## 1 Introduction  
During the past decades, nanomaterials have been intensively studied as promising new materials in the field of nanotechnology, catalysis, medicine and others [1]. As for medicine, nanomaterials are applied, among others, as drug carriers and in many diagnostics [2; 3; 4; 5; 6]. They are also important in electronics as nano-transistors, conductors, and displays [7; 8; 9; 10]. Nanosubstances may also be used to store energy (batteries or supercapacitors) and to improve efficiency of solar panels [11; 12; 13; 14; 15]. Nanomaterials also find wide application in the construction sector, mainly as elements of composites, because of their impact on the properties of the material, such as increased strength, lightness, and better thermal and electrical conductivity, as well as self-healing, corrosion protection capabilities, and many others [16; 17; 18; 19; 20; 21; 22]. In case of this work, an important application of nanomaterials is catalysis. They work well as catalysts because of their large specific surface area and a different structure compared to their macroscale equivalents [23; 24; 25; 26; 27; 28; 29]. Moreover, nanomaterials can potentially play a significant role in advancing quantum technologies. Research into them may open new possibilities for the development of next-generation quantum computers, sensors, and communication systems [30; 31; 32; 33; 34; 35; 36]. Nanomaterials create opportunities for the development of each of the above-mentioned areas and more, that conventional materials do not provide [37; 38; 39; 40; 41; 42; 43; 44]. However, it should be remembered that they may be potentially dangerous, as research into their negative impact, especially on health and the environment, is still ongoing [45; 46; 47; 48; 49; 50]. For some industrial processes and technologies, accurate determining of particle size distributions is a must, especially when dealing with nanomaterials (determining the mean size of nanocrystallites only is not sufficient) [51; 52; 53; 54; 55; 56; 57; 58]. Additionally, it is crucial to have a better understanding of phenomena occurring on nanoparticle surfaces under process conditions, in particular in heterogeneous catalysis [59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75]. The iron ammonia synthesis catalyst (a nanoFe-NH\({}_{3}\)-H\({}_{2}\) system), which contains less than 8% of promoters (apart from nano-iron), was the subject of numerous studies related to the determination of the mechanisms of ammonia synthesis and its decomposition [59; 60; 61; 62; 63; 64; 65; 66; 67], as well as the nitriding process [78; 79; 80]. Phenomena occurring in the iron catalyst at elevated temperatures were also studied [81]. The sintering of the catalyst that occurred at that time was analyzed in terms of the behavior of the promoters on the catalyst surface. On the basis of kinetic studies of a nitriding process of the above mentioned catalyst [79], a model of the reactions taking place between iron nanocrystallites and a gas phase was proposed [79; 80]. It was observed that \(\alpha\)-Fe nanocrystallites undergo a phase transition to \(\gamma^{\prime}\)-Fe\({}_{4}\)N nitride in their whole volume in the order of their sizes, from the smallest to the largest [79; 80]. These studies were enabled for elaborating two chemical methods for determining the iron nanocrystallite size distributions: (1) involving measurement of the average crystallite size [79] by the XRD technique; and (2) based on determining the rate of chemical reaction corresponding to given conversion degrees [79]. It was determined [78; 82; 83; 84] that during the nitriding of the iron ammonia synthesis catalyst (at a constant temperature) using hydrogen + ammonia gas mixtures of increasing nitriding potential, P, steady states were set up (at a given equilibrium nitriding potential, P\({}_{0}\), depending on the size of the nanocrystallites [58]). Under those conditions, the catalytic ammonia decomposition reaction runs at a constant rate but the nitriding reaction rate is zero.