A parallel beam of nitrogen molecules moving with velocity `v = 400 m s^(-1)` imprings on a wall at an angle `theta = 30^(@)` to its normal. The concentration of molecules in the beam is `n = 9 xx 10^(18) cm^(-3)`. Find the pressure exerted by the beam on the wall, assuming that collisions are perfectly elastic.

Explanation

Figure-2.27 shows a beam of molecules in cylindrical form striking the wall. According to elastic collision, each molecule is reflected from the wall at the same angle of incidence as shown with same speed. If `m_(0)` is the mass of each nitrogen molecule, momentum of each molecule is
`P_(0)=m_(0)v`
During collision change in momentum of each molecule or momentum imparted to wall by each molecule is
`DeltaP_(0)=2m_(0)v cos theta`
If S is the cross-sectional area of beam and v be the velocity of
molecules in the beam then number of molecules incident on wall per second are
`N=n_(0)"v S "[n_(0)="moleculear density"]`
Thus total momentum imparted to the wall per second by the beam or the force exerted on wall is
`F=(DeltaP_(0))/(Deltat)=2m_(0)v cos theta xxn_(0)vS" "...(2.60)`
As equation-(2.60) gives that total momentum imparted to the wall per second, this is the net force exerted by beam on wall.
Thus the pressure on wall by beam is
`P=(F)/(S)=2m_(0)v^(2)cos theta.n_(0)`
`=2xx(28xx10^(-3))/(6.023xx10^(23))xx9xx10^(24)xx(400)^(2)xx cos^(2)(30^(@))`
`=10^(5)Pa~=1" atm"`