The space between two large parallel plates separated by a distance `l = 5.0 mm` is filled with helium under a pressure `p = 1.0 Pa`. One plate is kept at a temperature `t_1 = 17 ^@C` and the other, at a temperature `t_2 = 37 ^@C`. Find the mean free path of helium atoms and the heat flow density.

Explanation

At this pressure and average temperature `= 27^@C = 300 K = T = ((T_1 + T_2))/(2)`
`lamda = (1 kT)/(sqrt(2) pi d^2 p) = 2330 xx 10^-5 m = 23.3 nmm gt gt 5.0 mm = l`
The gas is ultrathin and we write `lamda = (1)/(2) l` here
Then `q = k(dT)/(dx) = k(T_2 - T_1)/(l)`
where `k = (1)/(3) lt v gt xx (1)/(2) l xx (MP)/(RT) xx (R)/(gamma -1) xx (1)/(M) = (plt v gt)/(6 T(gamma -1))l`
and `q = (plt v gt)/(6T(gamma -1))(T_2 - T_1)`
where `lt v gt = sqrt((8 RT)/(M pi))`. We have used `T_2 - T_1 lt lt (T_2 + T_1)/(2)` here.