ISSN 10231935, Russian Journal of Electrochemistry, 2012, Vol. 48, No. 6, pp. 643–649. © Pleiades Publishing, Ltd., 2012. Published in Russian in Elektrokhimiya, 2012, Vol. 48, No. 6, pp. 714–720.

Measurement of the Change in Partial Molar Volume During Electrode Reaction by Gravity Electrode— II. Examination of the Accuracy of the Measurement1 Yoshinobu Oshikiria, Makoto Miurab, and Ryoichi Aogakic, z a

Yamagata College of Industry and Technology, Department of Environmental Engineering 221, Matsuei, Yamagata, Yamagata 9902473, Japan bPolytechnic College Akita, 61, Ohgidamichishita, Ohdate, Akita 0170805, Japan c Polytechnic University, 220121304, Ryogoku, Sumidaku, Tokyo 1300026, Japan Received June 11, 2011

Abstract—The reliability of the measurement of the change in partial molar volume between product and reactant ions measured by gravity electrode (GE) was examined by the thermodynamic measurement of pyc nometer (PM). Since the PM method requires an experimental equation of the apparent molar volume to calculate the partial molar volumes of the individual ions, the most suitable experimental equation must be first determined. As a test reaction for the experiment, oxidation of ferrocyanide (FERO) ion to ferricyanide (FERI) ion was adopted. After fitting several experimental equations to the data of the apparent molar vol umes by the PM method, the calculated changes in the partial molar volume were compared with the data of the GM method. Then, it is concluded that the polynomial with a degree of 3 of the logarithm of the molality of the FERO ion suggests the most suitable equation. As a result, the reliability of the GE method was also experimentally validated. Keywords: gravity field, gravity electrode, partial molar volume, convection, electrode reaction DOI: 10.1134/S1023193512060080

1. INTRODUCTION

The change in the partial molar volume VG measured by GE is expressed by [1]

Gravity electrode (GE) is operated in a high gravity field arising from a centrifugal force, and gives rise to the change in partial molar volume between the ions of the product and reactant [1]. On the other hand, as discussed in Part 1 [1], thermodynamic measurement such as pycnometer (PM) method leads to the partial molar volumes of the individual ions, which however requires an experimental equation for fitting into the experimental data of apparent molar volume. To ascertain the validity and accuracy of the GE method, it is necessary to compare the data obtained from the same reaction by both methods. In addition, such comparison should be done by the most suitable experimental equation for the PM method. In the present paper, the oxidation of ferrocyanide (FERO) ion to ferricyanide (FERI) ion is first taken up as a test reaction. 4–

Fe ( CN ) 6

3–

Fe ( CN ) 6 + e.

1 The article is published in the original. z

(1)

Corresponding author: [email protected] (Ryoichi Aogaki).

˜ VP – VR , ΔV G = α

(2)

where VP and VR are the partial molar volume of the product (FERI ion) and reactant (FERO ion), respec tively, and zR DR ˜ ≡  α , zP DP

(3)

where zR and zP are the charge number transferring in the reaction for FERO and FERI ions, respectively. DR and DP are the diffusion coefficients of FERO and FERI ions, respectively. As will be shown in Fig. 10, in ˜ = 1 can be the case of the oxidation of FERO, α adopted with sufficient accuracy. On the other hand, the PM method allows us to measure the apparent molar volumes ϕP and ϕR of FERI and FERO ions, respectively. Therefore, to eq eq obtain the partial molar volumes V P and V R of FERI and FERO ions, some proper experimental equation is required for fitting into the data. Determining the coefficients of the experimental equation by the least squares method, we calculate the partial molar vol

643

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YOSHINOBU OSHIKIRI et al. eq

eq

umes of V P and V R . Then, the difference ΔVpyc at the same molality of FERO and FERI ions is calculated as follows, ˜ V Peq – V Req. ΔV pyc = α

(4)

Furthermore, the relative error between ΔVG and ΔVpyc at the same molality is considered as follows, ε = ΔV G – ΔV pyc .

