Find the rate `v` with which helium flows out of a thermally insulated vessel into vacuum through a small hole. The flow rate of the gas inside the vessel is assumed to be negligible under these conditions. The temperature of helium in the vessel is `T = 1.000 K`.

Explanation

From energy conservation as in the derivation of Bernoullis theorem it reads
`(p)/(rho)+(1)/(2) v^2 + gz + u + Q_d = constant` ….(1)
In the Eq.(1) `u` is the internal energy per unit mass and in this case is the thermal energy per unit mass of the gas. As the gas vessel is thermally insulated `Q_d = 0`, also in our case.
Just inside the vessel `u = (C_VT)/(M) = (RT)/(M(gamma - 1)) also (p)/(rho) = (RT)/(M)`. inside the vessel `v = 0` also. Just outside `p = 0`, and `u = 0`. Ingeneral `gz` is not very significant for gases.
Thus applying Eq. (1) just inside and outside the hole, we get
`(1)/(2) v^2 = (p)/(rho) + u`
=`(RT)/(M) + (RT)/(M(gamma - 1)) = (gammaa RT)/(M(gamma - 1))`
Hence `v^2 = (2 gamma RT)/(M(gamma - 1))` or, `v = sqrt((2 gamma RT)/(M(gamma - 1)))= 3.22 km//s`.
The velocity here is the velocity of hydrodynamic flow of the gas into vaccum.This requires that the diameter of the hole is not too small (`D gt` mean free `l`). In the opposite case `(D lt lt l)` the flow is called effusion. Then the above result does not apply and kinetic theory methods are needed.