Question 2.168: (a) Show that the ratio of the pressure to the viscosity coe...
(a) Show that the ratio of the pressure to the viscosity coefficient gives approximately the number of collisions per unit time for a molecule in a gas.
(b) Calculate the number of collisions per unit time for a molecule in a gas at STP using the result of (a) above or by calculating it from the mean velocity, molecular diameter, and number density.
The coefficient of viscosity for air at STP is 1.8 \times 10^{-4} in cgs units.
Use values you know for other constants you need.
(a) The coefficient of viscosity is \eta=\frac{n}{3} m \bar{v} \bar{\lambda}, where n is the particle number density. The pressure of the gas is
p = nkT .
Hence
\frac{p}{\eta}=\frac{3 k T}{m \bar{v} \bar{\lambda}}.
The mean-square speed of the molecues is \overline{v^{2}}=\frac{3 k T}{m}. Neglecting the difference between the average speed and the rms speed, we have
\frac{p}{\eta}=\frac{\overline{v^{2}}}{\bar{v} \bar{\lambda}}\approx \frac{\bar{v}}{\bar{\lambda}}which is the average number of collisions per unit time for a molecule.
(b) At STP, the pressure is p=1.013 \times 10^{6} dyn / cm ^{2}. Hence the number of collisions per unit time is
\frac{p}{\eta}=\frac{1.013 \times 10^{6}}{1.8 \times 10^{-4}}=5.63 \times 10^{9} s ^{-1}.