First slide

Rms speed

Question

A closed vessel of fixed volume contains a mass m of an ideal gas, the root mean square speed being v. Additional mass m of the same gas is pumped into the vessel and the pressure rises to 2 P, the temperature remaining the same as before. The root mean square speed of the molecule now is

Moderate

A

$v / \sqrt{2}$
B

$v \sqrt{2}$
C

2V
D

V

Solution

$\begin{array}{l} PV = (m / M) RT (M = molecular mass) . . . . (1) \\ P^{'} V = (2 m / M) RT . . . . (2) \end{array}$
From eqs. (1) and (2),
$\frac{P^{'}}{P} = \frac{2}{1}$
Now, $V_{rms} = \sqrt{(\frac{3 P}{ρ})} = \sqrt{[\frac{3 P}{(m / V)}]} . . . . (3)$
and $v_{rms}^{'} = \sqrt{[\frac{3 P^{'}}{(2 m / V)}]} . . . (4)$
From eqs. (3) and (4),
$\begin{array}{l} \frac{v_{rms}^{'}}{v_{rms}} = \sqrt{\frac{1}{2} (\frac{P^{'}}{P})} = \sqrt{[\frac{1}{2} (2)]} = 1 \\ v_{rms}^{'} = v_{rms} \end{array}$

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