The current work aims at determining the standard free energy change (ΔG°) as a function of temperature (T), using the chemical equilibration technique. A novel approach was made to find the equilibrium state with the assistance of a Thermogravimetric Analyzer. The ΔG° for lead carbonate dissociation reaction PbCO₃(s) = PbO(s) + CO₂ (g) was calculated as an example. The weight of PbCO₃started decreasing at specific temperature termed as equilibrium temperature. At this temperature ΔG = 0 and the equilibrium relation between ΔG°and the equilibrium constant K applied. Below this temperature, when there was no change in weight, ΔG> 0, and above this temperature, when the weight kept changing for continued dissociation of PbCO₃ (s), ΔG< 0. The obtained ΔG° for PbCO₃(s) matched closely with the information found within in the literature.

Keywords: standard free energy, chemical equilibration technique, thermogravimetric analysis, thermodynamic data

A thermogravimetric analyzer (TGA) measures the change in the mass of a reacting sample as function of time at any specified temperature or as a function of both temperature and time, which are conventionally used in the kinetic analysis of the reaction under study. Commonly, the change in the mass is converted into the fractional (or, percentage) conversion of the sample. A set of data of fractional conversion vs time helps to graphically detect the correct rate controlling mechanism by referring to the governing equations of differentrate controlling mechanisms such as gas film diffusion, product layer diffusion, and the interfacial reaction. However, it is not much heard of that a thermogravimetric analyzer has ever been used for any thermodynamic measurements. In the last decades, researchers used different approach like thermo barometry, EMF, and calorimetry methods for the determination of thermodynamic data.^1-10 Berman¹ used thermo barometry techniques to measure the thermodynamic data of minerals.¹ Hugh et al⁸ studied the Gibbs free energies of formation of RuO₂, OsO₂ and IrO₂ using an electrochemical method with calcia-stabilized zirconia (CSZ) solid electrolytes.⁸

The current work aims at adapting the apparatus to the determination of the standard (Gibbs) free-energy change of a chemical reaction as a function of temperature, in the form ΔG° = A + BT, where A = ΔH° and B = -ΔS°, by using the chemical equilibration technique. An attempt was made to verify this new method by comparing the results obtained by it with the known, published data of the standard free-energy changes for the dissociation of two solid carbonates CaCO₃ and PbCO₃.Particularly, the reaction PbCO₃ (s) = PbO (s) + CO₂ (g) was studied in details. It is proposed that if the new method is found to be successful, it can be extended to measuring the standard free-energy change of many other reactions for which the data have not yet been available.

The principle of determining the standard free energy change (ΔG°) by the chemical equilibration technique in the thermogravimetric analyzer (TGA) is elucidated with the help of an example of the dissociation of lead carbonate expressed as

PbCO₃ (s) = PbO (s) +CO₂ (g); ΔG° = A + BT (1)

Here, both A and B, which are standard enthalpy change (ΔH°) and negative standard entropy change (-ΔS°), respectively, are treated as unknown, and it is required to evaluate them experimentally so that the measured value of ΔG° can be expressed in the above common form of two-term equation.

The two most important variables in the current method are the temperature (T) and the partial pressure of carbon dioxide (p_CO2). These are mutually dependent because of the fact that the thermodynamic equilibrium constant of Reaction (1), K₍₁₎, expressed by the ratio (α_PbO. aCO₂) / (α_PbCO3), is solely governed by p_CO2, the activities (ai) of the two pure condensed phases (PbCO₃ and PbO) being 1 and α_CO2being equal to p_CO2. In the thermogravimetric apparatus, a chosen pCO₂ (1 or less) is maintained by letting pure CO₂ (for pCO₂ = 1atm) or a mixture of CO₂ and Ar(for p_CO2< 1atm), Ar (argon) being the diluent, flow over the solid sample (PbCO₃) continuously to ensure equilibration between the gas and solid while the furnace is heating and the temperature is slowly rising. Now, there is a unique temperature at which Reaction (1) comes to equilibrium with respect to the applied p_CO2, satisfying the thermodynamic condition of ΔG = 0 for equilibrium. Below this temperature, ∆G is greater than 0 (criterion for the impossibility of reaction), and above this temperature, ΔG is less than 0 (criterion for the spontaneity of reaction). The manifestation of the condition ΔG< 0 in the TGA is the occurrence of Reaction (1) leading to the release of CO₂ from the sample and the accompanying decrease in the mass of the sample. Thus, starting from the room temperature up to this unique temperature there is no change in the mass of the sample and the mass (weight) vs temperature curve, a TGA output, will remain flat for this particular temperature span. Just above the unique temperature, the mass of the sample will start decreasing and there will be a consequent descent of the curve, which will continue descending with the increasing temperature till the sample lasts. Thus, at this unique temperature (T1, say), Reaction (1) will attain equilibrium (ΔG = 0) under the applied p_CO2 (p_CO2 (1), say). Alternatively, one can write, ΔG°_T1 = -RT1ln KT₁ = -R_T1ln p_CO2 (1), where the value of R, the universal gas constant, is 8.314 J/mol. K. Knowing both p_CO2 (1) and T₁ experimentally, the value of ΔG° at the temperature T₁, ΔG°T₁, is obtained. Similarly, by applying several other chosen partial pressures of CO₂ (with the help of CO₂-Ar mixtures produced at varying proportions) and by repeating the experiment, ΔG° at several other temperatures (T) can be found. These values of ΔG° when plotted against T will give ΔG° in the form of A + BT, where A and B are now known quantities.

