J. Chem. Thermodynamics 70 (2014) 39–45

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J. Chem. Thermodynamics

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Gibbs free energy of formation of rhodium sulfides

0021-9614/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jct.2013.10.011

⇑ Corresponding author. Tel.: +91 80 2293 2494; fax: +91 80 2360 0472.E-mail addresses: katob@materials.iisc.ernet.in, ktjacob@hotmail.com

(K.T. Jacob).

K.T. Jacob ⇑, Preeti GuptaDepartment of Materials Engineering, Indian Institute of Science, Bangalore 560012, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 September 2013Accepted 9 October 2013Available online 17 October 2013

Keywords:Sulfur potentialSystem (Rh + S)Emf measurementGibbs energy of formation

Using a solid-state electrochemical technique, thermodynamic properties of three sulfide phases(RhS0.882, Rh3S4, Rh2S3) in the binary system (Rh + S) are measured as a function of temperature overthe range from (925 to 1275) K. Single crystal CaF2 is used as the electrolyte. The auxiliary electrode con-sisting of (CaS + CaF2) is designed in such a way that the sulfur chemical potential converts into an equiv-alent fluorine potential at each electrode. The sulfur potentials at the measuring electrodes areestablished by the mixtures of (Rh + RhS0.882), (RhS0.882 + Rh3S4) and (Rh3S4 + Rh2S3) respectively. A gasmixture (H2 + H2S + Ar) of known composition fixes the sulfur potential at the reference electrode. Anovel cell design with physical separation of rhodium sulfides in the measuring electrode from CaS inthe auxiliary electrode is used to prevent interaction between the two sulfide phases. They equilibrateonly via the gas phase in a hermetically sealed reference enclosure. Standard Gibbs energy changes forthe following reactions are calculated from the electromotive force of three cells:2.2667Rh (s) + S2 (g) ? 2.2667RhS0.882 (s),

DrGo � 2330=ðJ �mol�1Þ ¼ �288690þ 146:18 ðT=KÞ;

4.44RhS0.882 (s) + S2 (g) ? 1.48Rh3S4 (s),

DrGo � 2245=ðJ �mol�1Þ ¼ �245596þ 164:31 ðT=KÞ;

4Rh3S4 (s) + S2 (g) ? 6Rh2S3 (s),

DrGo � 2490=ðJ �mol�1Þ ¼ �230957þ 160:03 ðT=KÞ:

Standard entropy and enthalpy of formation of rhodium sulfides from elements in their normal stan-dard states at T = 298.15 K are evaluated.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Processing ore concentrates containing platinum group metals(PGM) involves smelting to produce a (Fe + Co + Ni + Cu) matte thatacts as PGM collector [1]. Hence, accurate information of preciousmetal–sulfur systems is useful for optimizing PGM recovery. In re-cent years RhxSy, a balanced mixture of (Rh2S3 + Rh3S4 + Rh17S15)with Rh3S4 as an active phase, has emerged as a promising elec-tro-catalyst for the reduction of molecular oxygen to water inacidic medium [2–8]. It is the only commercial chalcogenide elec-tro-catalyst available for oxygen reduction reaction (ORR) applica-tions, such as depolarized electrolysis of HCl. The catalyst haspotential applications in fuel cells, especially direct methanol fuelcells (DFMCs). Accurate thermodynamic data for rhodium sulfides

will enable better characterization of the electrochemicalprocesses.

In the (Rh + S) binary system, four solid phases, Rh17S15

(RhS0.882), Rh3S4, Rh2S3 and RhS�3 are reported [9], but the phasediagram is incomplete. The diagram in the compilation of Massal-ski et al. [9] indicates the decomposition of Rh17S15 to a Rh–rich li-quid sulfide (matte) and sulfur–rich gas at T � 1373 K. Taylor [10]measured the liquidus and phase transformation temperatures of11 samples covering the composition range from (45 to 67) atompercent Rh by employing differential thermal analysis (DTA). Con-trary to the earlier phase diagram [9], the results of Taylor [10]show the existence of three-phase equilibrium consisting ofRh17S15, Rh3S4 and (Rh + S) liquid solution at T = 1386 K. Rh3S4

decomposes at T = 1403 K to form Rh2S3 and (Rh + S) liquid. Thedecomposition temperature of Rh2S3 was not determined.

