A thermally insulated vessel with gaseous nitrogen at a temperature of `27^(@)C` moves with velocity `100 m//s^(-1)`. How much (in percentage) and in what way will the gas pressrue change if the vessel is brought to rest suddenly ?

Explanation

The internal energy of the molecules are
`U = (1)/(2) mN lt (vec u - vec v)^2 gt =(1)/(2) mN lt u^2 - v^2 gt`
where `vec v`= velocity of the vessel, `N` = number of molecules, eacg of mass `m`. When the vessel to stopped, internal energy becomes `(1)/(2) mN lt u^2gt`
So there is an increase in internal energy of `Delta U = (1)/(2) mN v^2`. This will give rise to a rise in temperature of
`Delta T = ((1)/(2) mN v^2)/((i)/(2) R)`
=`(m Nv^2)/(i R)`
There being no flow of heat. This change of temperature will lead to an excess pressure
`Delta p = (R Delta T)/(V) = (m Nv^2)/(i V)`
and finally `(Delta p)/(p) = (M v^2)/(i R T) = 2.2 %`
where `M` = molecular weight of `N_2`, `i` = number of degrees of freedom of `N_2`.