Ruthenium redox equilibria: 1. Thermodynamic stability of Ru(III) and Ru(IV) hydroxides


doi:10.5599/jese.226  123 

J. Electrochem. Sci. Eng. 6(1) (2016) 123-133; doi: 10.5599/jese.226 

 
Open Access : : ISSN 1847-9286 

www.jESE-online.org 

Original scientific paper 

Ruthenium redox equilibria 
1. Thermodynamic stability of Ru(III) and Ru(IV) hydroxides 

Igor Povar, Oxana Spinu  

Institute of Chemistry of the Academy of Sciences of Moldova, 3 Academiei str., MD 2028, Chisinau, 
Moldova 

Corresponding Author: ipovar@yahoo.ca; Tel.: +373 22 73 97 36; Fax: +373 22 73 97 36 

Received: September 30, 2015; Accepted: February 18, 2016 
 

Abstract 
On the basis of the selected thermodynamic data for Ru(III) and Ru(IV) compounds in 
addition to original thermodynamic and graphical approach used in this paper, the 
thermodynamic stability areas of sparingly soluble hydroxides as well as the repartition 
of their soluble and insoluble chemical species towards the solution pH and initial 
concentrations of ruthenium in heterogeneous mixture solid phase–saturated solution 
have been investigated. By means of the ΔG–pH diagrams, the areas of thermodynamic 
stability of Ru(III) and Ru(IV) hydroxides have been established for a number of analytical 
concentrations in heterogeneous mixtures. The diagrams of heterogeneous and 
homogeneous chemical equilibria have been used for graphical representation of 
complex equilibria in aqueous solutions containing Ru(III) and Ru(IV). The obtained 
results, based on the thermodynamic analysis and graphic design of the calculated data 
in the form of the diagrams of heterogeneous chemical equilibria, are in good agre-
ement with the available experimental data.  

Keywords 
Heterogeneous chemical equilibria; Sparingly soluble hydroxides; Soluble and insoluble ruthenium 
chemical species; Thermodynamic analysis 

 

Introduction 

Chemistry of ruthenium is characterized by some specific properties, such as the existence of 

various valences from 0 to 8, the relatively easy formation of stable polynuclear compounds, 

occurrence of various disproportionation and comproportionation processes etc. Widespread 

application of ruthenium and its compounds in various practical purposes serves as an impulsion 

for the development of the ruthenium applied analytical chemistry. Currently, selective fast and 

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J. Electrochem. Sci. Eng. 6(1) (2016) 123-133 RUTHENIUM REDOX EQUILIBRIA: 1. THERMODYNAMIC STABILITY  

124  

efficient methods for determination of ruthenium were developed [1-5]. The choice of the appro-

priate method requires knowledge of the forms of existence of ruthenium in solution and the 

composition of its complex compounds. Depending on the solution pH, metal ion concentrations 

in solution, presence of oxidants and reductants as well as their concentration, ruthenium can 

exist in the form of several complexes, each exhibiting its own catalytic and voltammetric activities 

[6,7]. Within research of the reduction – oxidation mechanism, it is necessary to know if some 

coexisting species are reduced simultaneously or, conversely, the process has a “stepwise” 

character and one species reduces gradually [8]. The knowledge of the forms of metal ion in 

solution facilitates the choice of optimal sorption and extraction conditions. Therefore, the use of 

suitable analytical methods and procedures of ruthenium research involves a priori knowledge of 

its forms of existence for the interpretation of mechanisms of predominant reactions of certain 

species for selecting a suitable procedure. 

It is known that the assessment of solubility (S) based on the value of the solubility product can 

lead to large misinterpretation. This is explained by the fact that the precipitate (solid phase) 

components in solution are subjected to different reactions of hydrolysis, complex formation etc. 

Generally, these secondary processes contribute to increasing the precipitate solubility [9-15]. A 

particularly strong influence on the precipitate solubility (S) exercises the solution pH value. 

Determination of the dependence of S on pH requires extensive calculations, since this method 

involves solving systems of nonlinear equations for mass balance (MB). However, from the 

solubility diagrams one cannot draw conclusions on the thermodynamic stability of the solid phase 

when the precipitate solubility is relatively high. Authors [9-15] proposed a strict criterion for 

assessing the solid phase stability, based on the Gibbs energy change value of the precipitation - 

dissolution process of solid phases. In general, Gibbs energy variation calculations for these 

systems were examined in [9].  