In the PM method, as have been shown in Part 1 [1], the partial molar volume has the following rela tionship with the apparent molar volume. eq ∂ V k =  ( ν k m k ϕ k ) ∂m k

∑ ε /N , 2 i

(6)

i

where subscript “i” means the i’th experiment at a given molality mi (i = 1, 2, N), and N is the total data number. The experimental equation corresponding to the smallest E is determined as the most suitable equa tion. The sufficiently small rms value also guarantees the validity of the GE method.

µ·

(10)

n+1

νk mk ϕk =

∑ A*

j – 1 ( ln m k )

j–1

.

(11)

j=1

(b) Simple polynomial

2.1. GE Method in Vertical Mode (1) Plotting the limiting diffusion current density i lim (A m–2) against the 1/3rd power of gravity accel eration α1/3 (m s–2)1/3, we determine the slope Ldiff of the plot [2]. The value of the α is controlled by the angular velocity ω (s–1) of the rotor of GE, i.e., 2

µ·

where subscripts P and R imply the product and reac tant respectively, mk is the molality, and νk is the sto ichiometric coefficient of the active ion. The subscript μ· means that the physical quantities except for mk are kept constant. The experimental equation of ϕk with regard to mk must satisfy Eq. (10), so that in the present paper, the following three types of equation are considered. (a) Polynomial of logarithm

2. CALCULATION PROCEDURE

α = rω ,

∂ 1 =   ( ν k m k ϕ k ) m k ∂ ln m k

for k = P or R,

(5)

Finally, the root mean square (rms) value of the error is computed by E =

2.2. PM Method

(7)

n+1

νk mk ϕk =

∑ A*

j–1 j – 1 mk .

(12)

j=1

(c) Polynomial of square root n+1

νk mk ϕk =

∑ A*

( j – 1 )/2 , j – 1 mk

(13)

j=1

where r (m) is the distance between the working elec trode and the center of the rotor. (2) The density coefficient γlim in the limiting diffu sion is calculated by L diff ν γ lim = ± ⎛  ⎞ , ⎝ 0.0969z R Fc R ( s )⎠ D 2 R

(8)

where zR is the charge number transferring in the reac tion, F is Faraday constant, cR(s) is the bulk molar concentration (mol m–3), ν is the kinematic viscosity (m2 s–1), and DR is the diffusion coefficient (m2 s–1). The sign ± corresponds to the cases where the change in partial molar volume is positive and negative, respectively. (3) Finally, the simple equation of ΔVG is derived from Part 1 [1] γ lim ΔV G = , ρ s0 m R

(9)

where ρs0 is the solution density containing supporting electrolyte, and mR is the molality of the reactant, i.e., FERO ion.

where A *j – 1 are the arbitrary constants determined by the least squares fit of data to Eqs. (11) to (13). Equa tion (11) is expected in good agreement over the wide range of molality, however, at infinite dilution, the right hand side of Eq. (11) diverges. In view of this problem, Eq. (12) is chosen for the range of low mola lity. Based on the Debye–Hückel limiting law [3, 4], as shown in Eq. (13), the polynomial of square root of mk is adopted. In experiments, the polynomials with degrees of 2 to 6 were examined. From Eqs. (11), (12) and (13), according to Eq. (10), the partial molar vol umes of FERO and FERI ions are expressed by, (d) Polynomial of logarithm with a degree of n eq 1 V k =  mk

n+1

∑ ( j – 1 )A*

j – 1 ( ln m k )

j–2

.

(14)

j=2

(e) Simple polynomial with a degree of n n+1 eq

Vk =

∑ ( j – 1 )A*

j–2 j – 1 mk .

(15)

j=2

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(f) Polynomial of square root with a degree of n eq 1 Vk =  2

n+1

∑ ( j – 1 )A*

( j – 3 )/2 . j – 1 mk

645

g

(16)

j=2

After the preliminary formulation, the calculation procedure of the PM method is elucidated as follows: (1) The specific gravity of the solution d* is calcu lated by W '– w d* = , W–w

(17) b

c

where w is the weight of the PM at 27°C, W ' and W are the total weights of the PM filled with the solution and pure water at 27°C, respectively. (2) The specific gravity d* is converted to the den sity ρ with the density of pure water at 27°C ρw ρ = d*ρ w .

