**2.2.1 Methods based on <sup>o</sup>** *G***r x T diagrams**

It will be supposed that the oxide M2O5 can generate two gaseous chlorinated species, MCl4 and MCl5:

$$\begin{aligned} \text{M}\_2\text{O}\_5(\text{s}) + 5\text{Cl}\_2(\text{g}) &\to 2\text{MCl}\_5(\text{g}) + \frac{5}{2}\text{O}\_2(\text{g})\\ \text{M}\_2\text{O}\_5(\text{s}) + 4\text{Cl}\_2(\text{g}) &\to 2\text{MCl}\_4(\text{g}) + \frac{5}{2}\text{O}\_2(\text{g}) \end{aligned} \tag{21}$$

The first reaction is associated with a reduction of the number of moles of gaseous species (ng = -0.5), but in the second the same quantity is positive (ng = 0.5). If the gas phase is described as an ideal solution, the first reaction should be associated with a lower molar entropy than the second. The greater the number of mole of gaseous products, the greater the gas phase volume produced, and so the greater the entropy generated. By plotting the molar Gibbs energy of each reaction as a function of temperature, the curves should cross each other at a specific temperature (*T*C). For temperatures greater than *T*C the formation of MCl4 becomes thermodynamically more favorable (see Figure 4).

Fig. 4. Hypothetical <sup>o</sup> *G*<sup>r</sup> x T curves with intercept.

An interesting situation occurs, if one of the chlorides can be produced in the condensed state (liquid or solid). Let's suppose that the chloride MCl5 is liquid at lower temperatures.

On the Chlorination Thermodynamics 795

proposed reaction path. If so, the curves should lay one above the other. The standard reaction Gibbs energy would then grow in the following order: MCl, MCl2, MCl3, MCl4 and

Another possibility is that the curve for the formation of one of the higher chlorinated species is associated with lower Gibbs energy values in comparison with the curve of a lower chlorinated compound. A possible example thereof is depicted on Figure (7), where the <sup>o</sup> *G*<sup>r</sup> x *T* curve for the production of MCl3 lies bellow the curve associated with the

Fig. 6. Hypothetic <sup>o</sup> *G*r x T curves for successive chlorination reactions

Fig. 7. Successive chlorination reactions – direct formation of MCl3 from MCl

The formation of the species MCl2 would be thermodynamically less favorable, and MCl3 is preferentially produced directly from MCl (MCl + Cl2 = MCl3). In this case, however, for the diagram to remain thermodynamically consistent, the curves associated with the formation of MCl2 from MCl and MCl3 from MCl (broken lines) should be substituted for the curve associated with the direct formation of MCl3 from MCl for the entire temperature range. The same effect could originate due to the occurrence of a phase transition. Let's suppose that in the temperature range considered MCl3 sublimates at *T*s. Because of this

MCl5 (Figure 6).

formation of MCl2.

The ebullition of MCl5, which occur at a definite temperature (*T*t), dislocates the curve to lower values for temperatures higher than *T*t. Such an effect would make the production of MCl5 in the gaseous state thermodynamically more favorable even for temperatures greater than *T*c (Figure 5). Such fact the importance of considering phase transitions when comparing <sup>o</sup> *G*r x *T* curves for different reactions.

Fig. 5. Effect of MCl5 boiling temperature

Although simple, the method based on the comparison of <sup>o</sup> *G*r x *T* diagrams is of limited application. The problem is that for discussing the thermodynamic viability of a reaction one must actually compute the thermodynamic driving force (Eq. 15 and 16), and by doing so, one must fix values for the concentration of Cl2 and O2 in the reactor's atmosphere, which, in the end, define the value of the reaction coefficient.

If the <sup>o</sup> *G*r x *T* curves of two reactions lie close to one another (difference lower than 10 KJ/mol), it is impossible to tell, without a rigorous calculation, which chlorinated specie should have the highest concentration in the gaseous state, as the computed driving forces will lie very close from each other. In these situations, other methods that can address the direct effect of the reactor's atmosphere composition should be applied.

Apart from its simplicity, the <sup>o</sup> *G*r x *T* diagrams have another interesting application in relation to the proposal of reactions mechanisms. From the point of view of the kinetics, the process of forming higher chlorinated species by the "collision" of one molecule of the oxide M2O5 and a group of molecules of Cl2, and vise versa, shall have a lower probability than the one defined by the first formation of a lower chlorinated specie, say MCl2, and the further reaction of it with one or two Cl2 molecules (Eq. 22).

Let's consider that M can form the following chlorides: MCl, MCl2, MCl3, MCl4, and MCl5. The synthesis of MCl5 can now be thought as the result of the coupled reactions represented by Eq. (22).

$$\begin{aligned} \text{M}\_2\text{O}\_5 + \text{Cl}\_2 &\to 2\text{MCl} + 2.5\text{O}\_2\\ \text{MCl} + 0.5\text{Cl}\_2 &\to \text{MCl}\_2 &\text{MCl}\_2 + 0.5\text{Cl}\_2 &\to \text{MCl}\_3\\ \text{MCl}\_3 + 0.5\text{Cl}\_2 &\to \text{MCl}\_4 &\text{MCl}\_4 + 0.5\text{Cl}\_2 &\to \text{MCl}\_5 \end{aligned} \tag{22}$$

By plotting the <sup>o</sup> *G*r x *T* diagrams of all reactions presented in Eq. (22) it is possible to evaluate if the thermodynamic stability of the chlorides follows the trend indicated by the

