Pressure statistical thermodynamics formula

Beyond the basic applications of the Boltzmann distribution, partieularly in statistical thermodynamics or even in quantum applieations sueh as modeling populations of atoms in laser media, they are fimdamental also in Environmental Physics by the famous modeling of the pressure with the atmosphere known as the barometric formula. 

This approach is based on classical thermodynamics and statistical mechanics the latter makes the link to the microscopic picture of the adsorption processes. From the point of view of thermodynamics, adsorption can be treated like a chemical reaction. It will be shown that adsorption can proceed with or without a change of the number of gaseous molecules, so that the thermodynamic equilibrium constant expressed in partial pressures may or may not equal the constant expressed in concentrations. Notice that the standard values of some quantities accepted here when deriving formulae for the adsorption characteristics are not the standard states commonly used in chemical thermodynamics. In particular, it concerns the concentrations. 

Statistical mechanics may be used to derive practical microscopic formulae for thermodynamic quantities. A well-known example is the virial expression for the pressure, easily derived by scaling the atomic coordinates in the canonical ensemble partition function... 

In later chapters of this book there will be occasion to apply thermodynamic derivations to virial equations of state of a pure gas or gas mixture. These formulas accurately describe the gas at low and moderate pressures using empirically determined, temperature-dependent parameters. The equations may be derived from statistical mechanics, so they have a theoretical as well as empirical foundation. There are two forms of virial equations for a pure gas one a series in powers of 1/ Fm ... 

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