Thermodynamic Parameters for Hydrogen Storage in Metal Organic Frameworks

: The global energy crisis coupled with the rising demand to decarbonize the planet, has escalated research on alternative clean energy in the changing energy mix. Owing to the abundant availability and natural inexhaustibility of hydrogen in nature, the green hydrogen has turned out to be a promising and attractive energy carrier in cars and other mobile applications. However, hydrogen production still faces challenges on storage, distribution and usage. Microporous metal-organic frameworks have become the most promising materials for hydrogen storage since they have high surface areas and chemically tunable porous structures. Thus, this study is carried out to investigate the theoretically valid equations and thermodynamic parameters for hydrogen storage in metal organic frameworks. Our calculations revealed that, an efficient hydrogen storage metal organic framework should be tuned to high overall entropy above 484


Introduction
It is estimated that the global energy consumption will increase by approximately 56% by the year 2040 due to the growth of economies on the planet (Sieminski, 2013).This implies that, an economy depending on fossil fuels would emit more carbon dioxide to the atmosphere (Castro-Alvarez et al., 2018).In order to overcome this challenge, there is need to invest in clean energies.Extensive research has been done to affirm that hydrogen may be used as a clean alternative to hydrocarbon fuels in the transport sector.Thus, it is proposed that hydrogen may offer sufficient solution in decarbonizing the transport sector (Damman et al., 2020).Therefore, it is important to understand how hydrogen is stored, particularly in Metal organic Framworks (MOFs).In order to achieve this, this research aims at deriving the theoretically valid thermodynamic equations with particular emphasis on Gibbs free energy, enthalpy and overall entropy.

Materials and Methods
It is well known that hydrogen has strong propensity to bind with surfaces by either

Suggested Citation
Cherop, H. & Kanule, J. ( 2023).Thermodynamic Parameters for Hydrogen Storage in Metal Organic Frameworks.European Journal of Theoretical and Applied Sciences, 1(5), 615-621.DOI: 10.59324/ejtas.2023.1(5).51physisorption or chemisorption.Thus, materials with large surface areas and low densities such MOFs are ideal for hydrogen storage applications.Metal organic frameworks are a class of compounds consisting of an array of positively charged metal ions that form nodes that bind the arms of linkers together.MOFs exist mostly in the form of crystalline compounds but others exist as amorphous MOFs (Bennett & Cheetham, 2014).They possess two unique characteristics, namely, ultrahigh porosity and huge internal surface areas.Thus, MOFs have found several applications in science and technology.These applications include gas storage (hydrogen, carbon dioxide and methane), gas purification and storage, thin film devices, catalysis and biomedical imaging (Zhou et al., 2012).In hydrogen storage, MOFs have proved to be versatile since they have flexible and tunable porous structures in comparison to zeolites (Lochan & Head-Gordon, 2006).Some of the commonly used MOFs in hydrogen storage include  5  7 ,  2 ,  4 ,  2 ,  4 ,  4 , ,  4 , etc.
In order to use hydrogen for practical applications, it has to be compressed to very high pressures or stored at cryogenic temperatures.These conditions consume a lot of energy and increase the weight of the vehicle (Collins & Zhou, 2007).Currently, there are three modes of hydrogen storage; compressed gas tanks (350 to 700 atm.), liquid hydrogen tanks (20.3K) and solid state hydrogen tanks.All these modes of storage face some limitations.For instance, compressed gas tanks occupy relatively large volume at ambient temperatures.Liquid hydrogen evaporates easily and solid state hydrogen storage such as metal organic frameworks (MOFs) add significant weight to the vehicles and other mobile applications (Aceves et al., 2010).
The thermodynamic parameters that are indicators of the possible nature of hydrogen adsorption are change in Gibbs free energy (∆), change in enthalpy (∆) and change in entropy (∆).A simple thermodynamic equation that explains hydrogen adsorption as a thermodynamic process at a given temperature () can be written as: Hydrogen adsorption leads to decrease in entropy and hence ∆ < 0. Also, adsorption is a spontaneous process implying that ∆ < 0. Consequently, ∆ < 0 showing that hydrogen adsorption is an exothermic process.In the case of desorption, energy is absorbed hence ∆ > 0, ∆ > 0 and ∆ > 0. This shows that desorption is an endothermic process.
Metal hydrides are formed by a reversible reaction that is written as: where () is a metal,   is the corresponding hydride and  is the corresponding hydrogen to metal,  is the heat of reaction.
In the formation of hydride, entropy is reduced thus heat is lost (exothermic) while the reversible reaction of hydrogen release will be endothermic.
If we consider a system containing   number of particles in   states, the thermodynamic probability () of such a system can be written as: or The entropy of the system is written as; 617 where K is Boltzmann constant and W is the thermodynamic probability.
Similarly, by considering a system containing   number of hydrogen atoms in   interstitial sites, the change in entropy (Δ) of such system can be obtained by keeping   constant and varying   .Thus, change in entropy is given by (Tanui et al., 2022;Khanna, 1986); For variety of adsorbents the change in entropy is given by (Bhatia & Myers, 2006); Using Sterling's theorem and Maxwell Boltzmann equation in Equation ( 4) we obtain: where  is the translational partition function which is written as  =  Λ 3 and Λ is the thermal wavelength which is written as: The overall entropy of a system can be obtained by combining Equation (5) and Equation ( 8 Therefore, the values of pressure can be calculated from Equation ( 12) by varying the values of  between 0.01m 3 to 0.1m 3 and varying the values of  between 250 and 320.
If the values of ∆ and ∆ are known (from Equation (7) and Equation ( 13)), then the enthalpy of the metal hydrides can be calculated using Van't Hoff equation which is written as;

