Alternate Thermodynamic Characterization Technique

Currently, the only method used to determine a material’s reduction reaction enthalpy and entropy is thermogravimetric analysis. This process produces sparse plots of the nonstoichiometry at a given temperature and oxygen partial pressure. The experiments can be lengthy due to the long times required by the material to reach equilibrium. In principle, if cycling behavior occurs under the thermokinetic limit, then the nonstoichiometric profile and the enthalpy and entropy of reduction as a function of nonstoichiometry can be determined from the gas evolution rate data. Here, an appropriate experimental procedure is outlined.

7.2.1.1 Theory

From thermochemical cycling experiments, oxygen evolution and hydrogen production rates are measured as a function of time. By integrating the volumetric gas evolution profiles, the cumulative amount of gas produced as a function of time is obtained. If the material began the cycle from a known reference state, then the total amount of oxygen evolved (or hydrogen produced) during each time step can be used to calculate the nonstoichiometry of the material as a function of time via the equation below.

where is the equilibrium nonstoichiometry under the oxidizing conditions and refers to the reduction half cycle’s start time. An analogous equation can be used to determine the nonstoichiometry as a function of time for the oxidation half cycle.

( ) = ( ) − ( ) 7.2.1-2

where is the equilibrium nonstoichiometry under the reducing conditions and refers to the oxidation half cycle’s start time. Next, the oxide’s equivalent oxygen partial pressure as a function of time must be determined from the gas production data. The oxide’s nonstoichiometric rate of change during the reduction half cycle can be directly determined from the volumetric oxygen production rate data.

( ) = ( ) 7.2.1-3

For the portion of the oxygen evolution profile described by the thermokinetic limit, knowing the nonstoichiometric rate of change allows the material’s equivalent oxygen partial pressure as a function of time to be determined. By rearranging, Equation 6.2.1-7, the governing equation for the reduction reaction, the following relation is obtained.

( ) = ( )

( ) 7.2.1-4

An analogous set of equations can be derived to allow for the determination of the material’s equivalent oxygen partial pressure from the oxidation half reaction.

( ) = − ( ) 7.2.1-5

( ) = ( )

( ) −

,

where ( ) must be solved for numerically as an analytical expression is untenable. Now that ( ) and ( ) are known, the pairs at the same time become the nonstoichiometric data at the temperature for which the cycle is conducted. The free energy of reduction can be determined from the equilibrium expression for the reduction reaction.

( , ) = − ( ) 7.2.1-7

Note that here the free energy of reduction is expressed per oxygen atom. In order to determine the enthalpy and entropy of the reduction reaction explicitly, several cycles at different temperatures with overlapping nonstoichiometry values are necessary. The entropy and enthalpy could then be extracted by generating an Arrhenius plot of ΔG versus inverse temperature at a given nonstoichiometry.

7.2.1.2 Experimental

To demonstrate a proof of concept for this methodology, this analysis was performed on a previously published isothermal, pressure-swing cycle [17]. Figure 7.2-1

illustrates the comparison between the free energy of reduction for neat ceria at 1500⁰C as calculated from reference nonstoichiometry data [40] and the free energy of reduction for neat ceria at 1500⁰C as calculated from a thermochemical, isothermal, pressure-swing, cycling experiment. The experimental data shows moderate agreement with the nonstoichiometry data, but at this stage, results are too preliminary to make generalization about why the matching is better or worse.

0.03 0.04 0.05 0.06 0.07 240 260 280 300 320 340 360  Gre d ( kJ /m ol O ) Nonstoichimetry () Experimental Data Panlener Data

FIGURE 7.2-1Comparison between the free energy of reduction as calculated from Panlener’s neat ceria nonstoichiometric data and the free energy of reduction as calculated from thermochemical, isothermal, pressure-swing cycling experiments.

In order to fully determine the enthalpy and entropy of reduction the following experiments should be performed. One half gram of neat ceria should be isothermally, chemically cycled at multiple temperatures between 1100⁰C and 1500⁰C with flow rates at or below 200sccm for both the oxidizing and reducing streams. This will ensure that all operation occurs within the regime of the thermokinetic limit. The oxidizing gas should be pure argon passed through a humidifier held at 60⁰C (P of 0.19660 atm). The reducing gas should be 3% H2 in argon passed through a humidifier held at 20⁰C (P of

0.02308 atm). These two gas compositions will ensure that the ceria undergoes a sufficiently wide range of overlapping nonstoichiometries between each experiment. This

is the widest range possible due to experimental and safety limitation and should be used for other materials, unless there are known phase changes under these conditions. In between the reduction and oxidation half cycles, a 30 second 1500sccm pure argon purge should be conducted to remove hydrogen from the reactor. Table 7.2.1-1 outlines the nonstoichiometry ranges the cycles will reach at each temperature, based on the exact theoretical values calculated from the Panlener data.

TABLE 7.2.1-1 Relevant equilibrium nonstoichiometric conditions in neat ceria for isothermal, pressure-swing cycling.

Temperature

(⁰C) Reducing (atm)

Oxidizing (atm)

Reducing

Nonstoichiometry Nonstoichiometry Oxidizing 1100 4.974∗10-14 _9.329∗10-6 _{0.1153 0.001399}

1200 9.712∗10-13 _2.512∗10-5 _{0.1464 0.003712}

1300 1.304∗10-11 _5.969∗10-5 _{0.1689 0.008583}

1400 1.287∗10-10 _1.280∗10-4 _{0.1874 0.01694}

1500 9.833∗10-10 _2.518∗10-4 _{0.2057 0.03382}

This experimental method will allow for the determination of the reduction reaction’s entropy and enthalpy over a nonstoichiometry range of about 0.034 to 0.12. This range can be increased significantly by reducing in a more aggressive environment at lower temperatures, however, this would pass the lower combustible limit for hydrogen and pose significant safety concerns. Lower values of nonstoichiometry can be accessed by oxidizing in higher concentrations of steam at higher temperatures, but water management becomes more difficult as the gas lines will need to be held at a higher temperature to prevent premature condensation. Alternatively, if the need for five data points when fitting the Arrhenius plot is relaxed to four data points, then the range

increases to nonstoichiometric values of 0.017 to 0.15. Furthermore, more temperature steps could be explored within this range to provide better accuracy.

It is important to note that, in this cycling paradigm, the oxidation half cycle can be modeled with the current thermokinetic model to determine the nonstoichiometry thermodynamics, but a new reduction model is necessary to properly account for the introduction of hydrogen and water to the reducing gas. Additionally, an oxygen sensor should be downstream of the reactor for accurate determination of equilibrium oxygen partial pressure.