Gas in a Gravitational or Centrifugal Field

We consider an ideal gas (i.e., no interactions among the particles) consisting of N molecules at a temperature T contained in a vertical column of height H and cross-section a (Fig. 2.10). The system is subjected to an external field of force (either gravitational or centrifugal) such that at each height z, the interaction between the particle of mass m and the external field is 

g is the gravitational acceleration, z is the height of the particle measured relative to the bottom of the column in the case of the gravitational field or the distance from the axis of rotation in the case of a centrifugal field, and w is the angular velocity of rotation of the centrifuge. In general, g depends on the distance from the center of the earth. However, since //is very small compared to the radius of the earth, one can assume 

The canonical partition function for the system in the gravitational field is 

The new phenomenon of interest in this system is the density distribution along the 5 axis. To obtain this distribution, we divide the length of the system into M narrow strips or phases in such a way that in each strip of volume AF = aAz, the intensive variables are almost constant. But Az should be large enough so that macroscopic thermodynamics applies in the volume aAz. With these assumptions we rewrite the integral in (2.6.4) as 

This has the form of an equilibrium distribution of the N particles into M species, the species being distinguished by their different heights z, in the system. Each term corresponds to a specific set of Ai Nm, Ni being the number of particles in the Ith strip and A/ = A the total number of particles in the system. 

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