The ebullition of MCl5, which occur at a definite temperature (*T*t), dislocates the curve to lower values for temperatures higher than *T*t. Such an effect would make the production of MCl5 in the gaseous state thermodynamically more favorable even for temperatures greater than *T*c (Figure 5). Such fact the importance of considering phase transitions when

Although simple, the method based on the comparison of <sup>o</sup> *G*r x *T* diagrams is of limited application. The problem is that for discussing the thermodynamic viability of a reaction one must actually compute the thermodynamic driving force (Eq. 15 and 16), and by doing so, one must fix values for the concentration of Cl2 and O2 in the reactor's atmosphere, which, in

If the <sup>o</sup> *G*r x *T* curves of two reactions lie close to one another (difference lower than 10 KJ/mol), it is impossible to tell, without a rigorous calculation, which chlorinated specie should have the highest concentration in the gaseous state, as the computed driving forces will lie very close from each other. In these situations, other methods that can address the

Apart from its simplicity, the <sup>o</sup> *G*r x *T* diagrams have another interesting application in relation to the proposal of reactions mechanisms. From the point of view of the kinetics, the process of forming higher chlorinated species by the "collision" of one molecule of the oxide M2O5 and a group of molecules of Cl2, and vise versa, shall have a lower probability than the one defined by the first formation of a lower chlorinated specie, say MCl2, and the further

Let's consider that M can form the following chlorides: MCl, MCl2, MCl3, MCl4, and MCl5. The synthesis of MCl5 can now be thought as the result of the coupled reactions represented

2 22 2 3

(22)

3 2 44 2 5

By plotting the <sup>o</sup> *G*r x *T* diagrams of all reactions presented in Eq. (22) it is possible to evaluate if the thermodynamic stability of the chlorides follows the trend indicated by the

MCl 0.5Cl MCl MCl 0.5Cl MCl MCl 0.5Cl MCl MCl 0.5Cl MCl

 

direct effect of the reactor's atmosphere composition should be applied.

25 2 2

M O Cl 2MCl 2.5O

comparing <sup>o</sup> *G*r x *T* curves for different reactions.

Fig. 5. Effect of MCl5 boiling temperature

the end, define the value of the reaction coefficient.

reaction of it with one or two Cl2 molecules (Eq. 22).

by Eq. (22).

proposed reaction path. If so, the curves should lay one above the other. The standard reaction Gibbs energy would then grow in the following order: MCl, MCl2, MCl3, MCl4 and MCl5 (Figure 6).

Fig. 6. Hypothetic <sup>o</sup> *G*r x T curves for successive chlorination reactions

Another possibility is that the curve for the formation of one of the higher chlorinated species is associated with lower Gibbs energy values in comparison with the curve of a lower chlorinated compound. A possible example thereof is depicted on Figure (7), where the <sup>o</sup> *G*<sup>r</sup> x *T* curve for the production of MCl3 lies bellow the curve associated with the formation of MCl2.

Fig. 7. Successive chlorination reactions – direct formation of MCl3 from MCl

The formation of the species MCl2 would be thermodynamically less favorable, and MCl3 is preferentially produced directly from MCl (MCl + Cl2 = MCl3). In this case, however, for the diagram to remain thermodynamically consistent, the curves associated with the formation of MCl2 from MCl and MCl3 from MCl (broken lines) should be substituted for the curve associated with the direct formation of MCl3 from MCl for the entire temperature range.

The same effect could originate due to the occurrence of a phase transition. Let's suppose that in the temperature range considered MCl3 sublimates at *T*s. Because of this

On the Chlorination Thermodynamics 797

*g gg <sup>P</sup>*

*g g gg <sup>P</sup>*

Next, two intensive properties must be chosen, whose values are fixed, for example, the partial pressure of Cl2 and the temperature. The partial pressure of each chlorinated species

*P RT*

*P RT*

5/2

<sup>2</sup> , exp <sup>2</sup>

<sup>2</sup> , exp <sup>2</sup>

By fixing *T* and *P*(Cl2) the application of the natural logarithm to both sides of Eq. (24)

54 2 45 2

MCl MCl O MCl MCl O

The lines associated with the formation of MCl4 and MCl5 would have the same angular coefficient, but different linear coefficients. If the partial pressure of Cl2 is equal to one (pure Cl2 is injected into the reactor), the differences in the standard reaction Gibbs energy controls the values of the linear coefficients observed. If the lowest Gibbs energy values are associated with the formation of MCl5, its line would have the greatest linear coefficient

An interesting situation occurs if the curves obtained for the chlorinated species of interest cross each other (Figure 10). This fact would indicate that for some critical value of *P*(O2) there would be a different preference for the system to generate each one of the chlorides. One of them prevails for higher partial pressure values and the other for values of *P*(O2) lower than the critical one. Such a behavior could be exemplified if the chlorination of M also generates the gaseous oxychloride MOCl3 (M2O5 + 2Cl2 = 2MOCl3 +

ln ln 2.5ln ln ln 2.5ln *P f P P f P* 

*f TP P RT*

5/2

*f TP P RT*

5/2

2

g gg <sup>s</sup> <sup>5</sup> MCl O Cl M O

<sup>5</sup> 2g <sup>5</sup> 2

g gg <sup>s</sup> <sup>4</sup> MCl4 O2 Cl2 M2O5

<sup>5</sup> 2 4 2

2

exp

exp

2

becomes in this case a function of only the partial pressure of O2.

