`N` atoms of gaseous helium are enclosed in a cubic vessel of volume `1.0 cm^3` at room temperature. Find :
(a) the probability of atoms gathering in one half of the vesse ,
(b) the approximate numerical value of `N` ensuring the occurrence of this event within the time interval `~~ 10^10` years (the age of the Universe).

Explanation

(a) Each atom has probability `(1)/(2)` to be in either campartment. Thus
`p = 2^-N`
(b) Typical atomic velocity at room temperture is `~ 10^5 cm//s` so it takes an atom `10^-5 sec` to cross the vessel. This is the relevant time scale for our problem. Let `T = 10^-5 sec`, then in time `t` there will be `t//T` crossing or arrangements of the atoms. This will be large enough to produce the given arrangement if
`(t)/(tau) 2^-N ~ 1` or `N ~ (1n t//tau)/(1n 2) ~ 75`.