Crystallographic studies on Rh17S15 (RhS0.882) were reported byGeller [11,12]. Rh17S15 is iso-structural with Pd17Se15 and has cubicstructure with space group Pm3m and two formula units per unit

40 K.T. Jacob, P. Gupta / J. Chem. Thermodynamics 70 (2014) 39–45

cell [11,12]. Rh17S15 is a naturally occurring mineral known asmiassite and was first synthesized by Matthias et al. [13]. Narenet al. [14] have confirmed crystallographic data for Rh17S15 andfound superconductivity below T = 5.4 K caused by the presenceof high density of states of Rh d bands near Fermi level. The secondsulfide, Rh3S4, commonly known as kingstonite, crystallizes withmonoclinic structure in the space group C2/m [15]. Rh2S3, often re-ferred to bowieite, has orthorhombic crystal structure with spacegroup Pbcn [16].

The Gibbs energy of formation of rhodium sulfides aboveT = 1084 K has been measured using different techniques[10,17,18]. Taylor [10] used the Knudsen effusion technique formeasuring the sulfur pressures within the temperature range from(1213 to 1363) K. Larson and Elliott [17] employed a solid-state cellwith stabilized-zirconia as the electrolyte, pure oxygen as the ref-erence electrode and mixture of Rh and RhxS under 101.3 kPa pres-sure of SO2 gas as the measuring electrode in the temperaturerange from (1084 to 1357) K. They were unable to identify the sul-fide phase in equilibrium with Rh. According to the phase diagramsuggested by Taylor [10], metallic Rh coexists with liquid (Rh + S)solution (matte) above T = 1213 K. There is a significant differencein the temperature dependence of the Gibbs energy of formation ofthe metal saturated phase, Rh17S15 (RhS0.882) reported by Taylor[10] and RhxS reported by Larson and Elliott [17]. Juza et al. [18]used the manometric technique to measure the dissociation pres-sure two rhodium sulfides (Rh3S4 and Rh2S3) within the tempera-ture range from (1226 to 1356) K. A solid-state electrochemicalcell based on single crystal CaF2 as the electrolyte is used in thisstudy to extend the measurements to lower temperature so thatthermodynamic properties are better defined.

2. Experimental procedure

2.1. Materials

Purity and sources of the chemicals used in this study are givenin table 1. Optical grade single crystals of CaF2 in the form of disksof 1.5 cm in diameter and 0.2 cm thick, were obtained fromHarshaw Chemical Company. A high-purity gas mixture of (H2 +H2S + Ar) of constant composition containing 0.8 volume fractionAr, 0.16654 volume fraction hydrogen and 0.03346 volume frac-tion H2S supplied by Matheson is used to define the sulfur partialpressure at the reference electrode. The high concentration of Arin the gas mixture minimizes thermal segregation in the gas mix-ture caused by the large difference in the atomic mass of the con-stituents H2 and H2S.

Three rhodium sulfides Rh17S15 (RhS0.882), Rh3S4 and Rh2S3 areprepared by direct reaction between Rh and S in evacuated andsealed silica ampoules at high temperatures. Rh powder is first re-duced under H2 gas at T = 873 K to remove surface oxide. Powdersof Rh and S are mixed in the appropriate stoichiometric ratio and

TABLE 1Sources and purity of chemicals used in the experiment.

Chemicals Sources Mass fractionpurity

Rh (powder) Alfa aesar 0.9995CaF2 (single crystal) Harshaw chemicals 0.9999CaF2 (powder) Apache chemical company 0.99999CaS Ventron corporation 0.9999Au wire Johnson matthey and mallory

ltd.0.999

(H2 + H2S + Ar) gasmixture

Matheson 0.99999

S (powder) Alfa aesar 0.995

sealed in evacuated ampoules. The mixture is reacted initially atT = 673 K for 48 h. Temperature is then raised slowly. The finalreaction temperatures are T = 1200 K for Rh17S15 (RhS0.882) andT = 1373 K for Rh3S4 and Rh2S3. After holding at the highest tem-perature for 72 h, the ampoules are furnace cooled. Sulfur conden-sation on the ampoule indicates incomplete reaction. In such cases,additional heat treatment was done till all the sulfur is consumed.Formation of single phase sulfides is confirmed by X-ray diffraction(XRD) and their composition verified by energy dispersive spec-troscopy (EDS). The lattice parameters of the synthesized com-pounds are a = 0.9913 nm for Rh17S15, a = 1.031 nm, b = 1.069 nm,c = 0.6210 nm for Rh3S4 and a = 0.8464 nm, b = 0.5987 nm,c = 0.6141 nm for Rh2S3.