 

, ,

5 5 22 4 42 2

MCl MC Cl O

*P f TP P P f TP P*

 

MCl MCl Cl O

2

Cl MCl 5/2 O

2

Cl MCl 5/2 O

5

*P*

*P*

4

MCl Cl Cl

5 22

results in a linear behavior.

(Figure 9).

1.5O2).

MCl Cl Cl

4 22

5 22 2 5

(23)

(24)

522 2 5

g gg s MCl O Cl M O

*g g gg*

<sup>5</sup> 2 5

<sup>5</sup> 2 4

422 2 5

(25)

g gg s MCl O Cl M O

*g g gg*

phenomenon the curve for the formation of MCl2 crosses the curve for the formation of the last chloride at *T*c, so that for *T* > *T*c its formation is associated with a higher thermodynamic driving force (Figure 8). So, for *T* > *T*c, MCl3 is formed directly from MCl, resulting in the same modification in the reaction mechanism as mentioned above.

Fig. 8. Direct formation of MCl3 from MCl stimulated by MCl3 sublimation

For temperatures higher than *T*c, the diagram of Figure (8) looses its thermodynamic consistency, as, according to what was mentioned in the last paragraph, the formation of MCl2 from MCl is impossible in this temperature range. The error can be corrected if, for *T* > *T*c, the curves associated with the formation of MCl2 and MCl3 (broken lines) are substituted for the curve associated with the formation of MCl3 directly from MCl.

A direct consequence of that peculiar thermodynamic fact, as described in Figures (7) and (8), is that under these conditions, a predominance diagram would contain a straight line showing the equilibrium between MCl and MCl3, and the field corresponding to MCl2 would not appear.

### **2.2.2 Method of Kang and Zuo**

Kang Zuo (1989) introduced a simple method for comparing the thermodynamic tendencies of formation of compounds obtained by gas – solid reactions, in that each equilibrium equation is solved independently, and the concentration of the desired species plotted as a function of the gas phase concentration and or temperature. The method will be illustrated for the reactions defined by Eq. (21). The concentrations of MCl4 and MCl5 in the gaseous phase can be computed as a function of temperature, partial pressure of Cl2, and partial pressure of O2.

phenomenon the curve for the formation of MCl2 crosses the curve for the formation of the last chloride at *T*c, so that for *T* > *T*c its formation is associated with a higher thermodynamic driving force (Figure 8). So, for *T* > *T*c, MCl3 is formed directly from MCl, resulting in the

same modification in the reaction mechanism as mentioned above.

Fig. 8. Direct formation of MCl3 from MCl stimulated by MCl3 sublimation

for the curve associated with the formation of MCl3 directly from MCl.

would not appear.

partial pressure of O2.

**2.2.2 Method of Kang and Zuo** 

For temperatures higher than *T*c, the diagram of Figure (8) looses its thermodynamic consistency, as, according to what was mentioned in the last paragraph, the formation of MCl2 from MCl is impossible in this temperature range. The error can be corrected if, for *T* > *T*c, the curves associated with the formation of MCl2 and MCl3 (broken lines) are substituted

A direct consequence of that peculiar thermodynamic fact, as described in Figures (7) and (8), is that under these conditions, a predominance diagram would contain a straight line showing the equilibrium between MCl and MCl3, and the field corresponding to MCl2

Kang Zuo (1989) introduced a simple method for comparing the thermodynamic tendencies of formation of compounds obtained by gas – solid reactions, in that each equilibrium equation is solved independently, and the concentration of the desired species plotted as a function of the gas phase concentration and or temperature. The method will be illustrated for the reactions defined by Eq. (21). The concentrations of MCl4 and MCl5 in the gaseous phase can be computed as a function of temperature, partial pressure of Cl2, and

$$P\_{\rm MC1\_5} = \sqrt{\frac{P\_{\rm CI\_2}}{P\_{\rm O\_2}}} \exp\left(-\frac{\left(2\text{g}\_{\rm MC1\_5}^6 + \frac{5}{2}\text{g}\_{\rm O\_2}^6 - 5\text{g}\_{\rm CI\_2}^6 - \text{g}\_{\rm M\_2O\_5}^{\rm S}\right)}{RT}\right)$$

$$P\_{\rm MC1\_4} = \sqrt{\frac{P\_{\rm CI\_2}}{P\_{\rm O\_2}}} \exp\left(-\frac{\left(2\text{g}\_{\rm MC14}^6 + \frac{5}{2}\text{g}\_{\rm O2}^6 - 4\text{g}\_{\rm CI\_2}^6 - \text{g}\_{\rm MC205}^{\rm S}\right)}{RT}\right) \tag{23}$$

Next, two intensive properties must be chosen, whose values are fixed, for example, the partial pressure of Cl2 and the temperature. The partial pressure of each chlorinated species becomes in this case a function of only the partial pressure of O2.

