The objective of this section is to directly compute the statistics of the force exerted on a piston by gas pressure. We use the results of the section on a sequence of random impulses, where the force statistics are derived as a function of: (1) the rate of molecular collisions, and (2) the mean and variance of the collision velocities. We initially assume a monatomic molecule, but add results for a diatomic molecule in the final section.
The geometry is the same as for the blackbody analysis: an infinite surface at x=0, with all space in the region x < 0 occupied by an ideal gas, at equilibrium. We consider a tubular region of space perpendicular to the surface, of cross-sectional area A, and extending to from x=0 to x=- ∞. The piston is defined as the area formed by the intersection of the tube and the surface. The piston is immovable.
The analysis depends only on the statistics of the x-component of the molecular velocity. These statistics are the same for a tube with rigid walls, or a "tube" with imaginary walls. That is, the force is the same if the piston is at the end of rigid tube, or lying in a planar surface with no walls present.
Under equilibrium conditions we have:
Molecular Collision Rate
Consider all molecules with velocity v in the x-direction satisfying 0 < vo < v < vo+dvo. Statistically, the fraction of molecules within this velocity band is
In the time interval T these molecules will collide with the piston if they are within a distance voT from the piston. Within the tube, the expected total number of molecules within this distance is
Combining equations (1) and (2), the number of molecules in the selected velocity band expected to collide with the piston in the time interval T is
Where R(vo) is the Rayleigh probability distribution. The total number of expected collisions is obtained by integrating (3) over all positive values of vo. Then dividing by T yields the collision rate
Collision Velocity Statistics
The probability density function of vn is proportional to equation (3), so it is equal to the Rayleigh distribution. The mean and variance of vn are given by (e.g. see Papoulis page 148)
Application of Random Impulse Sequence Analysis
The results of the impulse sequence analysis section can then be used, showing that the expected value of the force is
This DC term, the atmospheric pressure, is the same as a similar derivation of pressure presented in another section. The expected value of the force averaged over any time interval is the same. The variance of the force averaged over the interval τ is
The collision mechanics of a diatomic molecule are analyzed in the Dumbbell Collision section. We assume that the molecular mass is the same. While the number of molecules colliding with the piston in a given time interval is the same, there are two significant changes: (1) for some collisions both ends of the molecule collide with the piston, and (2) the momentum change per collision is no longer 2mvn. As argued in the pulse sequence section, when both ends of the molecule collide, the two impulsive forces are combined into a single molecular collision impulse. The relative statistics for collision velocity for diatomic and monatomic molecules have been determined using a Monte-Carlo simulation, also described in the Dumbbell collision section. The change in momentum averaged over time is the same as for both molecules, so the expected value (DC component) of the force is the same. But the average of the square of the momentum change for a diatomic molecule is 0.9565 times the value for a monatomic molecule. Therefore the noise power level will be lower by this factor.
Back to Molecular Noise and "Blackbody" Radiation
Back to the Physics Contents