2.2. Electrochemical measurements

The electromotive force (e.m.f.) of the following solid-state elec-trochemical cells were measured,

Au;H2 þH2S=CaSþ CaF2==CaF2==CaF2 þ CaS=Rh

þ RhS0:882;Au ; ðIÞ

Au;H2 þH2S=CaSþ CaF2==CaF2==CaF2 þ CaS=RhS0:882

þ Rh3S4; Au; ðIIÞ

Au;H2 þH2S= CaSþ CaF2==CaF2== CaF2 þ CaS=Rh3S4

þ Rh2S3;Au: ðIIIÞ

Single crystal CaF2 used as the solid electrolyte is an F� ion con-ductor and responds to the difference in the chemical potential offluorine at the two electrodes. At each electrode, (CaS + CaF2) pelletis used to convert the sulphur chemical potential into an equiva-lent fluorine potential [19] by virtue of the exchange reaction;

CaF2 þ 1=2S2 ! CaSþ F2: ð1Þ

There is negligible solid solubility between CaF2 and CaS andboth phases are present at unit activity at the auxiliary electrodes.The cell e.m.f. is thus related to the difference in the sulfur chem-ical potential established at the electrodes.

Several examples of the use of auxiliary electrodes are reportedin the literature. Auxiliary electrode of CaS has been used to con-vert sulfur chemical to an equivalent oxygen potential in conjunc-tion with CaO-stabilized ZrO2 [20]. An auxiliary electrode of Na2Shas been used for sulfur potential measurement along with Nab-alumina, which is a Na+ ion conductor [21]. Na2SO4 has beenused as an auxiliary electrode in probes for SO2/SO3 based on Nab-alumina [22]. Similarly, a (CaSO4 + CaF2) auxiliary electrode hasbeen used to measure the partial pressure of SO3 using a solid-statecell based on CaF2 as electrolyte [23].

The gas reference electrode on the left-hand side of each cell isconnected to the negative terminal of a high impedance (>1012 )digital voltmeter. The (H2 + H2S + Ar) gas mixture establishes thesulfur potential at the reference electrode. A mixture of two adja-cent phases in the (Rh + S) system fixes the sulfur potential atthe measuring electrodes. When the sulfur chemical potential ofthe measuring electrode is higher than that of the reference, thecell e.m.f. is positive. The e.m.f. values for cell (I) is measured inthe range temperature from (925 to 1200) K, cell (II) and cell (III)from (925 to 1275) K. The upper temperature limit for cell (I) isset by the formation of eutectic liquid at T = 1213 K [11]. The upperlimit for cells (II) and (III) is set by the softening of CaF2 single crys-tal, which serves as an electrolyte.

A schematic outline of apparatus used for e.m.f. measurement isdisplayed in figure 1. Auxiliary electrode pellets composed of(CaS + CaF2) are spring-loaded on both sides of single crystal CaF2

K.T. Jacob, P. Gupta / J. Chem. Thermodynamics 70 (2014) 39–45 41

using a system of alumina rods and slabs, with a gauze made of Ausandwiched between the electrolyte and each auxiliary electrode.The electrolyte and electrode pellets are stacked vertically. Goldelectrical leads are spotwelded to the gauze on either side of theelectrolyte. An alumina crucible containing equimolar mixturesof (Rh + RhS0.882), (RhS0.882 + Rh3S4) or (Rh3S4 + Rh2S3) is placedover the top auxiliary electrode supported on a short silica tube.This alumina crucible with V-shaped notches cut at the top, is cov-ered by a second inverted alumina crucible, edges of which areground to form a knife edge. When spring loaded against theCaF2 single crystal at high temperatures, the knife edge cuts intothe single crystal making a gas-tight joint. The electric lead tothe auxiliary electrode adjacent to the measuring electrode passesthrough a small orifice in the inverted alumina crucible. The orificeis closed with a glass seal. Thus the measuring electrode is isolated.The sulfur partial pressure generated by the decomposition of rho-dium sulfide in the alumina crucible is transmitted to the auxiliaryelectrode inside the enclosure via the gas phase. The physical sep-aration of the auxiliary electrode and rhodium sulfides on the mea-suring side is designed to prevent interaction between calcium andrhodium sulfides. The interaction is via the gas phase. A constantsulfur partial pressure is established inside the inverted aluminacrucible

The cell assembly is mounted inside a vertical alumina tube.The ends of the tube are closed with brass caps, which have provi-sion for electrical leads and gas inlet and outlet. The alumina tubeenclosing the cell is evacuated and refilled with the (H2 + H2S + Ar)gas mixture three times. The gas mixture with controlled (pH2S=pH2