$$\begin{aligned} P\_{\text{MC}\_{1}} &= f\_{\text{MC}\_{5}} \left( T, P\_{\text{Cl}\_{2}} \right) P\_{\text{O}\_{2}}^{-5/2} \\ P\_{\text{MC}\_{4}} &= f\_{\text{MC}\_{4}} \left( T, P\_{\text{Cl}\_{2}} \right) P\_{\text{O}\_{2}}^{5/2} \\ f\_{\text{MC}\_{5}} \left( T, P\_{\text{Cl}\_{2}} \right) &= P\_{\text{Cl}\_{2}} \, ^{-5/2} \exp \left( -\frac{\left( 2 \text{g}\_{\text{NaCl}\_{5}}^{\text{g}} + \frac{5}{2} \text{g}\_{\text{O}\_{2}}^{\text{g}} - 5 \text{g}\_{\text{Cl}\_{2}}^{\text{g}} - \text{g}\_{\text{M}\_{2}\text{O}\_{5}}^{\text{s}} \right)}{2RT} \right) \\ f\_{\text{MC}\_{4}} \left( T, P\_{\text{Cl}\_{2}} \right) &= P\_{\text{Cl}\_{2}} \, ^{-2} \exp \left( -\frac{\left( 2 \text{g}\_{\text{MCl}\_{4}}^{\text{g}} + \frac{5}{2} \text{g}\_{\text{O}\_{2}}^{\text{g}} - 4 \text{g}\_{\text{Cl}\_{2}}^{\text{g}} - \text{g}\_{\text{M}\_{2}\text{O}\_{5}}^{\text{s}} \right)}{2RT} \right) \end{aligned} \tag{24}$$

By fixing *T* and *P*(Cl2) the application of the natural logarithm to both sides of Eq. (24) results in a linear behavior.

$$\begin{aligned} \ln P\_{\text{MCl}\_5} &= \ln f\_{\text{MCl}\_4} + 2.5 \ln P\_{\text{O}\_2} \\ \ln P\_{\text{MCl}\_4} &= \ln f\_{\text{MCl}\_5} + 2.5 \ln P\_{\text{O}\_2} \end{aligned} \tag{25}$$

The lines associated with the formation of MCl4 and MCl5 would have the same angular coefficient, but different linear coefficients. If the partial pressure of Cl2 is equal to one (pure Cl2 is injected into the reactor), the differences in the standard reaction Gibbs energy controls the values of the linear coefficients observed. If the lowest Gibbs energy values are associated with the formation of MCl5, its line would have the greatest linear coefficient (Figure 9).

An interesting situation occurs if the curves obtained for the chlorinated species of interest cross each other (Figure 10). This fact would indicate that for some critical value of *P*(O2) there would be a different preference for the system to generate each one of the chlorides. One of them prevails for higher partial pressure values and the other for values of *P*(O2) lower than the critical one. Such a behavior could be exemplified if the chlorination of M also generates the gaseous oxychloride MOCl3 (M2O5 + 2Cl2 = 2MOCl3 + 1.5O2).

On the Chlorination Thermodynamics 799

which is characterized by a proper phase ensemble, their amounts and compositions. This method is equivalent to solve all chemical equilibrium equations at the same time, so, that the compositions of the chlorinated species in each one of the phases present are calculated

For treating the equilibrium associated with the chlorination processes, two type of diagrams are important: *predominance diagrams*, and *phase speciation diagrams*. The first sort of diagram describes the equilibrium phase ensemble as a function of temperature, and or partial pressure of Cl2 or O2. The second type describes how the composition of individual

The first step is to change the initial constraint vector (*T*, *P*, *n*(O), *n*(M), *n*(Cl)), by modifying the definition of the components. Instead considering as components the elements O, M, and Cl, we can describe the global composition of the system by specifying amounts of M, Cl2

According to the phase-rule (Eq. 27) applied to a system with three components (M, Cl2 and O2), by specifying five degrees of freedom (intensive variables or restriction equations) the

(27)

*LC F F C L*

 

Where *F* denotes the number of phases present (as we do not know the nature of the phase ensemble*, F* = 0 at the beginning), *C* is the number of components, and *L* defines the number of degrees of freedom (equations and or intensive variables) to be specified. So, with *L* = 5,

In reality, the chlorination system is described as an open system, where a gas flux of definite composition is established. The constraint vector defined so far is consistent with the definition of a closed system, which by definition does not allow matter to cross its boundaries. The calculation can become closer to the physical reality of the process if we specify the chemical activities of Cl2 and O2 in the gas phase, instead of fixing their global molar amounts. Such a restriction would be analogous as fixing the inlet gas composition. Further, if the gas is considered to behave ideally, the chemical activities can be replaced by the respective values of the partial pressure of the gaseous components. So, the final

The two types of computation mentioned in the first paragraph can now be discussed. For generating a speciation diagram, only one of the parameters *T*, *P*(Cl2), or *P*(O2) is varied in a definite range. The composition of some phase of interest, for example the gas, can then be plotted as a function of the thermodynamic coordinate chosen. On the other hand, by systematically varying two of the parameter defined in the group *T*, *P*(Cl2), or *P*(O2), a predominance diagram can be constructed (Figure 11). The diagram is usually drawn in space *P*(Cl2) x *P*(O2) and is composed by cells, which describe the stability limits of individual phases. A line describes the equilibrium condition involving two phases, and a

Let's take a closer look in the nature of a predominance diagram applied to the case studied so far. In this situation, one must consider the gas phase, the solid metal M, and possible oxides, MO, MO2, and M2O5, obtained through oxidation of the element M at different oxygen potentials. The equilibrium involving two oxides defines a unique value of the

the constraint vector must have five coordinates (*T*, *P*, *n*(O2), *n*(M), *n*(Cl2)).

constraint vector should be defined as follows: *T*, *P*, *n*(M), *P*(Cl2), *P*(O2).

partial pressure of O2, which is independent of the Cl2 concentration.

point the equilibrium involving three phases.

phases varies with temperature and or concentration of Cl2 or O2.

equilibrium calculation problem has a unique solution:

simultaneously.

and O2 (*T*, *P*, *n*(O2), *n*(M), *n*(Cl2)).

