2 /8Πε0DkT )

(at the minimum)

Bjerrum in his theory of ion pairing suggested that an ionpair is defined as existing when the distance between the center of two ions of opposite sign is less than q.13 In water with D = 78.5 at 25 °C, the value of q is near 357 pm for z+ = 1 and z− = −1, while its value in 1,2-dichloroethane with D = 10 is near 2800 pm. It is useful to introduce the function θ as:

∫a

q

r 2e 2q / r dr

⎛ 4 3⎞ ⎛ −z+z −e 2 ⎞ ⎜ πa ⎟exp⎜ ⎟ 1000 N = A ⎝ 3 ⎠ ⎝ 4πε0DakT ⎠ (1 − θ )2 c 2 θc

(L·mol−1)

(18)

dNa(r )d(r ) = 4ΠNa*

(in terms of m 3· molecule−1)

where NA is Avogadro’s constant. From the thermodynamic view, the ion-pair constant, KIP, is only temperature-dependent, so the value of eq 23 may be used for other concentrations. The values of KIP can be calculated upon eq 23 numerically, and q is: z z q = ( −z+z −e 2 /8πε0DkT ) = −8.344·10−5 + − (m) DT

Solving eq 17 yields

q

θ 2c

KIP = 2000πNA

where z+ and z− represent cation and anion charges, respectively, e is the absolute value of electron charge, and ε0, D, r, k, and T are the vacuum permittivity, dielectric constant, the distance from the central cation, the Boltzmann constant, and the temperature in Kelvin, respectively. One of the interesting behaviors of dNa(r) is that its plot versus r passes through a minimum. So, at this minimum we have:

∫a

(21)

Inserting eq 19 into eq 22 and expressing KIP in tems of L·mol−1, we get

dNa(r ) = 4πr 2drNa(r )

θ=

f θ = IP2 2 2c(1 − θ ) f±

where c is the salt concentration in terms of molecule per cubic meter and f i is the respective activity coefficient of i. We may use eq 21 for a very dilute solution in the form of:

2. THEORETICAL ASPECTS OF ION-PAIRING It is obvious that, in the context of physicochemistry characteristics of an ionic solution, we should pay special attention to the ion-association phenomena, especially to ionpairing.1−15 In fact when the distance between two ions with opposite charges in the solution becomes less than certain amount, the ion-pair is formed.12−14 To extend this idea, we consider a central cation and denote the molecular density of anions around it by Na(r) and the bulk density of anions by Na*. Then, the number of anions, dNa, in a shell around the central cation may be defined from a Boltzmann distribution such as:

dNa(r )/dr = 0

2c(1 − θ )f+ c(1 − θ )f−

=

(L·mol−1)

Solving this cubic equation, we could obtain the value of x that represents the molarity of ion pairs (I−P) in the studied solution. The “K2Cr2O7 + water” system and other similar systems and ion-pair formation in their solutions are important from different aspects, such as improving electrolysis process, promotion of battery applications, controlling the corrosion process, and promoting the agricultural soil, and so forth.

= 4πr 2drNa*exp( −z+z −e 2 /4πε0DrkT )

cθfIP

(20)

(26)

By analogy, we propose the following equation for the reaction 20 in the same conditions KIP =

⎛ 4 ⎞ ⎛ −z+z − ⎞ 3 ·1000NA ⎜ πa3⎟exp⎜ ⎟ ⎝ 3 ⎠ ⎝ 4πε0DakT ⎠ 4

(L·mol−1) (27)

This equation can be simplified as:

(19)

that represents the number of counteranions in the sphere of radius q around the central cation and a is the minimum approach. The value of θ by definition must take a value between 0 and 1, that is, the fraction of anions around the central cations in the considered shell.