)ratio fixes the sulfur potential at reference electrode. Initially whenthe cell is cold and the knife edge has not formed a hermetic sealaround the reference electrode, the gas mixture also enters the ref-erence enclosure. However, when the cell is heated to T = 1200 K,the knife edge of the inverted alumina crucible penetrates theCaF2 single crystal to make a hermetic seal. The limited quantity

FIGURE 1. A line sketch of the solid-state electrochemical cell assembly used forthis study.

of H2S and H2 thus trapped inside the inverted alumina crucibleinteracts with the rhodium sulfides to establish a unique sulfur po-tential and H2S/H2 ratio inside the reference enclosure at constanttemperature.

The outer alumina tube enclosing the cell is suspended in a ver-tical resistance furnace such that the cell is situated in the constanttemperature zone (±1 K). An earthed Kanthal shield is placed be-tween the vertical alumina tube and the furnace to avoid the in-duced e.m.f. on cell leads from the furnace winding. Theassembled cell is first heated to T = 1200 K to from the hermeticenclosure around the reference electrode. The e.m.f. is then moni-tored as a function of time at different temperatures. The temper-ature of the cell is measured with a Pt/Pt-13%Rh thermocouplechecked against the melting point of Au. The temperature is con-trolled to (±1 K). The e.m.f. of the cell becomes steady in �6 h atT = 1200 K. To check the reversibility of the cells, a small current(�15 lA) is passed through the cell in each direction for 10 minusing an external D.C. source. The e.m.f. is then observed as a func-tion of time. It is verified that the e.m.f. returned to the same valueafter successive micro-coulometric titrations in opposite direc-tions, thus confirming electrochemical reversibility. The e.m.f. isfound to be reproducible when the temperature is approachedfrom higher and lower sides, confirming thermal reversibility.Changing the flow rate of the reference gas mixture in the rangefrom (150 to 300) ml �min�1 did not affect the e.m.f.

After the completion of each experiment, the cell is cooled andthe electrodes are checked by optical and scanning electronmicroscopy and X-ray diffraction. No significant change in thephase composition of the electrodes during electrochemical mea-surement is observed.

3. Results and discussion

The measured e.m.f.s of the three cells are listed in table 2 anddisplayed in figure 2. They show linear variation with temperature.In the temperature range of measurement, the e.m.f. of cell (I) isnegative, while the e.m.f. of the other two cells are positive.Expressions obtained from least-squares regression analysis are:

EI � 3:58=ðmVÞ ¼ �278:84þ 0:1902 ðT=KÞ; ð2Þ

EII � 3:18=ðmVÞ ¼ �167:18þ 0:2372 ðT=KÞ; ð3Þ

EIII � 4:22=ðmVÞ ¼ �129:25þ 0:2261 ðT=KÞ: ð4Þ

The uncertainty limits correspond to twice the standard deviation.The gas mixture (H2 + H2S + Ar) of known composition establishes

Table 2Reversible e.m.f. (E) of the solid-state electrochemical cells (I)–(III) at differenttemperatures (T).

Cell I Cell II Cell III

T/K E/mV T/K E/mV T/K E/mV

925 �100.5 ± 1.7 925 50.8 ± 1.5 925 80 ± 2975 �95.2 ± 1.5 975 64.4 ± 1.4 937.5 85 ± 2

1000 �89.6 ± 1.5 1025 76.7 ± 1.4 959 85.5 ± 1.91065 �76.1 ± 1.4 1075 89.7 ± 1.4 1024 104 ± 1.91125 �66.6 ± 1.3 1124 96.7 ± 1.3 1028.5 101 ± 1.91150 �58.5 ± 1.3 1150 106.8 ± 1.3 1075 114 ± 1.81200 �50 ± 1.2 1175 112.4 ± 1.3 1125 125 ± 1.8

1228 124.5 ± 1.3 1173 138 ± 1.81275 134 ± 1.2 1197 137 ± 1.8

1215 146 ± 1.71230 150 ± 1.71259 154 ± 1.71275 161.1 ± 1.7

FIGURE 2. Temperature dependence of the e.m.f. of the solid-state electrochemicalcells: ––d–– (orange online), cell (I); ––N–– (blue online), cell (II); ––�–– (magentaonline), cell (III). (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.)