Fig. 9. Concentrations of MCl4 and MCl5, as a function of *P*(O2)

Fig. 10. Concentrations of MOCl3, MCl4 and MCl5 as a function of *P*(O2)

$$\ln P\_{\text{MCCl}\_3} = \ln f\_{\text{MCCl}\_3} + 1.5 \ln P\_{\text{O}\_2} \tag{26}$$

The linear coefficient of the line associated with the MOCl3 formation is higher for the initial value of *P*(O2) than the same factor computed for MCl4 and MCl5. As the angular coefficient is lower for MOCl3, The graphic of Figure (10) depicts a possible result.

According to Figure (10), three distinct situations can be identified. For the initial values of *P*(O2), the partial pressure of MOCl3 is higher than the partial pressure of the other chlorinated compounds.

By varying *P*(O2), a critical value is approached after which *P*(MCl5) assumes the highest value, being followed by *P*(MOCl3) and then *P*(MCl4). A second critical value of *P*(O2) can be identified in the graphic above. For *P*(O2) values higher than this, the atmosphere should be more concentrated in MCl5 and less concentrated in MOCl3, MCl4 assuming a concentration value in between.

### **2.2.3 Minimization of the total gibbs energy**

The most general way of describing equilibrium is to fix a number of thermodynamic variables (physical parameters that can be controlled in laboratory), and to chose an appropriate thermodynamic potential, whose maxima or minima describe the possible equilibrium states available to the system.

By fixing *T*, *P*, and total amounts of the components M, O, and Cl (*n*(O), *n*(M), and *n*(Cl)), the global minimum of the total Gibbs energy describes the equilibrium state of interest,

Fig. 9. Concentrations of MCl4 and MCl5, as a function of *P*(O2)

Fig. 10. Concentrations of MOCl3, MCl4 and MCl5 as a function of *P*(O2)

is lower for MOCl3, The graphic of Figure (10) depicts a possible result.

chlorinated compounds.

**2.2.3 Minimization of the total gibbs energy** 

equilibrium states available to the system.

value in between.

The linear coefficient of the line associated with the MOCl3 formation is higher for the initial value of *P*(O2) than the same factor computed for MCl4 and MCl5. As the angular coefficient

According to Figure (10), three distinct situations can be identified. For the initial values of *P*(O2), the partial pressure of MOCl3 is higher than the partial pressure of the other

By varying *P*(O2), a critical value is approached after which *P*(MCl5) assumes the highest value, being followed by *P*(MOCl3) and then *P*(MCl4). A second critical value of *P*(O2) can be identified in the graphic above. For *P*(O2) values higher than this, the atmosphere should be more concentrated in MCl5 and less concentrated in MOCl3, MCl4 assuming a concentration

The most general way of describing equilibrium is to fix a number of thermodynamic variables (physical parameters that can be controlled in laboratory), and to chose an appropriate thermodynamic potential, whose maxima or minima describe the possible

By fixing *T*, *P*, and total amounts of the components M, O, and Cl (*n*(O), *n*(M), and *n*(Cl)), the global minimum of the total Gibbs energy describes the equilibrium state of interest,

ln ln 1.5ln *P*MOCl MOCl 3 32 *f P*O (26)

which is characterized by a proper phase ensemble, their amounts and compositions. This method is equivalent to solve all chemical equilibrium equations at the same time, so, that the compositions of the chlorinated species in each one of the phases present are calculated simultaneously.

For treating the equilibrium associated with the chlorination processes, two type of diagrams are important: *predominance diagrams*, and *phase speciation diagrams*. The first sort of diagram describes the equilibrium phase ensemble as a function of temperature, and or partial pressure of Cl2 or O2. The second type describes how the composition of individual phases varies with temperature and or concentration of Cl2 or O2.

The first step is to change the initial constraint vector (*T*, *P*, *n*(O), *n*(M), *n*(Cl)), by modifying the definition of the components. Instead considering as components the elements O, M, and Cl, we can describe the global composition of the system by specifying amounts of M, Cl2 and O2 (*T*, *P*, *n*(O2), *n*(M), *n*(Cl2)).

According to the phase-rule (Eq. 27) applied to a system with three components (M, Cl2 and O2), by specifying five degrees of freedom (intensive variables or restriction equations) the equilibrium calculation problem has a unique solution:

$$\begin{aligned} L &= \mathbf{C} + \mathbf{2} - F\\ F &= \mathbf{0} & \mathbf{C} &= \mathbf{3} \\ L &= \mathbf{5} \end{aligned} \tag{27}$$

Where *F* denotes the number of phases present (as we do not know the nature of the phase ensemble*, F* = 0 at the beginning), *C* is the number of components, and *L* defines the number of degrees of freedom (equations and or intensive variables) to be specified. So, with *L* = 5, the constraint vector must have five coordinates (*T*, *P*, *n*(O2), *n*(M), *n*(Cl2)).

