**2.1.1 Thermodynamic basis for the construction of** <sup>o</sup> *G*<sup>r</sup> **x T diagrams**

To construct the <sup>o</sup> *G*r x *T* diagram of a particular reaction we must be able to compute its standard Gibbs energy in the whole temperature range spanned by the diagram.

$$\begin{aligned} \Delta G^{\circ}\_{\rm r} &= \Delta H^{\circ}\_{\rm r} - T \Delta S^{\circ}\_{\rm r} \\ \Delta H^{\circ}\_{\rm r} &= \Delta H\_{298} + \int\_{298.15}^{T} \Delta C^{\circ}\_{\rm P} dT \\ \Delta S^{\circ}\_{\rm r} &= \Delta S\_{298} + \int\_{298.15}^{T} \frac{\Delta C^{\circ}\_{\rm P}}{T} dT \\ \Delta C^{\circ}\_{\rm P} &= \frac{d\Delta H^{\circ}\_{\rm P}}{dT} = 2C^{\circ}\_{\rm P, \rm MCl\_{5}} + \frac{5}{2}C^{\circ}\_{\rm P, \rm O\_{2}} - 5C^{\circ}\_{\rm P, Cl\_{2}} - C^{\circ}\_{\rm P, M\_{2}O\_{5}} \\ \Delta H\_{298} &= 2H^{\circ}\_{298, \rm MCl\_{5}} + \frac{5}{2}H^{\circ}\_{298, \rm O\_{2}} - 5H^{\circ}\_{298, \rm O\_{2}} - H^{\circ}\_{298, \rm M\_{2}O\_{5}} \\ \Delta S\_{298} &= 2S^{\circ}\_{298, \rm MCl\_{5}} + \frac{5}{2}S^{\circ}\_{298, \rm O\_{2}} - 5S^{\circ}\_{298, \rm O\_{2}} - S^{\circ}\_{298, \rm M\_{2}O\_{5}} \end{aligned} \tag{17}$$

For accomplishing this task one needs a mathematical model for the molar standard heat capacity at constant pressure, valid for each participating substance for *T* varying between 298.15 K and the final desired temperature, its molar enthalpy of formation ( <sup>o</sup> *<sup>H</sup>*<sup>298</sup> ) and its molar entropy of formation ( <sup>o</sup> <sup>298</sup> *S* )at 298.15 K

For the most gas – solid reactions both the molar standard enthalpy ( <sup>o</sup> *H*<sup>r</sup> ) and entropy of

reaction ( <sup>o</sup> <sup>r</sup> *S* ) do not depend strongly on temperature, as far no phase transformation among the reactants and or products are present in the considered temperature range. So, the observed behavior is usually described by a line (Fig. 1), whose angular coefficient gives us a measurement of <sup>o</sup> <sup>r</sup> *S* and <sup>o</sup> *H*<sup>r</sup> is defined by the linear coefficient.

Fig. 1. Hypothetical <sup>o</sup> *G*r x T diagram

To construct the <sup>o</sup> *G*r x *T* diagram of a particular reaction we must be able to compute its

oo o rr r o o r 298 P 298.15 K

*G H TS*

*H H C dT*

*T*

<sup>o</sup> <sup>o</sup> <sup>P</sup>

*<sup>C</sup> S S dT*

<sup>o</sup> o s <sup>r</sup> o,g g g P P,MCl P,O P,Cl P,M O o s g g 298 298, MCl 298,O 298,Cl 298,M O

*d H C C C CC*

*HH H H H*

<sup>5</sup> 2 5 2

298.15 K

*T*

5 2

For accomplishing this task one needs a mathematical model for the molar standard heat capacity at constant pressure, valid for each participating substance for *T* varying between 298.15 K and the final desired temperature, its molar enthalpy of formation ( <sup>o</sup> *<sup>H</sup>*<sup>298</sup> ) and its

For the most gas – solid reactions both the molar standard enthalpy ( <sup>o</sup> *H*<sup>r</sup> ) and entropy of

among the reactants and or products are present in the considered temperature range. So, the observed behavior is usually described by a line (Fig. 1), whose angular coefficient gives

<sup>r</sup> *S* and <sup>o</sup> *H*<sup>r</sup> is defined by the linear coefficient.

<sup>r</sup> *S* ) do not depend strongly on temperature, as far no phase transformation

2

5 22 2 5

g g <sup>s</sup> <sup>5</sup>*S S* 298,Cl 298,M O

2 2 5

(17)

5 2 2 2 5

*T*

**2.1.1 Thermodynamic basis for the construction of** <sup>o</sup> *G*<sup>r</sup> **x T diagrams** 

standard Gibbs energy in the whole temperature range spanned by the diagram.

r 298

<sup>5</sup> 2 5 2

o 298 298, MCl 298,O

molar entropy of formation ( <sup>o</sup>

us a measurement of <sup>o</sup>

Fig. 1. Hypothetical <sup>o</sup> *G*r x T diagram

reaction ( <sup>o</sup>

*SS S*

*dT*

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

<sup>298</sup> *S* )at 298.15 K

Fig. 2. Endothermic and exothermic reactions

Further, for a reaction defined by Eq. (1) the number of moles of gaseous products is higher than the number of moles of gaseous reactants, which, based on the ideal gas model, is indicative that the chlorination leads to a state of grater disorder, or greater entropy. In this particular case then, the straight line must have negative linear coefficient (- <sup>o</sup> <sup>r</sup> *S* < 0), as depicted in the graph of Figure (1).

