A box contains `N` molecules of a perfect gas at temperature `T_(1)` and temperature `P_(1)`. The number of molecule in the box is double keeping the total kinetic energy of the gas same as before. If the new pressure is `P_(2)` and temperature `T_(2)`, then

Options

A. `P_(2) = P_(1) , T_(2) = T_(1)`
B. `P_(2) = P_(1) , T_(2) = (T_(1))/(2)`
C. ` P_(2) = 2 P_(1) , T_(2) = T_(1)`
D. `P_(2) = 2P_(1) , T_(2) = (T_(1))/(2)`

Answer

B

Explanation

`E_("total") = (f)/(2) NkT`
`implies T prop (1)/(N) " " [because E_("total") , f, K = "constant"]`
`implies (T_(2))/(T_(1)) = (N_(1))/(N_(2)) = (1)/(2) implies T_(2) = (T_(1))/(2)`
Also from PV = NkT `implies P prop NT`
`implies (P_(1))/(P_(2)) = (N_(1))/(N_(2))* (T_(1))/(T_(2)) = (1)/(2) xx (2)/(1) = (1)/(1)`.