Question

One gram mole of oxygen at 27${}^{\circ }$C and one atmospheric
pressure is enclosed in a vessel, (a) Assuming the molecules
to be moving with v${}_{rms}$, find the number of collisions per
second which the molecules make with one square metre area
of the vessel wall, (b) The vessel is next thermally insulated
and moved with a constant speed v${}_0$. It is then suddenly
stopped. The process results in a rise of the temperature of
the gas by 1${}^{\circ }$C. Calculate the speed v${}_0$.

Answer 1

(a) $v_{rms}=\sqrt{\frac{3Rt}{M}}=\sqrt{\frac{3\times 8.31\times 300}{32\times 10^{-3}}}$
$=483.4m/s$
Given, p${}_0=1.01\times 10^{5}N/m^2$= Force per unit area.
Let n molecules of oxygen strike the wall per second per
m${}^2$ and recoil with same speed. Change in momentum is
(2 nmv${}_{rms}$).
The change in momentum per unit time is the force.
Hence, $p_0=2nm{v}_{rms}$
$\therefore n=\frac{p_0}{2m{c}_{rms}}=\frac{1.01\times 10^5}{2\left[\frac{32}{6.02\times 10^{26}}\right](483.4)}$
$=1.96\times 1^{27}/s$
(b) $\frac{1}{2}(m_{gas})v_0^2=n{C}_v\Delta T$
$\therefore v_0=\sqrt{\frac{2n{C}_v\Delta t}{m_{gas}}}=\sqrt{\frac{(2)(n)(\frac{5}{2}\times 8.31)(1)}{(n)(32\times 10^{-3})}}$
$=36m/s$