FIGURE 3. Standard Gibbs energy change for the reaction 2.2667Rh (s) + S2

(g) ? 2.2667 RhS0.882 (s) as a function of temperature: (black online), thisstudy; —N— (green online), Taylor [10]; (red online), RT ln PS2 in the two-phase (Rh + liquid) [10]; (blue online), standard Gibbs energy changefor the reaction xRh (s) + S2 (g) ? RhxS (s) [17]. (For interpretation of the referencesto color in this figure legend, the reader is referred to the web version of thisarticle.)

42 K.T. Jacob, P. Gupta / J. Chem. Thermodynamics 70 (2014) 39–45

the sulfur partial pressure over the (CaS + CaF2) electrode on the ref-erence side by virtue of the reaction:

2H2 ðgÞ þ S2 ðgÞ ! 2H2S ðgÞ: ð5Þ

The standard Gibbs energy of formation of H2S from NIST-JANAF[24] tables over the temperature range from (900 to 1300) K canbe represented by the equation:

Drð5ÞGo � 1880=ðJ �mol�1Þ ¼ �181074þ 99:46 ðT=KÞ: ð6Þ

For cell (I), the virtual cell reaction is:

2:2667RhS0:882 ðsÞ þ 2H2 ðgÞ ! 2:2667Rh ðsÞ þ 2H2S ðgÞ: ð7Þ

Similarly, for cells (II) and (III) the net cell reactions can be writ-ten as:

1:48Rh3S4 ðsÞ þ 2H2 ðgÞ ! 4:44RhS0:882 ðsÞ þ 2H2S ðgÞ; ð8Þ

6Rh2S3 ðsÞ þ 2H2 ðgÞ ! 4Rh3S4 ðsÞ þ 2H2S ðgÞ: ð9Þ

Gibbs energy changes for the reactions (7)–(9) can be computedfrom the e.m.f. using relation:

DG ¼ �gFE; ð10Þ

where g = 4 is number of electrons involved in the electrode reac-tions, F = 96485 J � V�1 is the Faraday constant and E/V is thee.m.f. measured. The standard Gibbs energy change for the cell reac-tions (7)–(9), can be computed from the e.m.f. and H2S/H2 ratio inthe gas mixture flowing over the reference electrode:

DGo ¼ DG� 2RT lnðpH2=pH2SÞ ¼ �gFE� 2RT lnðpH2S=pH2

Þ; ð11Þ

where R = 8.3144 J � K�1 �mol�1 is the gas constant and ratio of par-tial pressures of H2S to H2 in the reference gas mixture is(pH2S=pH2

) = 0.2009. Employing equation (11), the standard Gibbsenergy change for the cell reactions (7)–(9) are obtained as:

Drð7ÞGo � 1382=ðJ �mol�1Þ ¼ 107616� 46:72 ðT=KÞ; ð12Þ

Drð8ÞGo � 1227=ðJ �mol�1Þ ¼ 64522� 64:85 ðT=KÞ; ð13Þ

Drð9ÞGo � 1636=ðJ �mol�1Þ ¼ 49883� 60:57 ðT=KÞ: ð14Þ

The sulfur chemical potential defining reactions occurring at themeasuring electrodes of the cells (I)–(III) are:

2:2667Rh ðsÞ þ S2 ðgÞ ! 2:2667RhS0:882 ðsÞ; ð15Þ

4:44RhS0:882 ðsÞ þ S2 ðgÞ ! 1:48Rh3S4 ðsÞ; ð16Þ

4Rh3S4 ðsÞ þ S2 ðgÞ ! 6Rh2S3 ðsÞ: ð17Þ

The standard Gibbs energy change for the reactions (15)–(17)are evaluated by combining standard Gibbs energy change forthe cell reactions (12)–(14) with standard Gibbs energy of forma-tion for H2S (gas) from the NIST–JANAF tables [24] representedby equation (6). Thus,

Drð15ÞGo � 2330=ðJ �mol�1Þ ¼ �288690þ 146:18 ðT=KÞ; ð18Þ

Drð16ÞGo � 2245=ðJ �mol�1Þ ¼ �245596þ 164:31 ðT=KÞ; ð19Þ

Drð17ÞGo � 2490=ðJ �mol�1Þ ¼ �230957þ 160:03 ðT=KÞ: ð20Þ

Standard Gibbs energy change for the reactions (15)–(17) areplotted as a function of temperature and compared with prior mea-surements reported in the literature in figures 3, 4 and 5,respectively.