(18)

(3) Then, the apparent molar volume of the solute ion ϕk for each given mk is calculated by [1] ρ s0 – ρ M ν k ϕ k =   +  , m k ρρ s0 ρ

(19)

where М is the molar mass of the solute salt, and ρ is the density of the solvent containing only supporting electrolyte. (4) By means of the method of least squares, a set of the data of νkmkϕk vs. mk is fitted by Eqs. (11), (12) and (13), and the coefficients A *j – 1 are determined for FERO and FERI ions, respectively. (5) Finally, with Eqs. (14), (15) and (16), the partial eq eq molar volumes V R and V P of both ions are calcu lated, from which ΔVpyc in Eq. (4) is determined. 3. EXPERIMENTAL 3.1. GE Method in Vertical Mode As mentioned initially, as a test reaction, the oxida tion of FERO ion to FERI ion was adopted. The solu tions with 100 mol m–3 K2SO4 as a supporting electro lyte were prepared for the FERO ion molar concentra tions from 10 to 30 mol m–3. For the measurement of the density coefficient γlim, GE (GE01, Nikkou Keisoku Co.) was used in vertical mode. As shown in Fig. 1, a pair of circular Pt plates with a 5 mm diame ter, which were shielded by orings, were used for working and counter electrodes, where the active areas inside orings were 3.14 mm2. Oxidation of FERO ion decreases the density of the solution, so that hydrodynamic instability for convec tion is induced by downside electrode. Therefore, as shown in Fig. l, the downside electrode was used as working electrode for convection. As a reference elec trode, a Ag wire covered with a AgCl film of a 1 mm diameter was used. The reaction was performed at an RUSSIAN JOURNAL OF ELECTROCHEMISTRY

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a

Fig. 1. Electrolysis cell in vertical mode, a, working elec trode; b, counter electrode; c, reference electrode; d, solu tion; g, gravity field.

overpotential of +200 mV, i.e., at the limiting diffusion range. Prior to experiment, argon bubbling was per formed to evacuate dissolve oxygen. To calculate the density coefficient in Eq. (8), the kinematic viscosity ν was measured by Cannon–Fenske viscometer (Sibata Scientific Technology Ltd.), and the diffusion coeffi cients DP and DR were determined by rotating disk electrode (RDE) (RRDE1, Nikkou Keisoku Co.). Temperature was kept at 27 ± 1°C. The measurements of the diffusion coefficients were performed in a solu tion with 100 mol m–3 K2SO4 for each 10 mol m–3 of FERO and FERI ions. 3.2. PM Method Concerning FERO and FERI ions, each eigh teen samples of the solution with 100 mol m–3 K2SO4 was prepared for the molality from 1.0 × 10 ⎯ 3 to 3.0 × 10–2 mol kg–1. Hubbardtype PM (Specific Gravity Bottle, Sibata Scientific Technology Ltd.) was used. Temperature was also kept at 27 ± 1°C. No. 6

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i lim, А m

20

–2

ΔVG, m3 mol–1 0.0001 1 2 3 4 5

10

0 0.01 5

0

0.02

10

α1/3, (m s–2)1/3

Fig. 2. Dependence of the limiting diffusion current den sity i lim on the 1/3rd power of acceleration α1/3. [K2SO4] = 100 mol m–3, Overpotential = +200 mV. 4–

0.03 mR, mol kg–1

Fig. 3. Plot of the change in the partial molar volume ΔVG vs. molality of FERO ion mR by gravity electrode. DR = 6.983 × 10–10 m2 s–1, ν = 9.006 × 10–7 m2 s–1, ρs0 = 1010.23 kg m–3.

