A and B are closed flasks having the same volume. In flask A, `O_(2)` gas is present at TK and 1 atm pressure. In flask B, `H_(2)` gas is present at `(T)/(2)K` and 1 atm pressure. If these gases behave ideally, then compare their (1) total kinetic energies, (2) total number of molecules, (3) root mean square velocities.

Explanation

Suppose, `n_(1) and n_(2)` are the number of moles of the gases present in flask A and flask B, respectively. Applying ideal gas equation to the gases `O_(2) and H_(2)`, we obtain
`PV=n_(1)RT and PV=n_(2)R(T)/(2)" "therefore (n_(1))/(n_(2))=(1)/(2)`
(1) The total kinetic energy of the molecule of `n_(1)` mol `O_(2)` in flask A, `E_(1)=n_(1)xx(3)/(2)RT` and the total kinetic energy of the molecules of `n_(2)` mol `H_(2)` in flask B, `E_(2)=n_(2)xx(3)/(2)Rxx(T)/(2)` [`because` total kinetic energy of the molecules of 1 mol gas `=(3)/(2)Rxx`absolute temperature]
`therefore (E_(1))/(E_(2))=(2n_(1))/(n_(2))=1[becausen_(2)=2n_(1)]" "therefore E_(1)=E_(2)`
So, the total kinetic energy of themolecules of `O_(2)` gas =the total kinetic energy of the molecules of `H_(2)` gas.
(2) If `n_(1)'` and `n_(2)'` be the number of molecules of `O_(2)` gas and `H_(2)` gas respectively then `n_(1)'=n_(1)xxN and n_(2)'=n_(2)xxN` [N=Avogadro's number]
`therefore (n_(1)')/(n_(2)')=(n_(1)xxN)/(n_(2)xxN)=(1)/(2)`
`therefore`Number of molecules in `H_(2)` gas`=2xx`number of molecues in oxygen gas
(3) rms velocity of `O_(2),c_(rms)=sqrt((3RT)/(M))=sqrt((3RT)/(32))`
rms velocity of `H_(2),c_(rms)=sqrt((3R(T)/(2))/(2))=sqrt((3RT)/(4))`
`therefore (c_(rms)(O_(2)))/(c_(rms)(H_(2)))=sqrt((4)/(32))=sqrt((1)/(8))=(1)/(2sqrt(2))`.