For the formation of solid RhS0.882 according to reaction (15),the standard Gibbs energy change obtained directly in this studyat lower temperatures is in good agreement with data reportedby Taylor [10] at higher temperatures, as shown in figure 3. Taylor[10] used the Knudsen effusion technique to measure the sulfurpartial pressure. Since a liquid phase of variable composition sep-arates solid Rh and RhS0.882 phases above T = 1213 K, Taylor [10]had to measure sulfur pressure as a function of composition ofthe liquid phase at different temperatures, integrate GibbsDuhemequation to obtain the activity of Rh as a function of compositionand then compute Gibbs energy of formation of RhS0.882. Consider-ing the procedure involved, the agreement between the two sets ofdata is encouraging. It is to be noted that the standard Gibbs en-ergy change for reaction (15) is equal to the chemical potentialof sulfur for (Rh + RhS0.882) equilibrium. Also shown in figure 3,are the chemical potentials corresponding to (Rh + liquid) equilib-rium reported by Taylor [10], and (Rh + RhxS) equilibrium reportedby Larson and Elliott [17] using a cell and with stabilized-zirconia

FIGURE 4. Standard Gibbs energy change for the reaction 4.44RhS0.882 (s) + S2

(g) ? 1.48 Rh3S4 (s) as a function of temperature: (black online), this study; d

(red online), Juza et al. [18]. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

FIGURE 5. Standard Gibbs energy change for the reaction 4Rh3S4 (s) + S2

(g) ? 6Rh2S3 (s) as a function of temperature: — (black online), this study; d (redonline), Juza et al. [18]. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

K.T. Jacob, P. Gupta / J. Chem. Thermodynamics 70 (2014) 39–45 43

as the electrolyte in the temperature ranges from (1084 to 1357) K.Larson and Elliott [17] were unable to identify the sulfide phaseformed in their experiments since XRD data for the sulfide didnot match structure information in the literature [13,18]. In thehigher temperature reaches of their experiment, a liquid phase isexpected to coexist with Rh according to the phase diagram of Tay-lor [10]. Their reported sulfur potentials match well with that ofTaylor [10] for (Rh + liquid) equilibrium. Since the results of Larsonand Elliott [17] do not show a change of slope at the eutectic tem-perature, they probably encountered super-cooled liquid belowT = 1213 K. DTA analysis by Taylor [10] of a liquid (Rh + S) sampleduring cooling indicated suppression of the eutectic reaction byT = 180 K. This would indicate that the measurements of Larsonand Elliott [17] correspond to (Rh + liquid) equilibrium in wholerange of temperature. The composition of the liquid would varywith temperature, defined by the liquidus curve and its extensionbelow the eutectic.

Employing a manometric technique, Juza et al. [18] measuredthe decomposition pressures of Rh3S4 and Rh2S3 in the tempera-ture range from (1226 to 1356) K. Their data are compared withthe results of this study in figures 4 and 5, respectively. There isgood agreement, although temperature dependence of Gibbs en-ergy change shows some difference. In view of difficulties in accu-rate calibration of differential Bourdon-type gauge at moderatelyhigh temperatures required to prevent condensation of sulfur va-por, the results of this study are considered to be superior provid-ing more accurate thermodynamic information.

Molar thermodynamic properties for three rhodium sulfides canbe computed using the results presented above:

Rh ðsÞ þ 0:441S2 ðgÞ ! RhS0:882 ðsÞ; ð21Þ

Df GoðRhS0:882Þ � 1030=ðJ �mol�1Þ¼ �127361þ 64:49 ðT=KÞ; ð22Þ

3Rh ðsÞ þ 2S2 ðgÞ ! Rh3S4 ðsÞ; ð23Þ

Df GoðRh3S4Þ � 3445=ðJ �mol�1Þ ¼ �548026þ 304:5 ðT=KÞ; ð24Þ

2Rh ðsÞ þ 1:5S2 ðgÞ ! Rh2S3 ðsÞ; ð25Þ

Df GoðRh2S3Þ � 2335=ðJ �mol�1Þ ¼ �403844þ 229:67 ðT=KÞ: ð26Þ

In equations (22), (24), and (26), the temperature-independentterm gives the enthalpy of formation (Df H

oTav

) at an average tem-perature of experiment and the temperature-dependent term withreversed sign gives the corresponding entropy change (Df S

oTav

). Theaverage temperature for reaction (21) is Tav = 1063 K and for reac-tions (23) and (25) Tav = 1100 K.