Results and Discussion
The overall entropies of eight stable metal hydrides namely,  5  7 ,  2 ,  4 ,  2 ,  4 ,  4 ,  and  4 , were calculated using Equation ( 11) and their results are illustrated in Figure 1 and Figure 2.

Figure 1. Graphical Illustration of Overall Entropy against Volume at T=273K using Equation (11)
It is noted that, as temperature increases from T= 273K to T=298K, the entropy of the metal hydrides increases in agreement with the second law of thermodynamics.In this temperature range,  5  7 was noted to have higher values of overall entropy while  had the least amount of overall entropy.This is mainly because increase in molar mass of the hydrides leads to corresponding increase in the manner in which the particles can vibrate, thus, the greater the molar mass the greater the entropy.

Figure 2. Graphical Illustration of Overall
Entropy against Volume at T=298K using Equation ( 11).
Additionally, the overall entropies of the metal hydrides were noted to increase with increase in volume of the metal hydrides.This is attributed to the fact that larger volumes increases the ways in which the molecules can distribute themselves, hence, increase in the microstates.
As the volumes of the metal hydrides were increased from 0.01 3 to 0.1 3 , it was noted that the entropies increase rapidly between 0.01 3 to 0.05 3 in comparison with the region between 0.05 3 to 0.1 3 .This implies that the molecules distribute themselves faster in the smaller containers than in the larger containers.
By comparing the graphical illustrations in Figure 1 and Figure 2, it is noted that the overall entropy of the selected hydrides increase logarithmically with increase in volume and temperature.This is because of the increase in the macrostates that results from the increase in the volume of the container, mass and energy of the interacting particles.Thus, entropy is an extensive property and the combination of states in such a system is exponential.Furthermore, the overall entropy of the hydrides were plotted against temperature in different volumes.As temperature is increased from T=250K to T=320K, it was noted that the entropy rose steadily among all the hydrides even when their volumes were increased from 0.01 3 to 0.1 3 as shown in Figure 3 and Figure 4.It is noted that, the variation between the overall entropy and temperature is linear for all the hydrides.This indicates that, thermal energy within the metal hydrides is dispersed in the same rate within a given container, thus, confirming the standard volumes for effective and efficient storage systems for metal hydrides in mobile transport.
Gibbs free energy is the maximum amount of free energy available to do useful work.The plot of Gibbs free energy against pressure is shown in Figure 5.The graphical illustration of the Gibbs free energy against pressure shows that, Gibbs free energy increases logarithmically between  = 10  to = 40  , thereafter, it is followed by almost a linear variation with pressure between  = 40  to  = 100  as illustrated in Figure 5 below.
The changes in the volumes in Figure 5 illustrate the phase change from gaseous state to liquid state and finally to solid state.Between  = 10  to = 40  the graph exhibits a curvature showing that the volume changes with increase in pressure.This corresponds to a gaseous state since gases are compressible.
Between  = 40  to = 100 , the volumes does not change with pressure.This describes the region that is ideal storage frameworks for liquid and solid hydrides.The calculations of enthalpy changes was carried out using Equaiton ( 14) and their graphical illustration of enthalpy against pressure was plotted as shown in Figure 6.It is noted that enthalpy increases with increase in pressure.By definition, enthalpy is the heat absorbed or produced during any process that occurs at constant pressure.Therefore, our calculations reveal that at constant pressure, the enthalpy increases with increase in temperature.This is because changes in enthalpy are equal to the heat flow in the system or the heat content of a system as a function of entropy and pressurre.
Thus, increase in enthalpy describes the fundamental requirement of an efficient system.

Conclusion
This study investigates the theoretically valid equations and thermodynamic parameters for an efficient hydrogen storage system.The thermodynamic parameters investigated are overall entropy, Gibbs free energy and enthalpy changes.Our calculations revealed that an efficient hydrogen storage metal organic framework should be tuned to high overall entropy above 484 . −1  −1 at room temperature, ∆ ≅ 1650 / and ∆ ≅ −1800 . −1  −1 at () between 40  to 100  .Additionally, such storage systems should have volumes above 0.1 3 for heavy and large mobile transport applications such as trucks, trains, boats, submarines etc. at room temperature.
=  is the universal gas constant;  = 8.3145 . −1  −1 .translational partition function () can be written in terms of pressure such that,