Commercially available powder PbCO₃(s) and CaCO₃ (s) were used for this work. To measure the weight change in the powder material with temperature and pressure thermogravimetric analyze (Bysakh, Okay) was used. The powdered sample of PbCO₃(s) and CaCO₃(s) was placed into the crucible. A heating rate of 10K/min was used for the experiment. A gas mixture of Ar and CO₂ was used for experiments. Gas mixture is passed through silica gel moisture trap to get rid of the moisture. For PbCO₃, three different values of pCO₂ were considered: 0.4 atm, 0.5 atm, and 0.75 atm. These partial pressures were established by mixing CO₂ and Ar (pCO₂+p_Ar=1 atm) in the ratio 2:3 (for p_CO2 = 0.4 atm), 1:1 (for p_CO2 = 0.5 atm), and 3:1 (for p_CO2 = 0.75 atm). However, for the experiment with CaCO₃, pure CO₂ (p_CO2 = 1 atm) alone was considered. (Figure 1) shows the schematic of TGA experimental set-up.

Figure 1 Schematic diagram of the experimental set-up: 1. CO₂ cylinder; 2. Ar cylinder; 3. Flow meter; 4. Moisture trap; 5. Mixer and gas channeling arrangement; 6. TGA.

CaCO₃

The weight vs temperature plot obtained from the TGA run on the dissociation of CaCO₃, the dissociation reaction

CaCO₃ (s) = CaO (s) + CO₂ (g) (2)

Under 1 atm of CO₂, is shown in (Figure 2). Barring some small fluctuations in the curve because of gas flow disturbance, the temperature at which there is a monotonic descent of the curve is clearly noticeable and is found to be 853 °C (1126 K). Now, according to the published data,¹¹ the standard free-energy change for Reaction (2) is

ΔG°=178500 – 159.1 T J (3)

This gives T = 178500/159.1 = 1122 K = 849 °C for ΔG =ΔG° = 0, noting that this is a case of standard state equilibrium because not only the two solids (CaCO₃ and CaO) but CO₂ also is at the standard state (1 atm). This means that according to the published data, the onset of the descent of the curve should have been at 849 °C. The agreement between the TGA result and the published one is thus very close.

Figure 2 Weight versus temperature plot for the dissociation of CaCO₃under 1 atm CO₂.

PbCO₃

The weight vs temperature plots obtained from the TGA run on the dissociation of PbCO₃ (Reaction (1)) for the three different values of p_CO2, 0.4 atm, 0.5 atm, and 0.75 atm, are shown in (Figures 3a, b, c), respectively. The corresponding temperatures for the onset of the descent of the curve, in a monotonic and noticeable manner, are 280±4°C (553±4 K), 287±3°C (560±3 K), and 300±6°C (573±6 K), respectively. This allows one to find the values of ΔG° at these temperatures by using the relation ΔG° = -RT ln K = -RT ln p_{CO2 (applied)}. These values are listed in (Table 1).

Figure 3 Weight versus temperature plot for the dissociation of PbCO₃ under (a) 0.4 atmpCO₂ (b) 0.5 atmpCO₂ and (c) 0.75 atmpCO₂.

In (Figure 4), a plot of ΔG° vs T is made and a good linear fit is found, which yields the following expression for the standard free energy change of Reaction (1):

ΔG°=82955-142.37 T J (4)

In comparison, the published data¹¹ for Reaction (1) is

ΔG°= 86700-149 T J (5)

The two results differ by only about 4%, which may be considered quite satisfactory, and the standard free-energy data thus measured by the thermogravimetric analyzer may be considered reliable. This method may be extended to measure the standard free energy changes for the dissociation of CrCO₃(s), NiCO₃(s), TiCO₃(s), and WCO₃(s) and for similar other gas-solid reactions (dissociation of oxides, nitrides, etc) for which data are not yet available in the literature.

Figure 4 Plot of standard free energy change (ΔG°) vs temperature (T) for the reactionPbCO₃(s) = PbO (s) + CO₂ (g).

The salient features of the current work are as follows:

1. The thermogravimetric analyzer (TGA) has been traditionally used for kinetic studies. The current work aims at adapting the apparatus to the determination of the standard (Gibbs) free-energy change of a chemical reaction as a function of temperature, by using the chemical equilibration technique.
2. An attempt was made to verify this new method by comparing the results obtained by it with the known, published data of the standard free-energy changes for the dissociation of two solid carbonates CaCO₃ and PbCO₃.
3. The standard free energy change for the reaction PbCO₃ (s) = PbO (s) + CO₂ (g) was measured by the current method and the result obtained compared well with the published data:

ΔG° = 82955-142.37 T J (current work)
ΔG°= 86700 - 149 T J (published in data book)¹¹.
The two results differ by only about 4%. Hence, the standard free-energy data thus measured by the thermogravimetric analyzer may be considered reliable.

4. This method may be extended to measure the standard free energy changes for the dissociation of CrCO₃(s), NiCO₃(s), TiCO₃(s), and WCO₃(s) and for similar other gas-solid reactions (dissociation of oxides, nitrides, etc) for which data are not yet available in the literature.

None.

The author declares no conflict of interest.

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