The paper presents a thermodynamic approach for the complex chemical equilibria 

investigation of two-phase systems containing Ru(III) and Ru(IV) hydroxides. This approach utilizes 

thermodynamic relationships combined with original mass balance constraints, where the solid 

phases are explicitly expressed. The factors influencing the distribution and concentrations of 

various soluble ruthenium species were taken into account. The new type of diagrams, based on 

thermodynamics, graphical and computerized methods, which quantitatively describe the 

distribution of soluble and insoluble, monomeric and polymeric ruthenium species in a large range 

of pH values was used.  

The developed thermodynamic approach is based on: 

1. Analysis of thermodynamic stability areas for the solid phase; 

2. Determination of the molar fractions of chemical species in heterogeneous systems 

precipitate-solution based on the MB equations of the method of residual concentrations 

(MRC). 

Theory and calculations 

Thermodynamic stability areas of sparingly soluble Ru(III) and Ru(IV) hydroxides 

In the present paper the relations, obtained in [10], will be applied to determine the 

thermodynamic stability of hydroxides Ru(III) and Ru(IV) under real conditions, different from 

standard ones. We will expose the quintessence of this method on the example of the equilibrium 

of Ru(III) hydroxide with aqueous solution: 



I. Povar et al. J. Electrochem. Sci. Eng. 6(1) (2016) 123-133 

doi:10.5599/jese.226 125 

+ 3+
3 2 2 S0

Ru(OH) ×H O+3H =Ru +4H O, lg 1.64K    (1) 

where KS is the equilibrium constant of reaction (1). 

The variation of the Gibbs energy of reaction (1) under standard conditions is equal to 

0 0 3+ 0 0
r f f 2 f 3 2 S(Ru ) 4 (H O) (Ru(OH) ×H O (s)) lnG G G G RT K         (2) 

where 
0
f
( )G i  is standard Gibbs energy of formation of species i. The selected values for 

0
f
( )G i  

[16] are shown in Table 1.  

Table 1. The values of standard Gibbs energy of formation of ruthenium chemical species at 298.15 K 

Chemical species 0
f
( )G i  / kJ mol

-1
 Chemical species 

0
f
( )G i / kJ mol

-1
 

2 5
H RuO  (aq)  -391.2 4+

4 12
HRu (OH)  (aq)  -1877 

-
5HRuO  (aq)  -325.4 

3+
Ru  (aq)  173.4 

-
4RuO  (aq)  -250.1 

2+
Ru(OH)  (aq)  -51.0 

2-
4

RuO  (aq)  -306.6 
+
2Ru(OH)  (aq)  -280.9 

2 2
RuO ×2H O (am)  -691.0 8+

4 4Ru (OH)  (aq)  -193.5 

3 2
Ru(OH) ×H O (am)  -766.0 2+Ru  (aq)  150.3 

2 5
Ru O  (aq)  -445 

2
H O (aq)  -237.19 

2+
2Ru(OH) (am)  -221.8 

-
OH  (aq)  -157.33 

 

But the 0rG  value cannot serve as a characteristic of thermodynamic stability of Ru(III) 

hydroxide under real conditions. In this case, the equilibrium is described by the equation of 

isothermal reaction that for the scheme (1) takes the form: 

0 + 3
r r Ru (s)  (s) ln( /[H ] )G G RT C       (3) 

where 0
Ru

C  denotes the concentration of ruthenium in heterogeneous mixture. It is easy to see 

that the Gibbs energy change of reaction (1) depends strongly on the pH value. At the same time 

the equation (3) can also be applied for the determination of thermodynamic stability of the 

ruthenium(III) hydroxide as the Ru
3+

 ion, depending on the pH and the initial concentration of the 

ruthenium, 0
Ru

C , is subjected to complex chemical transformations in solution: 

3+ 2+ +
2 10

3+ + +
2 2 20

3+ 8+ +
2 4 4 44

1.  Ru +H O=Ru(OH) +H , lg 2.24

2.  Ru +2H O=Ru(OH) +2H , lg 3.52

3.  4Ru +4H O=Ru (OH) +4H , lg 10.80

K

K

K

 

 

 

  (4) 

(The hydrolysis constants were calculated from 0
r

G  values using the equation 
0
ij ijlnG RT K   ). 