The same can not be said about the molar reaction enthalpy. In principle the chlorination reaction can lead to an evolution of heat (exothermic process, then <sup>o</sup> *H*<sup>r</sup> < 0) or absorption of heat (endothermic process, then <sup>o</sup> *H*<sup>r</sup> > 0). In the first case the linear coefficient is positive, but in the later it is negative. Hypothetical cases are presented in Fig. (2) for the chlorination of two oxides, which react according to equations identical to Eq. (1). The same molar reaction entropy is observed, but for one oxide the molar enthalpy is positive, and for the other it is negative.

Finally, it is worthwhile to mention that for some reactions the angular coefficient of the straight line can change at a particular temperature value. This can happen due to a phase transformation associated with either a reactant or a product. In the case of the reaction (1), only the oxide M2O5 can experience some phase transformation (melting, sublimation, or ebullition), all of them associated with an increase in the molar enthalpy of the phase. According to classical thermodynamics, the molar entropy of the compound must also increase (Robert, 1993).

$$
\Delta S\_{\text{t}} = \frac{\Delta H\_{\text{t}}}{T\_{\text{t}}} \tag{18}
$$

Where <sup>t</sup> *S* , *H* <sup>t</sup> and *T*t represent respectively, the molar entropy, molar enthalpy and temperature of the phase transformation in question. So, to include the effect for melting of M2O5 at a temperature *T*t, the molar reaction enthalpy and entropy must be modified as follows.

On the Chlorination Thermodynamics 793

In many situations the reaction of a metallic oxide with Cl2 leads to the formation of a family of chlorinated species. In these cases, multiple reactions take place. In the present section three methods will be described for treating this sort of situation, the first of them is of qualitative nature, the second semi-qualitative, and the third a rigorous one, that reproduces

The first method consists in calculating <sup>o</sup> *G*r x *T* diagrams for each reaction in the temperature range of interest. The reaction with the lower molar Gibbs energy must have a greater thermodynamic driving force. The second method involves the solution of the equilibrium equations independently for each reaction, and plotting on the same space the concentration of the desired chlorinated species. Finally, the third method involves the calculation of the thermodynamic equilibrium by minimizing the total Gibbs energy of the system. The

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

25 2 5 2

<sup>5</sup> M O s 5Cl <sup>g</sup> 2MCl <sup>g</sup> <sup>O</sup> <sup>g</sup> <sup>2</sup>

25 2 4 2

<sup>5</sup> M O s 4Cl <sup>g</sup> 2MCl <sup>g</sup> <sup>O</sup> <sup>g</sup> <sup>2</sup>

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

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.

MCl4 becomes thermodynamically more favorable (see Figure 4).

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

(21)

concentrations of all species in the phase ensemble are then simultaneously computed.

**2.2 Multiple reactions** 

and MCl5:

the equilibrium conditions quantitatively.

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

$$\begin{aligned} \Delta H\_{\mathbf{r}}^{\rm o} &= \int\_{298.15}^{T\_{\rm t}} \Delta C\_{\rm P}^{\rm o} dT - \Delta H\_{\rm t, M\_2O\_5} + \int\_{T\_{\rm t}}^{T} \Delta C\_{\rm P}^{\rm o} dT\\ \Delta S\_{\rm r}^{\rm o} &= \int\_{298.15}^{T\_{\rm t}} \frac{\Delta C\_{\rm P}^{\rm o}}{T} dT - \frac{\Delta H\_{\rm t, M\_2O\_5}}{T\_t} + \int\_{T\_{\rm t}}^{T} \frac{\Delta C\_{\rm P}^{\rm o}}{T} dT \end{aligned} \tag{19}$$

It should be observed that the molar entropy and enthalpy associated with the phase transition experienced by the oxide M2O5 were multiplied by its stoichiometric number "-1", which explains the minus sign present in both relations of Eq. (19).

