First slide

Kinetic theory of ideal gases

Question

The molecules of a given mass of a gas have a rms velocity of 200 m/s at $27^{0} C$ , pressure 1.0 x $10^{5}$ N/ $m^{2}$ at $127^{0}$ C and pressure is 0.5 x $10^{5}$ N/ $m^{2}$ , the rms velocity in m/sec will be

Moderate

A

$200 \sqrt{3}$
B

$\frac{200}{\sqrt{3}}$
C

$400 \sqrt{3}$
D

$\frac{400}{\sqrt{3}}$

Solution

$V_{r m s} = \sqrt{\frac{3 R T}{M}} \Rightarrow V_{r m s} α \sqrt{T} g i v e n t e m p e r a t u r e T_{1} = 27 + 273 = 300 K; V_{300^{o}} = 200 m / s T_{2} = 127 + 273 = 400 K; s u b s t i t u t e i n t h e e q u a t i o n \frac{V_{1}}{V_{2}} = \frac{\sqrt{T_{1}}}{\sqrt{T_{2}}} \Rightarrow \frac{V_{300^{o}}}{V_{400^{o}}} = \sqrt{\frac{300}{400}} = \sqrt{\frac{3}{4}} \Rightarrow \frac{200}{V_{400}} = \sqrt{\frac{3}{4}} \Rightarrow V_{400} = \frac{(200) (2)}{\sqrt{3}} = \frac{400}{\sqrt{3}} m / s = V_{r m s}$

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