KIP =

⎛ −1.67· 10−3z z ⎞ 3 + − ⎟ ·2.522· 1021a3 exp⎜ 4 DaT ⎝ ⎠

(28) −1

where a is in centimeters, T in Kelvin, and KIP in L·mol . A more rigorous expression for KIP with respect to reaction 25 is: 384


Journal of Chemical & Engineering Data ⎛ 4 ⎞ ⎛ −z+z −e 2 ⎞⎛ 1 ⎞ ⎟⎜ KIP = 1000NA ⎜ πa3⎟exp⎜⎜ ⎟ ⎝ 3 ⎠ ⎝ 4πε0DakT ⎟⎠⎝ 1 + Ka ⎠

Article

Table 1. Solubilities, s, of K2Cr2O7 in Water at Various Ionic Strengths at 25 °C

(29)

c(NaNO3)

where the expression of K is:

−1

mol·L

⎛ 2000F 2I ⎞ C ⎟ K=⎜ ⎝ ε0DRT ⎠

1/2

0 0.0500 0.100 0.200 0.300 0.400 0.500 0.700 0.800 1.000 1.200 1.400 1.600 1.800 2.000 2.200 2.400 2.600

(30)

and F = 96485 C·mol−1; Ic = 1/2∑Miz2i or K = 2.91·1010(Ic /D)1/2

(in SI units and at 298 K) (31)

So the analogue expression for the reaction 20 is KIP =

⎛ 4 ⎞ ⎛ −z z e 2 ⎞ 3 1000NA ⎜ πa3⎟exp⎜⎜ + − ⎟⎟ 2 ⎝ 3 ⎠ ⎝ 4πε0DkT ⎠ 2(2 + θ + θ + ...) ⎛ 1 ⎞ ⎜ ⎟ ⎝ 1 + Ka ⎠

(32)

where it diminishes to eq 27 when θ→0, or in a very dilute solution. If we express “a” in angstrom, Å, eq 32 will be reduced to:

s mol·L

KSP(c) = 4s3

I −1

0.451 0.462 0.475 0.482 0.489 0.499 0.511 0.529 0.555 0.556 0.578 0.593 0.607 0.611 0.620 0.602 0.555 0.483

−1

mol·L

1.353 1.437 1.524 1.646 1.767 1.898 2.031 2.286 2.406 2.661 2.933 3.180 3.421 3.634 3.831 4.007 4.064 4.048

mol3·L−3 0.367 0.395 0.428 0.448 0.467 0.497 0.532 0.591 0.614 0.679 0.771 0.836 0.895 0.913 0.945 0.874 0.682 0.450

⎛ ⎞ ⎜ −1.637·105z+z − ⎟ KIP = 1.891·10 (a /Å)exp⎜ ⎟ ⎜ DT (a /Å) ⎟ ⎝ ⎠ −3

⎛ 1 ⎜ ⎜⎜ ⎝ 1 + 5.030·10(a /Å)

I DT

⎞ ⎟ ⎟⎟ ⎠

3. EXPERIMENTAL SECTION K2Cr2O7 and other chemicals were purchased from Merck Company with a high degree of purity and used without further purification. The test solutions with respect concentrations were prepared by using deionized water. Then, the solubilities, s, of K2Cr2O7 in the presence of various concentrations of NaNO3 (0, 0.0500, 0.100, 0.200, 0.300, 0.400, ...) mol·L−1 were determined at 25 °C by using the solvent evaporating method1−4 (Table 1 and Figure 1).

Figure 1. Plot of s versus I for the solubilities of K2Cr2O7 in water in the presence of various molarities of NaNO3 at 25 °C.

4. DISCUSSION The solubility of K2Cr2O7 in water is affected by the presence of NaNO3. As we can see from Table 1 and Figure 1, the solubility of K 2 Cr 2 O 7 increases with increasing KNO 3 concentration up to 2.000 and then decreases. Figures 2 to 5 show nearly linear plots of s, ln s, Ksp, and ln Ksp versus √I, respectively; where s (mol·L−1) is the solubility of K2Cr2O7 in water at 25 °C and Ksp is the concentration solubility product, Ksp(c) = 4s3. We focus on Figure 2, because it shows better linearity with the equation, s = [(0.0203 ± 0.03)√I + (0.223 ± 0.004)] mol·L−1, on a wide range of ionic strength. Then the interception of the line with the y-axis for √I → 0 gives: s0 = (0.223 ± 0.004) mol·L−1

(33)

where s0 is the solubility of K2Cr2O7 in water at 25 °C for I→0. In this condition, ions in solution may be assumed to be ideal, and then Ksp(th) for an ionic compound with the formula of B2A or AB2 such as K2Cr2O7 is:

Figure 2. Plot of s versus I1/2 for the solubilities of K2Cr2O7 in water in the presence of various molarities of NaNO3 at 25 °C.