4–

(1) [Fe(C N ) 6 ] = 10 mol m–3; (2) [Fe(C N ) 6 ] = 4–

16 mol m–3; (3) [Fe(C N ) 6 ] 4–

= 20 mol m–3; 4–

(4) [Fe(C N ) 6 ] = 26 mol m–3; (5) [Fe(C N ) 6 ] = 30 mol m–3.

4. RESULTS AND DISCUSSION The data obtained from the RDE is as follows; for FERO ion, DR = 6.983 × 10–10 m2 s–1; for FERI ion, ˜ = 0.843 DP = 8.281 × 10–10 m2 s–1. With zR = zP = 1, α is derived, which allows us to calculate Eq. (4). According to the GE method, in Fig. 2, the limit ing diffusion current density is plotted against the 1/3rd power of the acceleration. Then, as shown in Fig. 3, the change in partial molar volume ΔVG in Eq. (9) is calculated for each given molality mR. On the other hand, concerning the PM method, in Fig. 4, as an example, the results of the least squares fit of Eq. (11) in the case of logarithm are exhibited. As mentioned above, the results from the polynomials with degrees of 2 to 6 are represented. The forms of the curves of apparent molar volumes of FERO and FERI ions are quite different each other, and also different from the plots in the case of pure solvent where the apparent molar volume follows a linear function of the square root of the molality [4, 5]. Figure 5 shows the partial molar volumes of FERO and FERI ions derived from the experimental equa

tions of the apparent molar volume in Fig. 4. As shown in Fig. 5, at low molalities, due to the uncertainty of PM method and the divergence of the logarithmic equation in the measurement of the apparent molar volume shown in Fig. 4, the plots and data indicate abrupt changes, so that the plots should be cut off at a proper value of the molality. Based on the data obtained in Figs. 5, 6 represents the differences between the molar volumes ΔVpyc’s of FERI and FERO ions at given molalities. As men tioned above, in view of the uncertainty of PM method and the logarithmic divergence at low molality together with the limit of GE method, the data below 0.01 mol kg–1 were cut off. As shown in the literature [5], conventional PM does not have sufficient preci sion for measuring apparent molar volume; less satis factory for measuring the small density difference (between solvent and solution) at low solute concen trations that are necessary for the determination of reliable values of apparent molar volume. However, in the present case, the cutoff level corresponds to the order of 1000 ppm, whereas in the above discussion, the precision limit required for PM is ±5 ppm, much smaller than the present level. Therefore, in the present case, it is concluded that PM does its duty. From the normalization, in other cases, the data below the same molality were also disregarded. Then, in Fig. 7 the rms relative errors between ΔVG and ΔVpyc against the degrees of the polynomial of logarithm are

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–νk mkϕk, m3 kg–1 2[1×10–6]

FERO FERI

1[1×10–6]

0 2 –logmk [mol kg–1]

3

Fig. 4. Plots of the apparent molar volume against the molality in the case of polynomial of logarithm, FERO ion; solid, a degree of 2; dot, a degree of 3; short dash, a degree of 4; dash, a degree of 5; dot dash, a degree of 6. eq

3

V k , m mol

–1

1 2 3 4 5

0.0002

0.0001

1' 2' 3' 4' 5'

0

–0.0001

10–2 mk, mol kg–1

10–3

Fig. 5. Plots of the partial molar volume against the morality for different degrees of the polynomial of logarithm (derived from the data in Fig. 4). FERO ion: (1) a degree of 2; (2) a degree of 3; (3) a degree of 4; (4) a degree of 5; (5) a degree of 6. FERI ion: (1') a degree of 2; (2') a degree of 3; (3') a degree of 4; (4') a degree of 5; (5') a degree of 6.

plotted by bar graph. The degree of the polynomial cor responding to the minimum value suggests the most suit able equation in the case of logarithmic plot for the apparent molar volume. The polynomial with a degree of 3 with a rms error value of 1.48 × 10–5 m3 mol–1 was cho sen as the most suitable one. RUSSIAN JOURNAL OF ELECTROCHEMISTRY

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In Figs. 8 and 9, the rms relative errors in the cases of the simple polynomial and square root polynomial are exhibited, respectively. Among the results in Figs. 7, 8 and 9, the polyno mial of logarithm with a degree of 3 does not bring the smallest rms error value; the smallest one corresponds No. 6

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YOSHINOBU OSHIKIRI et al.