An analogous procedure can be applied if other phase transition phenomena take place. One must only be aware that the mathematical description for the molar reaction heat capacity at constant pressure ( <sup>o</sup> <sup>P</sup> *<sup>C</sup>* ) must be modified by substituting the heat capacity of solid M2O5 for a model associated with the most stable phase in each particular temperature range. If, for example, in the temperature range of interest M2O5 melts at *T*t, for *T* > *T*t, the molar heat capacity of solid M2O5 must be substituted for the model associated with the liquid state (Eq. 20).

$$\begin{aligned} \Delta \mathbf{C}\_{\rm P}^{\rm o} &= 2 \mathbf{C}\_{\rm P,M\rm Cl\_5}^{\rm o,g} + \frac{5}{2} \mathbf{C}\_{\rm P,O\_2}^{\rm g} - 5 \mathbf{C}\_{\rm P,Cl\_2}^{\rm g} - \mathbf{C}\_{\rm P,M\_2O\_5}^{\rm s} \left( T < T\_{\rm t} \right) \\ \Delta \mathbf{C}\_{\rm P}^{\rm o} &= 2 \mathbf{C}\_{\rm P,M\rm Cl\_5}^{\rm o,g} + \frac{5}{2} \mathbf{C}\_{\rm P,O\_2}^{\rm g} - 5 \mathbf{C}\_{\rm P,Cl\_2}^{\rm g} - \mathbf{C}\_{\rm P,M\_2O\_5}^{\rm l} \left( T > T\_{\rm t} \right) \end{aligned} \tag{20}$$

The effect of a phase transition over the geometric nature of the <sup>o</sup> *G*r x *T* curve can be directly seen. The melting of M2O5 makes it's molar enthalpy and entropy higher. According to Eq. (19), such effects would make the molar reaction enthalpy and entropy lower. So the curve should experience a decrease in its first order derivative at the melting temperature (Figure 3).

Fig. 3. Effect of M2O5 melting over the <sup>o</sup> *G*r x T diagram

Based on the definition of the reaction Gibbs energy (Eq. 17), similar transitions involving a product would produce an opposite effect. The reaction Gibbs energy would in these cases dislocate to more negative values. In all cases, though, the magnitude of the deviation is proportional to the magnitude of the molar enthalpy associated with the particular transition observed. The effect increases in the following order: melting, ebullition and sublimation.

### **2.2 Multiple reactions**

792 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

o o o r Pt,M O P

*H C dT H C dT*

*T T*

o o <sup>o</sup> P P t, M O

It should be observed that the molar entropy and enthalpy associated with the phase transition experienced by the oxide M2O5 were multiplied by its stoichiometric number "-1",

An analogous procedure can be applied if other phase transition phenomena take place. One must only be aware that the mathematical description for the molar reaction heat

solid M2O5 for a model associated with the most stable phase in each particular temperature range. If, for example, in the temperature range of interest M2O5 melts at *T*t, for *T* > *T*t, the molar heat capacity of solid M2O5 must be substituted for the model associated with the

5 22 2 5

P P P,MCl P,O P,Cl ,M O t

<sup>5</sup> 2 5

*C C C C C TT*

<sup>5</sup> 2 5

*C C C C C TT*

P P P,MCl P,O P,Cl ,M O t

The effect of a phase transition over the geometric nature of the <sup>o</sup> *G*r x *T* curve can be directly seen. The melting of M2O5 makes it's molar enthalpy and entropy higher. According to Eq. (19), such effects would make the molar reaction enthalpy and entropy lower. So the curve should experience a decrease in its first order derivative at the melting temperature (Figure 3).

Based on the definition of the reaction Gibbs energy (Eq. 17), similar transitions involving a product would produce an opposite effect. The reaction Gibbs energy would in these cases dislocate to more negative values. In all cases, though, the magnitude of the deviation is proportional to the magnitude of the molar enthalpy associated with the particular transition observed. The effect increases in the following order: melting, ebullition and

o s o,g g g

2

2

Fig. 3. Effect of M2O5 melting over the <sup>o</sup> *G*r x T diagram

o l o,g g g

5 22 2 5

*T T*

298.15 T

298.15

*t*

which explains the minus sign present in both relations of Eq. (19).

*t*

r

capacity at constant pressure ( <sup>o</sup>

liquid state (Eq. 20).

sublimation.

2 5

*t*

(19)

(20)

*T*

*t*

<sup>P</sup> *<sup>C</sup>* ) must be modified by substituting the heat capacity of

2 5

*C C H S dT dT T T*

*t T*

In many situations the reaction of a metallic oxide with Cl2 leads to the formation of a family of chlorinated species. In these cases, multiple reactions take place. In the present section three methods will be described for treating this sort of situation, the first of them is of qualitative nature, the second semi-qualitative, and the third a rigorous one, that reproduces the equilibrium conditions quantitatively.

The first method consists in calculating <sup>o</sup> *G*r x *T* diagrams for each reaction in the temperature range of interest. The reaction with the lower molar Gibbs energy must have a greater thermodynamic driving force. The second method involves the solution of the equilibrium equations independently for each reaction, and plotting on the same space the concentration of the desired chlorinated species. Finally, the third method involves the calculation of the thermodynamic equilibrium by minimizing the total Gibbs energy of the system. The concentrations of all species in the phase ensemble are then simultaneously computed.
