Question

A $25 \times {10^{ - 3}}$ volume cylinder is filled with 1mol of gas at room temperature (300 K). The molecular diameter of ​, and its root mean square speed, are found to be 0.3nm, and 200m/s, respectively. What is the average collision rate (per second) for a molecule ?

Answer 1

Here, we will first find the relation between the average collision rate of oxygen molecules and volume, from ideal gas equation. Further by putting the given values and the Avogadro number, we will get the required result.

Formula used:
\eqalign{ & v = \dfrac{{{V_{av}}}}{\lambda } \cr & P = \dfrac{{RT}}{V} \cr}

Complete step by step answer:
We know that collision frequency is defined as the average rate in which two reactants collide for a given system and is used to express the average number of collisions per unit of time in a defined system. Here, we will use the ideal gas expression and also use the relation between the average collision rate and volume.
\eqalign{& v = \dfrac{{200 \times \sqrt 2 \pi \times {\sigma ^2}{N_A}}}{{25 \times {{10}^{ - 3}}}} \cr & \Rightarrow v = 17.68 \times {10^8}/sec \cr & \therefore v = 0.1768 \times {10^{10}}/\sec \sim {10^{10}} \cr}
We also have the equation for pressure, temperature and volume, which is given by:
$P = \dfrac{{RT}}{V}$
Substituting this value in the above equation of wavelength we get:
$\lambda = \dfrac{V}{{\sqrt 2 \pi {\sigma ^2}{N_A}}}$
We also have an equation of average velocity, which is given by:
${V_{av}} = \sqrt {\dfrac{8}{{3\pi }}} \times {V_{rms}}$
Now, by substituting these values in equation (1), we get:
\eqalign{& v = \dfrac{{200 \times \sqrt 2 \pi \times {\sigma ^2}{N_A}}}{{25 \times {{10}^{ - 3}}}} \cr & \Rightarrow v = 17.68 \times {10^8}/sec \cr & \therefore v = 0.1768 \times {10^{10}}/\sec \sim {10^{10}} \cr}
Therefore, the required average collision rate of the oxygen molecule is given by v which is of the order of ${10^{10}}$.

Additional information:
Oxygen is a chemical element with the symbol O and it has atomic number 8. Oxygen is a member of the halogen group in the periodic table. It is a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as well as with other compounds.
Now, as we know about the ideal gas law, also called the general gas equation, it is the equation of a hypothetical ideal gas. It is a good approximation of the behavior of many under many conditions, although it has several limitations. In the ideal gas equation, R is the gas constant. It is the same for all gases.

Note:
Ideal gas equation is only for ideal gas, which is practically not possible. For the practical approach we have the real gas equation. Also, we must note that an ideal gas may not change temperature, but most gases like air are not ideal and follow the joule Thomson effect.