Question

The rms velocity of hydrogen is $\sqrt 7 $ times the rms velocity nitrogen. If $T$ is the temperature of the gas.
(A). $T\left( {{H_2}} \right) = T\left( {{N_2}} \right)$
(B). $T\left( {{H_2}} \right) > T\left( {{N_2}} \right)$
(C). $T\left( {{H_2}} \right) < T\left( {{N_2}} \right)$
(D). $T\left( {{H_2}} \right) = \sqrt 7 T\left( {{N_2}} \right)$

Answer 1

Hint: The RMS velocity depends upon the temperature of the gas. It is represented by the equation:
${V_{rms}} = \dfrac{{\sqrt {3RT} }}{{\sqrt M }}$ . It is clearly visible from the formula that RMS velocity is directly proportional to temperature.

Complete step by step answer:
RMS velocity means the root means square velocity. The RMS velocity is defined as the square root of the mean square of the velocity of individual gas molecules.
The formation for calculating RMS velocity is :
${V_{rms}} = \sqrt {\dfrac{{3RT}}{M}} $
Where : ${V_{rms}} \to {\text{RMS Velocity}}$
$M{\text{ }} \to {\text{ molar mass of the gas }}$
$R{\text{ }} \to {\text{molar gas constant}}$
$T{\text{ }} \to {\text{temperature in Kelvin}} $
RMS velocity is also the velocity, so it has units of velocity .
In general we use the RMS velocity instead of arranging velocity of the gas because for a gas sample the net velocity is zero since the particles move in all the directions .
Observing the formula for RMS velocity it is clear that RMS velocity is directly proportional to the square root of the temperature and inversely proportional to the square root of molar mass . Thus changing the temperature will change the RMS velocity of the gas .
Now in the question it is given that the RMS velocity of hydrogen is $\sqrt 7 $ lines the RMS velocity of nitrogen .
That is , $\mu {H_2} = \sqrt 7 \mu {N_2}$
As said earlier , the RMS velocity is directly proportional to the root of temperature .
So,$\mu \propto \sqrt T $
This implies that temperature will also vary in the same ratio as rms velocity:
$\sqrt {{T_{{H_2}}}} = \sqrt 7 \sqrt {{T_{{N_2}}}} $
Squaring both sides ,
${T_{{H_2}}} = 7{T_{{N_2}}}$
So ,${T_{{H_2}}} > {T_{{N_2}}}$
Hence , option B is the correct one .

Note:
As all gas particles move with random speed and direction . So , calculating average velocity will be zero . The RMS velocity takes into account both molecular mass and temperature that both affect a material’s kinetic energy .