In reality, the chlorination system is described as an open system, where a gas flux of definite composition is established. The constraint vector defined so far is consistent with the definition of a closed system, which by definition does not allow matter to cross its boundaries. The calculation can become closer to the physical reality of the process if we specify the chemical activities of Cl2 and O2 in the gas phase, instead of fixing their global molar amounts. Such a restriction would be analogous as fixing the inlet gas composition. Further, if the gas is considered to behave ideally, the chemical activities can be replaced by the respective values of the partial pressure of the gaseous components. So, the final constraint vector should be defined as follows: *T*, *P*, *n*(M), *P*(Cl2), *P*(O2).

The two types of computation mentioned in the first paragraph can now be discussed. For generating a speciation diagram, only one of the parameters *T*, *P*(Cl2), or *P*(O2) is varied in a definite range. The composition of some phase of interest, for example the gas, can then be plotted as a function of the thermodynamic coordinate chosen. On the other hand, by systematically varying two of the parameter defined in the group *T*, *P*(Cl2), or *P*(O2), a predominance diagram can be constructed (Figure 11). The diagram is usually drawn in space *P*(Cl2) x *P*(O2) and is composed by cells, which describe the stability limits of individual phases. A line describes the equilibrium condition involving two phases, and a point the equilibrium involving three phases.

Let's take a closer look in the nature of a predominance diagram applied to the case studied so far. In this situation, one must consider the gas phase, the solid metal M, and possible oxides, MO, MO2, and M2O5, obtained through oxidation of the element M at different oxygen potentials. The equilibrium involving two oxides defines a unique value of the partial pressure of O2, which is independent of the Cl2 concentration.

On the Chlorination Thermodynamics 801

Finally by walking along a vertical line associated with the coexistence of two metallic oxides, for example MO and MO2, a condition is achieved where the gaseous chlorides are formed. The equilibrium between the two oxides and the gas phase is defined by a point. In other words by fixing *T* and *P*, all equilibrium properties are uniquely defined. The equation associated with the coexistence of MO and MO2 (Eq. 28) is added and the partial pressure of

Equations (30) and (31) were presented here only with a didactic purpose. In praxis, the majority of the thermodynamic software (*Thermocalc*, for example) are designed to minimize the total Gibbs energy of the system. The algorithm varies systematically the composition of the equilibrium phase ensemble until the global minimum is achieved. By doing so the same algorithm can be implemented for dealing with all possible equilibrium conditions, eliminating at the end the difficulty of proposing a group of linear independent chemical equations, which for a system with a great number of components can become a

<sup>2</sup> 4 5

0.5 0

 

 

 

g g g

4 52 2

5 2 2 5 2 4 2 2 5 2

*g g*

 

0.5 0

2 2.5 5 0 2 2.5 4 0

gg g s MCl O M O Cl gg g s MCl O M O Cl

54 2

gg g MCl MCl Cl

M MO MCl MCl g g gg MCl MCl O Cl

*x x xx*

*n n nx x*

2 2

s s g MO MO O

*g g*

Fig. 12. Hypothetical predominance diagram: pure gaseous chlorides

MCl4 *<sup>x</sup>* , <sup>g</sup>

MCl5 *x* , <sup>g</sup>

Cl2

(31)

*x* ), indicating

So, we have five equations and five unknowns ( *<sup>g</sup> <sup>n</sup>* , MO2 *<sup>n</sup>* , <sup>g</sup>

O2 is allowed to vary, resulting in six variables and six equations (Eq. 31).

that the equilibrium calculation admits a unique solution.

1

complicated task.

Fig. 11. Hypothetical predominance diagram chlorides mixed in the gas phase

For the equilibrium between MO and MO2, for example, Eq. (28) enables the determination of the *P*(O2) value, which is fixed by choosing *T* and is independent of the Cl2 partial pressure. As a consequence, such equilibrium states are defined by a vertical line.

$$P\_{\rm O\_2} = \exp\left(-\frac{\left(\mathbf{g}\_{\rm MO\_2}^s - \mathbf{g}\_{\rm MO}^s - 0.5\mathbf{g}\_{\rm O\_2}^g\right)}{RT}\right) \tag{28}$$

The equilibrium when the phase ensemble is defined by the gas and one of the metal oxides, say MO2, is also defined by a line, whose inclination is determined by fixing *T*, *P*, *n*(M) and *P*(O2). This time the concentration of Cl2, MCl4 and MCl5 are computed by solving the group of non-linear equations presented bellow (Eq. 29). The first equation defines the restriction that the molar quantity of M is constant (mass conservative restriction). The second equation represents the conservation of the total mass of the gas phase (the summation of all mol fractions must be equal to one).

$$\begin{aligned} m\_{\rm M} &= n\_{\rm MO\_2} + n^{\rm g} \left( \mathbf{x}\_{\rm MC1\_4}^{\rm g} + \mathbf{x}\_{\rm MC1\_5}^{\rm g} \right) \\ 1 &= \mathbf{x}\_{\rm MC1\_4}^{\rm g} + \mathbf{x}\_{\rm MC1\_5}^{\rm g} + \mathbf{x}\_{\rm O\_2}^{\rm g} + \mathbf{x}\_{\rm C1\_2}^{\rm g} \\ \mu\_{\rm MC1\_5}^{\rm g} - \mu\_{\rm MC1\_4}^{\rm g} - \frac{\mu\_{\rm C1\_2}^{\rm g}}{2} &= 0 \\ 2\,\mu\_{\rm MC15}^{\rm g} + 2.5\,\mu\_{\rm O\_2}^{\rm g} - g\_{\rm M\_2O\_5}^{\rm s} - 5\,\mu\_{\rm C1\_2}^{\rm g} &= 0 \\ 2\,\mu\_{\rm MC1\_4}^{\rm g} + 2.5\,\mu\_{\rm O\_2}^{\rm g} - g\_{\rm M\_2O\_5}^{\rm s} - 4\,\mu\_{\rm C1\_2}^{\rm g} &= 0 \end{aligned} \tag{29}$$