Obviously, the ΔG calculation requires accounting for all equilibria (4). A rigorous thermo-

dynamic analysis shows that the change in Gibbs energy of reaction (1) in real conditions is 

described by the equation [13]: 

0 0 3
r r Ru Ru (s) ln ln( / [ ] )G G RT RT C H


     .  (5) 

Here αi represents the hydrolysis coefficient of reactions (4) [13], defined as:  



J. Electrochem. Sci. Eng. 6(1) (2016) 123-133 RUTHENIUM REDOX EQUILIBRIA: 1. THERMODYNAMIC STABILITY  

126  

+ -1 + -2 3+ + -3
Ru 10 20 441 [H ] [H ] 4 [Ru ][H ]K K K       (6) 

In relation (6) [Ru
3+

] is the equilibrium concentration of the ruthenium(III) ion, which is 

calculated for the fixed values of pH and 0
Ru

C  from the MB equation:  

0 3+ 2+ + 8+
Ru 2 4 4

3+ 3+ + -1 3+ + -2 3+ 4 + -4 3+
10 20 44 Ru

=[Ru ]+[R(OH) ]+[Ru(OH) ]+4[Ru (OH) ]

[Ru ]+ [Ru ][H ] [Ru [H ] 4 [Ru ] [H ] =[Ru ]×

C

K K K 



  
 (7) 

The changes in equation (7) are made by using the consequence of mass action law: 

3+ i + -j
i j ij[Ru (OH) ] [Ru ] [H ]K  

In the case of ruthenium(IV) hydroxide similar relationships are obtained. We present here only 

the basic equations: 

+ 2+ +
2 2 2 2 S

RuO ×2H O+2H =Ru(OH) +2H O , lg 0.91K   (8) 

0 0 + 2
r r Ru Ru (s) ln( /[H ] ) lnG G RT C RT       (9) 

0
r SlnG RT K  

                 

 

where αRu is coefficient of hydrolysis reactions of 
2+
2

Ru(OH)  : 

2+ 4+ +
2 2 4 12 44

4Ru(OH) +4H O=Ru (OH) +4H , lg 7.19K   

calculated by the equation: 

2+ 4 + -4
Ru 44 2=1+4K [Ru(OH) ] [H ]   (10) 

The equilibrium concentration of 2+
2

Ru(OH)  ion is determined for respective values of pH and 
0
Ru

C  from MB conditions: 

0 2+ 4+ 2+ 2+ + -4 2+
Ru 2 4 12 2 44 2 2=[Ru(OH) +4[Ru (OH) ]=[Ru(OH) ](1+4K [Ru(OH) [H ] =[Ru(OH) ] RuC   

Within this method, when ΔGr < 0 the solid phase is thermodynamically unstable towards 
dissolution according the schemes (1) and (8) and, vice-versa, for the values ΔGr > 0 the formation 

of solid phase takes place. Results of calculation of the ΔGr dependence on pH for 
0 6
Ru

10C


  and 

10
-4

 mol L
-1

 for both the hydroxides are presented graphically in Figs. 1 and 2. From these graphs it 
is observed that in acidic solutions Ru(III) and Ru(IV) hydroxides are thermodynamically unstable in 

regard to dissolution. From the 0
Ru

r
(pH)

C
G , diagrams the pH of beginning of precipitation (pH0) 

corresponds to the condition ΔGr = 0. 

In the case of ruthenium(III) hydroxide, Ru(OH)3∙H2O(s), one obtains for 
0
Ru

C  = 10
-4

 mol L
-1

, 

pH 2.24 and for 0
Ru

C  = 10
-6

 mol L
-1

, pH 4.42 (Table 2). Therefore, by increasing 0
Ru

C
 
in mixtures, the 

pH0 value shifts to region of acidic solutions.  