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phenomenon in the context of physicochemistry of ionic solution is of great importance.16−28 The modeled solubility product constant, Ksp(m), in turn, is also of great importance (eq 7). To estimate f±, we may use a modified Debye−Hückel equation such as: log f± =

Az+z − I + 0.2 I 1 + aB I

(35) −1

where A = 0.5091 and B = 0.324 Å for aqueous solution at 25 °C, and a is in angstroms. Now, if at the first approximation, we neglect the ion-association phenomenon and taking a = [(d+ + d−)/2] = 4.155 [Å], we get f± = 0.592 and Ksp (m) = 0.076. The difference between this value and Ksp(th) may be attributed to the ion-association phenomenon.1−4 Therefore, solving eq 15 yields the extension, x, of ion-pair formation in the saturated solution of K2Cr2O7 in water at 25 °C (Table 2).

Figure 3. Plot of ln s versus I1/2 for the solubilities of K2Cr2O7 in water in the presence of various molarities of NaNO3 at 25 °C.

Table 2. Values of I−P Molarity in the Saturated Solution of K2Cr2O7 in Water at 25 °C (Using an Iteration Procedure) iteration

I/mol·L−1

f±

x = [I−P]/mol·L−1

1 2 3

1.353 1.078 1.078

0.592 0.585 0.585

0.1 0.107 0.107

Then the equilibrium constant of ion-pair formation is: − K +aq + Cr2O27,aq → [KCr2O7 ]−IP

KIP = =

1/2

Figure 4. Plot of Ksp versus I for the solubilities of K2Cr2O7 in water in the presence of various molarities of NaNO3 at 25 °C.

0.107·0.585 = 0.6665 0.795· 0.344· (0.585)2

(36)

( f is are estimated from eq 35). At the first approximation, the value of KIP for the reaction 20, after the Fuoss model with respect to a very dilute solution (eq 28) is KIP = 4.251; somewhat larger than experimental value. But, if we use the eq 29, which is more accurate, we will get the value of KIP ≈ 0.555 for the equilibrium 20 which is more comparable with experimental value, 0.6665.

5. CONCLUSION The solubility of K2Cr2O7 in water and in the presence of NaNO3 increases nearly linearly with the increasing NaNO3 molarity in a wide range of 0.0500 up to 1.800 M. A straight line with the equation s (mol·L−1) = (0.203 ± 0.03)I1/2 + 0.223 ± 0.004 can fairly represent the variation of solubility of K2Cr2O7 with I1/2 at constant temperature. The interception of the line is the solubility of K2Cr2O7 at I→0. From the interception, s0, we evaluated thermodynamic solubility product constant, Ksp(th), of the salt (Ksp(th) = 4s30). Once we do this, the calculation of ion pairing extension is quite straightforward. The saturated solution of K2Cr2O7 is highly nonideal. This comes, partly, from the nonideal behavior of free ions in the saturated solution and partly from the ion pairing phenomenon. The latter could be treated upon the Bjerrum model or the Fuoss model of ion association. The Fuoss model of contact ion pair fits the results of our work better.

Figure 5. Plot of ln Ksp versus I1/2 for the solubilities of K2Cr2O7 in water in the presence of various molarities of NaNO3 at 25 °C.

K sp(th) = 4s03

[I − P]fIP aIP = + a+a− [K ][Cr2O27 −]f±2

(34)

so K sp(th) , K 2Cr2O7 /mol3· L−3 = 4· (0.223)3 = 0.0445

This value is much smaller than Ksp(c) = 0.367 in water (see Table 1). Indeed, part of the difference is due to the nonideality of ions in the solution, and part of it is due to the ion-pairing phenomenon. In fact, focusing on the ion-association 386



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AUTHOR INFORMATION

Corresponding Author

*E-mail: m_aghaie@iau-tnb, [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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Thermodynamic Study of the Solubility Product Constant of K2Cr2O7

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