ΔVpyc, m3 mol–1 0.0001

E, m3 kg–1 1×10–4 1 2 3 4 5

5×10–5

0

0 0.01

0.02

2

0.03 mk, mol kg–1

Fig. 6. Difference between the molar volumes ΔVpyc of FERI and FERO ions vs. molality mk derived from the data ˜ = 0.843 (for different degrees of the poly in Fig. 5 and α nomial of logarithm): (1) a degree of 2; (2) a degree of 3; (3) a degree of 4; (4) a degree of 5; (5) a degree of 6.

Fig. 8. Comparison of the rms relative errors E against the degree of the simple polynomial.

E, m3 kg–1 4×10–5

E, m3 kg–1 4×10–5

2×10–5

2×10–5

0

4 6 Degree of polynomial

0 2

4 6 Degree of polynomial

Fig. 7. Comparison of the rms relative error E against the degree of the polynomial of logarithm (derived from the data in Figs. 3 and 6).

to the simple polynomial of a degree of 2, of which error value is 1.36 × 10–5 m3 mol–1, but as shown in Fig. 8, in the case of simple polynomial, the error value drastically changes with its degree, whereas the error level in the polynomial of logarithm does not remark ably depend on the degree. Therefore, it is concluded that the polynomial of logarithm with a degree of 3 yields the most suitable experimental data so far. Figure 10 represents the relative percentage of ΔVG to the best ΔVpyc mentioned above against the molality

2

4 6 Degree of polynomial

Fig. 9. Comparison of the rms relative errors E against the degree of the polynomial of square root.

mR, where the break line of 100% indicates the values of the ΔVpyc itself. The data points around the 100% line shows good agreement of the ΔVG with the ΔVpyc. At the same time, the relative percentage of the ΔVG to ˜ = 1.0 is also plotted. The dif the simple ΔVpyc with α ference between the two kinds of plots is quite small, which means that the difference of the mass transfer between FERO and FERI ions can be neglected. The good agreement between ΔVG and ΔVpyc also assures us the validity of the GE method.

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Relative percentage of ΔVG/ΔVpyc, % 200 1 2

100

0 0.02

0.01

0.03 mR, mol kg–1

Fig. 10. Plots of the relative percentage of ΔVG to ΔVpyc vs. the molality of FERO ion mR. Break line indicates the ΔVpyc. To cal culate the ΔVpyc, the polynomial of logarithm with a degree of 3 is used, of which relative error is E = 1.48 × 10–5 m3 mol–1: (1) the ˜ = 0.843; (2) the case of α ˜ = 1.0. case of α

5. CONCLUSIONS

REFERENCES

As the experimental equation for the PM method, the polynomial of logarithm with a degree of 3 gives the best fit for the oxidation of FERO ion, and the rms relative error between the data by the PM and GE methods was 1.48 × 10–5 m3 mol–1, where in view of the uncertainty of the PM method and logarithmic divergence at low molality together with the limit of the GE method, the data below 0.01 mol kg–1 were cut off. From the good agreement with the data of the PM method, it is concluded that the data of the GE method also have the accuracy of the same level.

1. Aogaki, R., Oshikiri, Y., and Miura, M., submitting to Russian J. Electrochem.

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2. Sato, M., Oshikiri, Y., Yamada, A., and Aogaki, R., Jpn. J. Appl. Phys., 2003, vol. 42, pp. 4520–4528. 3. Moore, W.J., Physical Chemistry, New Jersey: Prentice Hall, 1962, 3rd ed., p. 330. 4. Marcus, Y. and Hefter, G., Chem. Rev., 2004, vol. 104, pp. 3405–3452. 5. Hepler, L.G, Stokes, J.M., and Stokes, R.H., Trans. Faraday Soc., 1965, vol. 61, pp. 20–29.

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