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Statistical Mechanics and Thermodynamics-PDF

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1Partition Functions of GasesAuthor’s NameInstitutional AffiliationSubjectDate
PARTITION FUNCTIONS OF GASES2Partition Functions of GasesStatistical thermodynamics is defined as a theory that predicts the macroscopic quantitiesbehavior of compounds through the use of molecular properties (Safran, 2018). Statisticalthermodynamics is also referred to as equilibrium statistical mechanics. While statisticalthermodynamics origins predate quantum physics development, the modern statisticalthermodynamics development makes the assumption that the levels of quantized energyassociated with a certain system are known. The primary aim of statistical thermodynamics is todevelop the classical thermodynamics of materials in regards to their constituent particlesproperties and their interactions. Therefore, statistical thermodynamics offers a connectionbetween the motions and microscopic behaviors happening inside the material, and the materialsmacroscopic properties in thermodynamic equilibrium (Safran, 2018).According to Gibbs (1902), “Ensembles” (also statistical ensembles) are collections ofvery large numbers of identical systems, which could be microscopic or macroscopic (Peliti, &Rechtman, 2017). An ensemble is an idealizati0n c0nsisting0f a large number0f virtual c0pies0f a system, c0nsidered all at0nce, each0f which represents a p0ssible state that the real systemmight be in. In0ther w0rds, a statistical ensemble is a pr0bability distributi0n f0r the state0f thesystem. 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 asingle 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 andthe number and type0f particles in the system are each fixed t0particular values; it can beviewed as a large heat bath carefully insulated from the surrounding objects so that the
PARTITION FUNCTIONS OF GASES3temperature and therefore the total energy is nearly constant;. The system must remain t0tallyis0lated (unable t0exchange energy0r particles with its envir0nment) in0rder t0stay instatistical 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 inthe Canonical ensemble is constant, while the energy varies. In place0f energy, the temperatureis specified; the system is inthermal contactwith a very large heat reservoir, which fixes thetemperature T. The can0nical ensemble is appr0priate f0r describing a cl0sed system, which isin,0r has been in, weak thermal c0ntact with a heat bath. In0rder t0be in statistical equilibriumthe system must remain t0tally cl0sed (unable t0exchange particles with its envir0nment), andmay c0me int0weak thermal c0ntact with0ther systems that are described by ensembles with thesame temperature (Furtenbacher et al., 2016). The equilibrium state0f such an ensemble isc0mpletely described by N, V, and T. We can use mathematical techniques to obtain averagequantities about a canonical ensemble.The Grand Canonical Ensemble (E varies , N varies).The system consists of the materialcontained in a volume V. His temperature T is also fixed by its thermal contact with a large heatreservoir. 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 f0rdescribing an0pen system:0ne which is,0r has been in, weak c0ntact with a reserv0ir (thermalc0ntact, chemical c0ntact, electrical c0ntact, etc.). The GCE remains in statistical equilibrium ifthe system c0mes int0weak c0ntact with0ther systems that are described by ensembles with thesame 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 themathematical point of view, the GCE is the most useful and flexible of the three. Three Types of
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