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Molecular Speeds - Vrms, Vavg, Vmps

If for two ga...

Question

If for two gases of moleculer weights $M_{A}$ and $M_{B}$ at temperature $T_{A}$ and $T_{B}$ , $T_{A}$ $M_{B} = T_{B} M_{A},$ then which property has the same magnitude for both the gases.

Density

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Pressure

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K.E. per mole

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V_{r m s}

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Solution

The correct option is D $V_{r m s}$
(a) Density of a gas ( $d$ ) = $\frac{P M}{R T}$
Since, $\frac{M_{B}}{T_{B}} = \frac{M_{A}}{T_{A}}$
$∴$ At the same pressure, $d_{A} = d_{B}$ But if the pressure is different, then $d_{A} \neq d_{B}$

(b) Pressure of the gases would be equal if their densities are equal otherwise this wouldn't be the case. The given condition does not imply this.

(c) $K . E .$ per mole is $\frac{3}{2}$ RT
$∴$ It will be different for the two gases unless their masses are the same.

(d) $V_{r m s} = \sqrt{\frac{3 R T}{M}}$ , since $\frac{T_{A}}{M_{A}} = \frac{T_{B}}{M_{B}}$ ; both the velocities are the same. This is the correct answer.

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