For ruthenium(IV) hydroxide, RuO2 ∙ H2O(s), from the ΔGr(pH) diagram one can see that pH0 is 

3.93 for 0
Ru

C  = 10
-4

 mol L
-1

, pH0 is 3.93 for 
0
Ru

C  = 10
-6

 mol L
-1

 (see also Table 2). 

 

 



I. Povar et al. J. Electrochem. Sci. Eng. 6(1) (2016) 123-133 

doi:10.5599/jese.226 127 

Table 2. Areas of thermodynamic stability of solid phases,  
ΔGr > 0 of the Ru(III) and Ru(IV) hydroxides in the system Ru

n+
-H2O 

The precipitate composition Area of thermodynamic stability of solid phase 

 0 4
Ru

10C


  mol L
-1

 0 -6
Ru

=10C  mol L
-1

 

Ru(OH)3 ∙ H2O (s) 2.42 < pH < 14.0 4.42 < pH < 14.0 

RuO2 ∙ H2O (s) 3.93 < pH < 14.0 4.43 < pH < 14.0 

 
Figure 1. The dependence of the Gibbs energy change on solution pH for the precipitation-

dissolution process of Ru(III) hydroxide. 
0
Ru

C : 1 – 10-4 mol L-1; 2 - 10-6 mol L-1 

 
Figure 2. The curves ΔGr(S)(pH) of the dissolution-precipitation process of Ru(IV) hydroxide, 

0
Ru

C : 1 – 10
-4 

mol L
-1

; 2 - 10
-6

 mol L
-1

 



J. Electrochem. Sci. Eng. 6(1) (2016) 123-133 RUTHENIUM REDOX EQUILIBRIA: 1. THERMODYNAMIC STABILITY  

128  

Repartition of Ru(III) and Ru(IV) soluble and insoluble chemical species towards the solution pH and 
initial concentrations of ruthenium in heterogeneous mixture solid phase–saturated solution 

In formulating MB conditions for the precipitate components in heterogeneous mixture we 

have taken into consideration the quantity of each component in the solid and liquid phases 

(residual quantity). Knowing the residual concentration of i-th ion ( r
i

C ) in solution, its quantity in 

precipitate in a unit volume is easily calculated as the difference between the analytical 

concentration in the mixture ( 0
i

C ) and that in solution. Therefore, in terms of molar 

concentrations, 
0 r

i i i
C C C    

where ΔCi denotes the quantity of i ion in precipitate (moles) in 1 L of solution. If this quantity is 

recalculated related to the mixture volume (Vam), then  
0 r

i i i
m m m  

  
where 0

i i
,m m  and r

i
m denote the quantity of ion (in moles) in precipitate, in mixture and in 

volume of liquid phase, respectively. For several systems it was established [17], that this 

approximation can be applied for solutions up to 0.5 mol L
-1

. 
Initially, a set of possible reactions [18] in the system Ru(IV)–H2O is taken into account: 

+ 2+
2 2 2 2

2+ + -2
S 2

RuO ×H O (s) +2H =Ru(OH) +2H O,

=[Ru(OH) ][H ]K
   (11) 

2+ 4+ +
2 2 4 12

4+ + 4 2+ 4
4 ,4 4 12 2

4Ru(OH) +4H O=Ru (OH) +4H ,

[Ru (OH) ][H ] /[Ru(OH) ]K 
  (12) 

The MB equation for Ru(IV) is written as: 

0 r 2+ 4+
Ru Ru Ru Ru 2 4 12+[Ru(OH) ]+4[Ru (OH) ]C C C C       (13) 

Taking into consideration equation (12), the expression for r
Ru

C  has the form: 
r 2+ 4+ 4 + -4 2+
Ru 2 4,4 4 12 2=[Ru(OH) ]+4 [Ru (OH) ] [H ] =[Ru(OH) ] RuC K   

Here αi is the coefficient of hydrolysis reactions for ruthenium: 
2+ 4 +

Ru 4,4 2=1+4 [Ru(OH) ] [H ]K  

From equation (11) it follows  
2+ + 2
2[Ru(OH) ] [H ]SK  

Substituting this expression in (13), one gets: 

0 + 2 4 + 4
Ru Ru S 4 ,4 S[H ] 4 [H ]C C K K K     (14) 

Results and discussion 

From obtained relations, ΔCRu for the set of values (
0
Ru

C , [H
+
] = 10

-pH
), can be easily determined. 