The other three relations define, respectively, the equilibrium conditions for the following group of reactions:

$$\begin{aligned} \text{MCl}\_{4}\text{(g)} + 0.5\text{Cl}\_{2}\text{(g)} &\rightarrow \text{MCl}\_{5}\text{(g)}\\ \text{MO}\_{2}\text{(s)} + 2.5\text{Cl}\_{2}\text{(g)} &\rightarrow \text{MCl}\_{5}\text{(g)} + \text{O}\_{2}\text{(g)}\\ \text{MO}\_{2}\text{(s)} + 2\text{Cl}\_{2}\text{(g)} &\rightarrow \text{MCl}\_{4}\text{(g)} + \text{O}\_{2}\text{(g)} \end{aligned} \tag{30}$$

Fig. 11. Hypothetical predominance diagram chlorides mixed in the gas phase

pressure. As a consequence, such equilibrium states are defined by a vertical line.

2

exp

5 4

 

g g Cl MCl MCl

*n n nx x <sup>M</sup>*

*x x xx*

O

*P*

1

fractions must be equal to one).

group of reactions:

For the equilibrium between MO and MO2, for example, Eq. (28) enables the determination of the *P*(O2) value, which is fixed by choosing *T* and is independent of the Cl2 partial

<sup>2</sup> <sup>2</sup>

*RT*

*gg g*

The equilibrium when the phase ensemble is defined by the gas and one of the metal oxides, say MO2, is also defined by a line, whose inclination is determined by fixing *T*, *P*, *n*(M) and *P*(O2). This time the concentration of Cl2, MCl4 and MCl5 are computed by solving the group of non-linear equations presented bellow (Eq. 29). The first equation defines the restriction that the molar quantity of M is constant (mass conservative restriction). The second equation represents the conservation of the total mass of the gas phase (the summation of all mol

<sup>2</sup> 4 5

g g g MO MCl MCl g g gg MCl MCl O Cl g

4 52 2

2 2 2.5 5 0 2 2.5 4 0

gg g s MCl5 O M O Cl gg g s MCl O M O Cl

2

*g g*

 

4 2 2 5 2

The other three relations define, respectively, the equilibrium conditions for the following

2 2 52 2 2 42

 

MO s 2.5Cl g MCl g O g MO s 2Cl g MCl g O g

42 5

MCl 0.5Cl g MCl g

*g*

 

2 2 2 5

 

 

0

0.5

(28)

(29)

(30)

s s g MO MO O So, we have five equations and five unknowns ( *<sup>g</sup> <sup>n</sup>* , MO2 *<sup>n</sup>* , <sup>g</sup> MCl4 *<sup>x</sup>* , <sup>g</sup> MCl5 *x* , <sup>g</sup> Cl2 *x* ), indicating that the equilibrium calculation admits a unique solution.

Finally by walking along a vertical line associated with the coexistence of two metallic oxides, for example MO and MO2, a condition is achieved where the gaseous chlorides are formed. The equilibrium between the two oxides and the gas phase is defined by a point. In other words by fixing *T* and *P*, all equilibrium properties are uniquely defined. The equation associated with the coexistence of MO and MO2 (Eq. 28) is added and the partial pressure of O2 is allowed to vary, resulting in six variables and six equations (Eq. 31).

Equations (30) and (31) were presented here only with a didactic purpose. In praxis, the majority of the thermodynamic software (*Thermocalc*, for example) are designed to minimize the total Gibbs energy of the system. The algorithm varies systematically the composition of the equilibrium phase ensemble until the global minimum is achieved. By doing so the same algorithm can be implemented for dealing with all possible equilibrium conditions, eliminating at the end the difficulty of proposing a group of linear independent chemical equations, which for a system with a great number of components can become a complicated task.

$$\begin{aligned} \eta\_{\rm M} &= \mu\_{\rm MO\_{2}} + \mu^{\rm g} \left( \mathbf{x}\_{\rm MC1\_{4}}^{\rm g} + \mathbf{x}\_{\rm MC1\_{5}}^{\rm g} \right) \\ \mathbf{1} &= \mathbf{x}\_{\rm MC1\_{4}}^{\rm g} + \mathbf{x}\_{\rm MC1\_{5}}^{\rm g} + \mathbf{x}\_{\rm O\_{2}}^{\rm g} + \mathbf{x}\_{\rm C1\_{2}}^{\rm g} \\ \mu\_{\rm MC1\_{5}}^{\rm g} &= \mu\_{\rm MC1\_{4}}^{\rm g} - 0.5 \mu\_{\rm C1\_{2}}^{\rm g} = 0 \\ 2 \mu\_{\rm MC1\_{5}}^{\rm g} &+ 2.5 \mu\_{\rm O\_{2}}^{\rm g} - g\_{\rm M\_{2}O\_{5}}^{\rm g} - 5 \mu\_{\rm C1\_{2}}^{\rm g} = 0 \\ 2 \mu\_{\rm MC1\_{4}}^{\rm g} &+ 2.5 \mu\_{\rm O\_{2}}^{\rm g} - g\_{\rm M\_{2}O\_{5}}^{\rm g} - 4 \mu\_{\rm C1\_{2}}^{\rm g} = 0 \\ g\_{\rm MO\_{2}}^{\rm s} - g\_{\rm MO}^{\rm s} - 0.5 \mu\_{\rm O\_{2}}^{\rm g} &= 0 \end{aligned} \tag{31}$$

