Question

A gas bulb of $1litre$ capacity contains $2.0\times {10}^{21}$ molecules of nitrogen exerting a pressure of $7.57\times {10}^{3}N{m}^{-2}$. Calculate the root mean square speed and temperature of the gas molecules. If the ratio of most probable speed to the root mean square speed is $0.82$, calculate the most probable speed of these molecules at this temperature.

Answer 1

Amount of the gas $=\frac{2.0\times {10}^{21}}{6.023\times {10}^{23}}mole$

Given :

$V=1litre={10}^{-3}{m}^3$

$P=7.57\times {10}^{3}N{m}^{-2}$

$R=8.314J{K}^{-1}mo{l}^{-1}$.

Applying $PV=nRT$ or $T=\frac{PV}{nR}$

$T=\frac{{10}^{-3}\times 7.57\times {10}^3}{\frac{2.0\times {10}^{21}}{6.023\times {10}^{23}}\times 8.314}$

rms speed $=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3\times 8.314\times 274.2}{28\times {10}^{-3}}}=494.22m{s}^{-1}$

Most probable speed $=0.8\times rms$

$=0.8\times 494.22m{s}^{-1}=395.376m{s}^{-1}$