Further, the concentrations of the chemical species in solution are calculated. Then the molar 

fractions of chemical species in solution are computed. Finally, the molar fractions of chemical 

species in heterogeneous mixture, in function of pH for constant values of 
0
Ru

C  are determined, 

using the equations: 

0 2+ 0 4+ 0
s Ru Ru 10 2 Ru 44 4 12 Ru sum 10 44

/ ; [Ru(OH) ]/ ; 4[Ru (OH) ]/ ;C C C C            (15) 



I. Povar et al. J. Electrochem. Sci. Eng. 6(1) (2016) 123-133 

doi:10.5599/jese.226 129 

The subscript index (sum) symbolizes the sum of the all soluble species fractions containing 

Ru(IV). It is easily to observe that 
sum S

1   . 

The molar fraction of the metal ion from precipitate is defined also as the degree of 
precipitation [12,13]. For heterogeneous equilibria, the molar fractions of chemical species 

depend on the mixture initial composition, in this case of 
0
Ru

C , even in the absence of polynuclear 

complexes, thus 0
i Ru( , pH).f C  Consequently, the diagrams of repartition in heterogeneous 

systems (DRHS) can be plotted in coordinates 0
Ru

i =co
( , pH)

C nst
  or 

0
i Ru pH=const( , )C . It should be 

mentioned that the relationships derived above is valid only within the thermodynamic stability 
area of the precipitate (∆Gr < 0) [9]. 

Analogically, DRHS are calculated for the system Ru(III)–H2O, where the following equilibria are 

possible [11]: 

+ 3+
3 2 2 S

Ru(OH) ×H O (s) +3H Ru +4H O, lg 1.64K   (16) 

3+ 2+ +
2 10

Ru +H O=Ru(OH) +H , lg 2.24K    (17) 

3+ + +
2 2 20

Ru +2H O=Ru(OH) +2H , lg 3.52K    (18) 

3+ 8+ +
2 4 4 44

4Ru +4H O=Ru (OH) +4H , lg 10.80K    (19) 

Here, the particular equilibrium constants represent: 

0 3+ + -3
S

=[Ru ][H ]K  (20) 

2+ + 3+
10 =[Ru(OH) ][H ]/[Ru ]K  (21) 

+ + 2 3+
20 2=[Ru(OH) ][H ] /[Ru ]K  (22) 

8+ + 4 3+ 4
44 4 4=[Ru (OH) [H ] /[Ru ]K  (23) 

The MB equations for this system are following: 

0 r
Ru Ru RuC C C     (24) 

r 3+ 2+ + 8+
Ru 2 4 4

3+ + -1 + -2 3+ 3 + -4
10 20 44

=[Ru ]+[Ru(OH) ]+[Ru(OH) ]+4[Ru (OH) ]

[Ru ](1+ [H ] + [H ] +4 [Ru ] [H ]

C

K K K




 (25) 

The equilibrium concentration of Ru
3+

 ion is calculated using the relation: 

3+ + 3
S

[Ru ] [H ]K
 (26) 

From equations (24) and (25) one obtains the expression for calculating ΔCRu, then the 

equilibrium concentrations of the free Ru(III) ion and its hydroxocomplexes are determined. 

Finally, on the diagram of repartition the γij(pH) functions are plotted for fixed 
0
Ru

C values: 

0 3+ 0 2+ 0
S0 Ru Ru 00 10 Ru

+ 0 8+ 0
20 2 Ru 44 4 4 Ru

00 10 20 44

/ ; [Ru ]/ ; =[Ru(OH) ]/

=[Ru(OH) ]/ ; =4[Ru (OH) ] /

Ru

Sum

C C C C

C C

  

 

    

  

   

 

Therefore, the procedure of DRHS construction includes several stages: 



J. Electrochem. Sci. Eng. 6(1) (2016) 123-133 RUTHENIUM REDOX EQUILIBRIA: 1. THERMODYNAMIC STABILITY  

130  

1. The determination of the thermodynamic stability of solid phases (ΔCRu > 0) in function of 

solution pH or 
0
Ru

C . 