Fig. 12. Hypothetical predominance diagram: pure gaseous chlorides

On the Chlorination Thermodynamics 803

phases will not be included in the data-base used for the following computations. Additionally, it was considered that the concentration of the oxides in gas phase is low enough to be neglected. Further, on what touches the computations that follows, the software *Thermocalc* was used in all cases, and it will always be assumed that equilibrium is

The relative stability of the possible vanadium oxides can be assessed through construction of a predominance diagram in the space *T* – *P*(O2) (see Figure 13). As thermodynamic constraints we have *n*(V) (number of moles of vanadium metal – it will be supposed that *n*(V) =1), *T*, *P* and *P*(O2). The reaction temperature will be varied in the range between 1073 K and 1500 K and the partial pressure of O2 in the range between 8.2.10-40atm and 1atm.

The total pressure was fixed at 1atm. It can be seen that for the temperature range considered and a partial pressure of O2 in the neighborhood of 1atm, V2O5 is formed in the liquid state. Through lowering the oxygen potential, crystalline vanadium oxides precipitate, VO2 being formed first, followed by V2O3, VO, and finally V. The horizontal line between fields "5" and "6" indicates the melting of V1O2, which according to classical thermodynamics must occur at a fixed temperature. Next it will be considered the species

The already identified species formed between vanadium, chlorine and oxygen are: VCl,

On Table (1) it was included information regarding the physical states at ambient conditions and some references related to phase equilibrium studies conducted on samples of specific

Only a few studies were published in literature in relation to the thermodynamics of vanadium chlorinated phases. On Table (1) some references are given for earlier

achieved, or in other words, kinetic effects can be neglected.

Fig. 13. Predominance diagram for the system V – O

formed by vanadium, chlorine and oxygen.

VCl2, VCl3, VCl4, VOCl, VOCl2, VOCl3, VO2Cl.

**3.1 Vanadium oxides and chlorides** 

vanadium chlorinated compounds.

A simplified version of the predominance diagram of Figure (11) can be achieved through considering each possible gaseous chloride as a pure substance. In this case, the field representing the gas phase will be divided into sub-regions, each one representative of the stability of each gaseous chlorinated compound. By considering, that, besides MCl5 and MCl4, gaseous MOCl3 can also be formed, a diagram similar to the one presented on Figure (12) would represent possible stability limits found in equilibrium.

The diagram of Figure (12) is associated with a temperature value where gaseous MCl5 can not be present in equilibrium for any suitable value of *P*(Cl2) and *P*(O2) chosen. It is interesting to note, that in this sort of diagram, there is a direct relation between the inclination of a line representative of the equilibrium between a gaseous chloride or oxychloride and an oxide, with the stoichiometric coefficients of the chemical reaction behind the transformation.

According to Eq. (32), the inclination of the line associated with the equilibrium between MOCl3 and MO2 should be lower than the one associated with the equilibrium between MOCl3 and M2O5. On the other hand, in the case of the equilibrium between MO and MOCl3, the line is horizontal (does not depend on *P*(O2)), as the same number of oxygen atoms is present in the reactant and products, so O2 does not participate in the reaction.

$$\begin{aligned} \ln P\_{\text{Cl}\_2} &= \frac{1}{3} \ln P\_{\text{O}\_2} - \frac{2}{3} \ln K\_{\text{MO}\_2} \text{ ( $T$ )}\\ \ln P\_{\text{Cl}\_2} &= \frac{1}{2} \ln P\_{\text{O}\_2} - \frac{1}{3} \ln K\_{\text{M}\_2\text{O}\_5} \text{ ( $T$ )}\\ \ln P\_{\text{Cl}\_2} &= -\frac{2}{3} \ln K\_{\text{MO}} \text{ ( $T$ )} \end{aligned} \tag{32}$$

Where, MO2 *<sup>K</sup>* , OM <sup>52</sup> *<sup>K</sup>* and *K*MO represent respectively the equilibrium constants for the formation of MOCl3 from MO2, M2O5 and MO (Eq. 33).

$$\begin{aligned} \text{MO}\_2 + 1.5\text{Cl}\_2 &\rightarrow \text{MOCl}\_3 + 0.5\text{O}\_2\\ \text{M}\_2\text{O}\_5 + 3\text{Cl}\_2 &\rightarrow 2\text{MOCl}\_3 + 1.5\text{O}\_2\\ \text{MO} + 1.5\text{Cl}\_2 &\rightarrow \text{MOCl}\_3 \end{aligned} \tag{33}$$

The diagrams of Figures (11) and (12) depict a behavior, where no condensed chlorinated phases are present. For many oxides, however, there is a tendency of formation of solid or liquid chlorides and or oxychlorides, which must appear in the predominance diagram as fields between the pure oxides and the gas phase regions. Such a behavior can be observed in the equilibrium states accessible to the system V – O – Cl.