To evaluate thermodynamic properties of three rhodium sul-fides at T = 298.15 K heat capacity of reactants and products as afunction of temperature is required. Since heat capacity of rhodiumsulfides has not been measured, an average value for the change inheat capacity suggested by Kubaschewski and Alcock [25] may beused. The change in heat capacity (DCP) for reactions involving agas phase such as:

A ðsÞ þ bS2 ðgÞ ! AS2b ðsÞ; ð27Þ

aAxSy ðsÞ þ bS2 ðgÞ ! AaxSayþ2b ðsÞ; ð28Þ

can be approximated as:

DCP=J � K�1 �mol�1 ¼ 12:552b: ð29Þ

Thus the calculated average values for DCP for reactions (21),(23), and (25) are 5.534, 25.106 and 18.829 J � K�1 �mol�1, respec-tively. The standard entropy and enthalpy of formation for threerhodium sulfides according to reactions (21), (23), and (25) atT = 298.15 K can be calculated using the relations:

Df So298:15K ¼ Df S

oTav�Z Tav

298:15 KðDCP=TÞdT

¼ Df SoTav� DCPðln Tav � ln 298:15Þ; ð30Þ

Df Ho298:15 K ¼ Df H

oTav�Z Tav

298:15 KDCPdT

¼ Df HoTav� DCPðTav � 298:15Þ: ð31Þ

Values of Df So298:15 K for RhS0.882, Rh3S4, Rh2S3 corresponding to

reactions (21), (23), and (25) are �(71.53 ± 3.45),�(337.27 ± 11.42) and �(254.25 ± 7.68) J � K�1 �mol�1, respec-tively. Standard entropy (So

298:15 K ) of rhodium sulfides atT = 298.15 K can be calculated by combining the standard entropychange for the reactions (21), (23), and (25) at T = 298.15 K derived

FIGURE 7. Composition dependence enthalpy of mixing for the system (Rh + S) atT = 298.15 K relative to solid Rh and solid S (ortho): —d— (black online), this study;—j— (red online), Diéguez and Marzari [27]; —N— (green online), recalculated fromDiéguez and Marzari [27]. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

44 K.T. Jacob, P. Gupta / J. Chem. Thermodynamics 70 (2014) 39–45

from equation (30) and standard entropies of elements of Rh(31.56 ± 0.21) J � K�1 �mol�1 and S2 (g) (228.1 ± 0.42) J � K�1 �mol�1

from the thermodynamic data compilation of Pankratz [26].Thus, So

298:15K for RhS0.882, Rh3S4 and Rh2S3 are (60.62 ± 3.46),(213.61 ± 11.47) and (151.02 ± 7.71) J � K�1 �mol�1, respectively.

The values of Df Ho298:15 K calculated using equation (31) corre-

sponding to reactions (21), (23), and (25), where the standard statefor sulfur is diatomic gas, are �(131.6 ± 3.8), �(568.2 ± 12.3) and�(418.9 ± 8.3) kJ �mol�1, respectively. Since the standard state forsulfur at T = 298.15 K is solid sulfur with orthorhombic structure,the enthalpy for the change of standard state of sulfur from dia-tomic gas to solid sulfur needs to be considered. Pankratz[26] gives for the reaction, 2S (ortho) ? S2 (g), DrH

o298:15K�

0:9=ðkJ �mol�1Þ ¼ 128:5: Thus, the standard enthalpy of formationof rhodium sulfides from solid Rh and solid orthorhombic sulfurare Df H

o298:15K = �(74.93 ± 3.9) kJ �mol�1 for RhS0.882, Df H

o298:15K =

�(311.2 ± 12.4) kJ �mol�1 for Rh3S4 and Df Ho298:15K = �(226.2 ±

8.4) kJ �mol�1 for Rh2S3.Figure 6 shows the Gibbs energy of mixing (DmixG) at T = 1100 K

and enthalpy of mixing at T = 298.15 K as a function of compositionfor the system (Rh + S); the standard state of rhodium is solid me-tal and sulfur is diatomic gas. The values for three rhodium sulfidesare calculated by dividing the Gibbs energy of formation fromequation (22), (24), and (26) by the total number of atoms of rho-dium and sulfur present in each rhodium sulfide. The minimum forboth Gibbs energy and enthalpy of mixing occurs at the composi-tion corresponding to Rh2S3. The enthalpy of solid sulfur relativeto diatomic gas is also shown in the diagram. The length of the ar-rows in the figure is a measure of the enthalpy of mixing (DmixH)with respect to solid rhodium and solid sulfur.

