Question

A gas bulb of 1 L capacity contains $20\times {10}^{21}$ molecules of nitrogen exerting a pressure of 7.57$\times$10${}^3$Nm${}^{-2}$.

Calculate the root mean square (rms) speed and the temperature of the gas molecules. If the ratio of the most probable speed to root mean square speed is 0.82, calculate the most probable speed for these molecules at this temperature.

Answer 1

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

$Mass=\frac{2\times {10}^{21}}{6.023\times {10}^{23}}\times 28\times {10}^{-3}kg$

$Volume=1L={10}^{-3}{m}^3$

$Density=\frac{Mass}{Volume}=\frac{2\times {10}^{21}}{6.023\times {10}^{23}}\times \frac{28\times {10}^{-3}}{{10}^{-3}}$

$=0.09297kg{m}^{-3}$

$V_{rms}=\sqrt{\frac{3P}{d}}=\sqrt{\frac{3\times 7.57\times {10}^3}{0.09297}}=494.23m{s}^{-1}$

$\frac{V_{mostprobable}}{V_{rms}}=0.82$ (given)

$\therefore V_{mostprobable}=0.82\times V_{rms}$

$=0.82\times 494.23$

$=405.269\approx 405m{s}^{-1}$