The Perfect Gas in a Force Field

The Perfect Gas in a Force Field.—For two sections we have been considering the distribution of velocities included in the Maxwell-Boltz-mann distribution law. Next, we shall take up the distribution of coordinates in cases where there is an external field of force. First, we should observe that on account of the form of Eqs. (1.4) and (1.5), the distributions of coordinates and velocities are independent of each other. These 

Formula (4.1) is often called the barometer formula, since it gives the variation of barometric pressure with altitude. It indicates a gradual decrease of pressure with altitude, going exponentially to zero at infinite height. 

The barometer formula can be derived by elementary methods, thus checking this part of the Maxwell-Boltzmann distribution law. Consider a column of atmosphere 1 sq. cm. in cross section, and take a section of this column bounded by horizontal planes at heights ft and ft + dh. Let the pressure in this section be P we are interested in the variation of P with ft. Now it is just the fact that the pressure is greater on the lower face of the section than on the upper one which holds the gas up against gravity. That is, if P is the upward pressure on the lower face, P + dP the downward pressure on the upper face, the net downward force is dP, 

The derivation which we have given for the barometer formula in Eq. (4.3) can be easily extended to a general potential energy. Let the potential energy of a molecule be t . Then the force acting on it is d t /ds, where ds is a displacement opposite to the direction in which the force acts. Take a unit cross section of height ds in this direction. Then, as before, we have 

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