2. Calculation of the molar fractions of all species γi in the solid phase and solution within the pH 

(or 
0
Ru

C ) range, established at the first stage.  

3. For the image integrity, outside this range, in the case of homogeneous solution, the molar 

fractions of chemical species are calculated by typical equations for plotting the diagrams of 

distribution in solutions. Usually, in the γi(pH) diagrams the pH values vary between 0 and 14. 

By the developed diagrams, the precipitate quantity (in moles or grams), in 1 L of solution for 

certain pH and 
0
Ru

C
 
values, is easily determined. For example, from Fig. 3(b) it results that for  

pH 3.95 and 
0
Ru

C =10
-4

 mol L
-1 

the degree of precipitation S0 = 0.5, whence it follows that  

ΔCRu=S0 
0
Ru

C =5·10
-5

 mol L
-1

 or quantity of precipitate P (in g L
-1

) PRuM(RuO2·2H2O)=5·10
-5

 mol L
-

1
·169 g mol

-1
=

 
=8.45·10

-3
 g L

-1
. Thus, under these conditions in one liter of solution 8.45 mg of 

ruthenium(IV) hydroxide is deposited. 
a 

 
b 

 

Figure 3. Repartition for the system Ru(OH)3(S)·H2O – saturated aqueous solution, 
0

CRu : a) 10
-4

; b) 10
-6

 mol L
-1

 



I. Povar et al. J. Electrochem. Sci. Eng. 6(1) (2016) 123-133 

doi:10.5599/jese.226 131 

In Fig. 3 (a,b) and 4 (a,b) there are shown DRHS for both systems for two values of 
0
Ru

C : 10
-4

 and 

10
-6

 mol L
-1

. From these diagrams, the pH regions of predominance of different Ru(III) and Ru(IV) 

species are depicted in Table 3. 

 
a 

 
b 

 

Figure 4. Repartition for the system RuO2·2H2O(S) – H2O ; 
0

CRu : a) 10
-4

, b) 10
-6

 mol L
-1 



J. Electrochem. Sci. Eng. 6(1) (2016) 123-133 RUTHENIUM REDOX EQUILIBRIA: 1. THERMODYNAMIC STABILITY  

132  

Table 3. pH regions of predominance of chemical species in  A - Ru(OH)3(S)·H2O – H2O  
and B - RuO2·2H2O(S) – H2O system 

System Chemical species pH regions of predominance 

  0
Ru

C = 10
-4

 mol L
-1

 
0
Ru

C  = 10
-6

 mol L
-1

 

A 

Ru
3+ 

0.00 < pH < 1.76 0.00 < pH < 1.76 
+
2

Ru(OH)  1.76 < pH < 2.42 1.76 < pH < 4.42 

Ru(OH)3∙H2O (s) 2.42< pH < 14.0 4.42 < pH < 14.0 

B 

2+
2

Ru(OH)  0.00 < pH < 1.05 0.00 < pH < 2.55 

4+
4 12

Ru (OH)  1.05 < pH < 3.93 2.55 < pH < 4.43 

RuO2∙H2O (s) 3.93 < pH < 14.0 4.43 < pH < 14.0 

Conclusions 

From the obtained results the following conclusions can be drawn: 

1. By means of the diagrams ΔG–pH, the areas of thermodynamic stability of Ru(III) and Ru(IV) 

hydroxides have been established for a number of analytical concentrations in heterogeneous 

mixtures. The diagrams of heterogeneous and homogeneous chemical equilibria have been 

used for graphical representation of complex equilibria in aqueous solutions containing Ru(III) 

and Ru(IV). 

2. The ruthenium (III) polynuclear hydroxocomplex 
8+

4 4Ru (OH)  (aq) for the considered values is not 

formed. 

3. At increasing 0
Ru

C  from 10
-6

 to 10
-4

 mol L
-1

, the region of predominance of the ruthenium(IV) 

polynuclear complex 4+
4 12

Ru (OH)  shifts to the acidic solutions and is increased by one pH unit. 

4. The ruthenium(III) monohydroxide +
2

Ru(OH)  is formed in insignificant amounts under analyzed 

conditions. 

5. The thermodynamic stability area of solid phases increases with growing the initial 

concentration of ruthenium in mixture. 

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