Recently, Diéguez and Marzari [27] have performed first-princi-ples calculations of structural, electronic and thermodynamicproperties of the three rhodium sulfides using density-functionaltheory (DFT) in the Kohn-Sham framework. They have reportedcohesive energies and derived enthalpies of formation of Rh17S15

(RhS0.882), Rh3S4 and Rh2S3. Unfortunately, the two sets of dataare not internally consistent. Assuming that the results of their pri-mary calculations of cohesive energies (�5.05 eV � atom�1 forRh17S15, �4.83 eV � atom�1 for Rh3S4 and �4.73 eV � atom�1 for

FIGURE 6. Composition dependence Gibbs energy of mixing (DmixG) and enthalpyof mixing (DmixH) for the system (Rh + S): —j— (black online), DmixG at T = 1100 K;– – –d– – – (green online), DmixH at T = 298.15 K; X (red online), enthalpy of solid(ortho) sulfur relative to diatomic gas at T = 298.15 K; vertical arrows (red online)indicate the enthalpy of mixing (DmixH) with respect to Rh (s) and sulfur (ortho).(For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

Rh2S3) are correct, enthalpies of formation have been recalculated,using cohesive energies of solid Rh and S from Pankratz [26]. Dié-guez and Marzari [27] have specified neither the values of cohesiveenergies of pure Rh and S used in their calculations nor the sourceof their data. Since the literature data on cohesive energies of Rhand S are fairly consistent, the calculated enthalpies of formationare not significantly dependent on the source of data. The enthalpyof mixing at T = 298.15 K obtained in this study is compared withthe original data reported by Diéguez and Marzari [27] and valuesrecalculated from their cohesive energies in figure 7. Althoughtheir reported values appear to be in fair agreement with the re-sults of this study, it is the product of an incorrect calculation[27]. The values correctly calculated from their cohesive energiesdiffer substantially from the results of this study. An important dif-ference between the results of first-principles calculations andexperiment is the position of the minimum. Experimental datashow a minimum at the composition corresponding to Rh2S3,whereas calculations place the minimum at the composition ofRh3S4. The comparison illustrates that despite recent claims of suc-cess of first-principles approach for generating thermodynamicdata, experiments still remain the most reliable route.

4. Conclusions

The standard Gibbs energies of formation of three rhodium sul-fides, RhS0.882, Rh3S4 and Rh2S3, have been determined over thetemperature range from (925 to 1275) K using a solid-state elec-trochemical cell incorporating single crystal CaF2 as the solid elec-trolyte, and (CaF2 + CaS) as auxiliary electrodes to convert thesulfur chemical potential to equivalent fluorine potential. A novelcell design is used for measurement. The results can be expressedby the following equations:

Rh (s) + 0.441S2 (g) ? RhS0.882 (s),

Df GoðRhS0:882Þ � 1030=ðJ �mol�1Þ ¼ �127361þ 64:49 ðT=KÞ;

3Rh (s) + 2S2 (g) ? Rh3S4 (s),

Df GoðRh3S4Þ � 3445=ðJ �mol�1Þ ¼ �548026þ 304:5 ðT=KÞ;

2Rh (s) + 1.5S2 (g) ? Rh2S3 (s),

K.T. Jacob, P. Gupta / J. Chem. Thermodynamics 70 (2014) 39–45 45

Df GoðRh2S3Þ � 2335=ðJ �mol�1Þ ¼ �403844þ 229:67 ðT=KÞ:

The results obtained in this study are in reasonable accord withexperimental data reported in the literature above T = 1213 K. Thestandard entropy (So

298:15 K ) of the three rhodium sulfides estimatedfrom the results are (60.62 ± 3.46) J � K�1 �mol�1 for RhS0.882,(213.61 ± 11.47) J � K�1 �mol�1 for Rh3S4 and (151.02 ±7.71) J � K�1 �mol�1 for Rh2S3. The enthalpy of formation from solidRh and orthorhombic sulfur at T = 298.15 K are �(74.9 ±3.9) kJ �mol�1 for RhS0.882, �(311.16 ± 12.4) kJ �mol�1 for Rh3S4

and�(226.2 ± 8.4) kJ �mol�1 for Rh2S3. These results do not supportdata obtained from recent first-principles calculations [27].

Acknowledgements

K.T. Jacob is indebted to the Indian National Academy of Engi-neering for support as INAE Distinguished Professor. Preeti Guptaacknowledges the University Grants Commission, India, for theaward of Dr. D.S. Kothari Postdoctoral Fellowship.

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JCT 13–350