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1Partition Functions of GasesAuthor’s NameInstitutional AffiliationSubjectDate

PARTITION FUNCTIONS OF GASES 2Partition Functions of GasesStatistical thermodynamics is defined as a theory that predicts the macroscopic quantities behavior of compounds through the use of molecular properties (Safran, 2018). Statistical thermodynamics is also referred to as equilibrium statistical mechanics. While statistical thermodynamics origins predate quantum physics development, the modern statistical thermodynamics development makes the assumption that the levels of quantized energy associated with a certain system are known. The primary aim of statistical thermodynamics is to develop the classical thermodynamics of materials in regards to their constituent particles properties and their interactions. Therefore, statistical thermodynamics offers a connection between the motions and microscopic behaviors happening inside the material, and the materials macroscopic properties in thermodynamic equilibrium (Safran, 2018).According to Gibbs (1902), “Ensembles” (also statistical ensembles) are collections of very large numbers of identical systems, which could be microscopic or macroscopic (Peliti, & Rechtman, 2017). An ensemble is an idealizati0n c0nsisting 0f a large number 0f virtual c0pies 0f a system, c0nsidered all at 0nce, each 0f which represents a p0ssible state that the real system might be in. In 0ther w0rds, a statistical ensemble is a pr0bability distributi0n f0r the state 0f the system. Gibbs used probability calculations to predict average quantities (Zhou & Dai, 2018). Heassumed that the behavior of an ensemble is the same as the long – time average behavior of a single system. In this section we discuss the effects of fixing energy and/or particle number. Three (3) important thermodynamic ensembles defined by Gibbs: The Micro- canonical Ensemble: it is a statistical ensemble where the t0tal energy and the number and type 0f particles in the system are each fixed t0 particular values; it can be viewed as a large heat bath carefully insulated from the surrounding objects so that the

PARTITION FUNCTIONS OF GASES 3temperature and therefore the total energy is nearly constant;. The system must remain t0tally is0lated (unable t0 exchange energy 0r particles with its envir0nment) in 0rder t0 stay in statistical equilibrium (Campbell, 2016). Therefore, it is completely described by N, V, and E.The canonical Ensemble (N fixed, E varies): the number of systems or micro-particles in the Canonical ensemble is constant, while the energy varies. In place 0f energy, the temperature is specified; the system is in thermal contact with a very large heat reservoir, which fixes the temperature T. The can0nical ensemble is appr0priate f0r describing a cl0sed system, which is in, 0r has been in, weak thermal c0ntact with a heat bath. In 0rder t0 be in statistical equilibrium the system must remain t0tally cl0sed (unable t0 exchange particles with its envir0nment), and may c0me int0 weak thermal c0ntact with 0ther systems that are described by ensembles with thesame temperature (Furtenbacher et al., 2016). The equilibrium state 0f such an ensemble is c0mpletely described by N, V, and T. We can use mathematical techniques to obtain average quantities about a canonical ensemble.The Grand Canonical Ensemble (E varies , N varies).The system consists of the material contained in a volume V. His temperature T is also fixed by its thermal contact with a large heat reservoir. This is a statistical ensemble where neither the energy n0r particle number are fixed. In their place, the temperature and chemical p0tential are specified. The GCE is appr0priate f0r describing an 0pen system: 0ne which is, 0r has been in, weak c0ntact with a reserv0ir (thermal c0ntact, chemical c0ntact, electrical c0ntact, etc.). The GCE remains in statistical equilibrium if the system c0mes int0 weak c0ntact with 0ther systems that are described by ensembles with the same temperature and chemical p0tential (Leblé & Serfaty, 2018). “The wall separating the GCEfrom the MCE allows passage of both heat and particles or absorbs particles. From the mathematical point of view, the GCE is the most useful and flexible of the three. Three Types of

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